Hindawi Discrete Dynamics in Nature and Society Volume 2019, Article ID 8959681, 22 pages https://doi.org/10.1155/2019/8959681 Research Article Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays M. De la Sen ,1 R. Nistal,1 S. Alonso-Quesada ,1 and A. Ibeas 2 1 Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa (Bizkaia), P.O. Box 48940, Spain 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193-Barcelona, Spain Correspondence should be addressed to M. De la Sen; manuel.delasen@ehu.eus Received 20 December 2018; Revised 20 May 2019; Accepted 16 June 2019; Published 8 July 2019 Academic Editor: Cengiz Çinar Copyright © 2019 M. De la Sen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study. 1. Introduction endemic equilibrium point, as well as the stability properties of the equilibrium points, is definitely linked with the value Epidemic models have been widely studied in the last decades of the so-called reproduction number (sometimes referred to involving several inter-actuating subpopulations with mutual as “basic reproduction number” or “basic reproduction ratio” coupled dynamics. Important properties which have to be in the literature [4, 5]). If such a reproduction number is less required to the epidemic models for their well-posedness are than unity then the disease-free equilibrium point is locally their solution positivity under any given nonnegative initial asymptotically stable and the endemic one is unattainable conditions as well as their stability conditions around one of (or unreachable) since it has some negative component for the equilibrium points. Thus, relevant background literature some infective subpopulation. This fact implies the global has dealt with the study of these issues for different types asymptotic stability to the disease-free equilibrium point of epidemic models, in both continuous and discrete-time provided that all the infective subpopulations are asymptot- [1–5]. A general result is that the disease-free equilibrium ically stable. However, if the reproduction number exceeds point is locally asymptotically stable when the endemic one unity, then the disease-free equilibrium point is unstable and is not reachable and if this last one is reachable (i.e., allocated the endemic equilibrium point is reachable (or attainable) within the closed first orthant of the real space including any so that the disease becomes permanent through time. The trajectory solution) then it is locally stable while the disease- frontier between both situations typically occurs when the free equilibrium one is unstable. The reachability of the reproduction number is unity. See, for instance, [6–12] and 2 Discrete Dynamics in Nature and Society some references therein. In those studies, the first orthant of (−𝑉) being the disease transition matrix, i.e., its maximum the space state is an invariant subspace as a result of the non- real positive eigenvalue, [2, 3, 16], is less than unity then the negativity properties of the state trajectory solutions which is linearized system around the disease-free equilibrium point also an invoked property of other biological problems related is proved to be locally asymptotically stable and conversely. It to species evolution dynamics. See, for instance, [13] and is also proved that the reproduction number is linked to the some references therein. Since the reproduction number is a Perron root of the above auxiliary matrix, which coincides very important tool to have a biological insight about if the with its spectral radius in typical examples of epidemic disease is either permanent (more than one contagion per models. The presence of delays is an important modeling tool infectious individual, in average) or it extinguishes (less than in epidemiology [11, 17] in cases when there are successive one contagion per infectious individual, in average) many outbreaks and regrowths of the disease intensity caused by efforts have been done to calculate it for different epidemic increase of the transmission vector numbers or external models, [4, 5]. The calculation is usually performed in a immigration to the environment under study in the model. case-by-case fashion providing the value of the reproduction Therefore, once the above general algebraic framework is set number for each particular considered model. It turns out for delay-free models, the above study is extended to epidemic that there are no general analysis tools available in the models under, in general, incommensurate (in the sense that background literature to discuss those properties based on they are not all integer multiple of the smaller, or base, delay) general reasonable and generic assumptions independent of state point delays, [18]. The case of commensurate delays is the particular epidemic model. Although these kinds of results a particular case of the above one when all the delays are are known from the background literature in a variety of epi- integer multiples of the base delay [19–23]. Two auxiliary demic models, they are revisited and presented here in a general delay-free systems are defined related to the linearization framework as a fruitful combination of algebraic results based processes around the equilibrium points which correspond on positivity and stability of the linearized systems around the to the zero delay, i.e., delay-free, and infinity delay cases. equilibrium points. Thus, a general technique to obtain the Those auxiliary systems are used to formulate stability and reproduction number and discuss the stability properties of positivity properties, independent of the delay sizes, for the equilibrium points for any type of compartmental epidemic whole epidemic model. As a result of the developed formal models is proposed and discussed. The formal study is based treatment, the reproduction number in the delayed case is on linear algebraic tools to investigate the positivity and the not directly given by the spectral radii of certain matrices stability of the linearized dynamics around the disease-free related to the delay-auxiliary free and infinity delay linearized and endemic equilibrium points. Consequently, our results systems around the equilibrium points. Those spectral radii unify previous research in this respect. To this end, the origi- are, in general, lower bounds of the reproduction number. nal state vector is split into the dynamically coupled infective In particular, they give a guaranteed worst-case measure of and noninfective compartments each one containing the guaranteed stability of the disease-free equilibrium point. corresponding state components. The infective compartment The remaining main body is organized in two more includes the dynamics of all the infective subpopulations sections referred to the model analysis in the delay-free case in the model, typically, the exposed, asymptomatic and and under the presence of delays. The positivity and stability symptomatic infectious and dead-infective, if considered in of the solutions in both cases are formally discussed in a the model. The noninfective compartment is composed of general context, rather than for specific models, based on the subpopulations being free of the disease and typically the linearization analysis on the infective compartment for contains the dynamics of the susceptible and the recovered. For the subsequent study development, the characteristics the disease-free and endemic equilibrium points. Illustrative of the state transition matrices of the linearized infective examples referred to the formal links of the presented compartments around the equilibrium points of the epidemic mathematical framework to particular epidemic models are models, which are Metzler matrices, [14, 15], as well as discussed. Two given appendices are given which contain their stability properties and the relations of their properties auxiliary technical results on stability of Metzler and 𝑀- to those of their opposite 𝑀 – matrices, are seen to be matrices and on the uniqueness of the equilibrium points. crucial mathematical tools. The main reason is that the linearized epidemic system versions around the equilibrium 1.1. Notation points have also to possess nonnegative solution trajectories since the whole epidemic model has nonnegative solution 𝑝 = {1, 2, . . . , 𝑝}, trajectories under any given nonnegative initial conditions. In R0+ = R+ ∪{0} = {𝑟 ∈ R : 𝑟 ≥ 0}; R+ = {𝑟 ∈ R : 𝑟 > 0}, particular, such state transition matrices define the so-called R0− = R− ∪{0} = {𝑟 ∈ R : 𝑟 ≤ 0}; R− = {𝑟 ∈ R : 𝑟 < 0}, next generation matrix [4] of the infective compartment. The maximum modulus of such a matrix is the relevant parameter C0+ = C+ ∪ {i𝜔 : 𝜔 ∈ R} = {𝑧 ∈ C : Re 𝑧 ≥ 0}; to characterize the stability of the infective compartment C+ = {𝑧 ∈ C : Re 𝑧 > 0}; i = √−1 is the complex and it often determines the disease reproduction number. unity; 𝐶(0, 1) is the closed circle in C centred at 0 It is also a crucial fact in the analysis the property that the and radius one of boundary 𝜕𝐶(0, 1) (that is, the unit transmissions matrices 𝐹 of such linearized systems around circumference), the equilibrium points are nonnegative. If the Perron root C0− = C− ∪ {i𝜔 : 𝜔 ∈ R} = {𝑧 ∈ C : Re 𝑧 ≤ 0}; of the auxiliary matrix (next generation matrix) 𝐹𝑉−1 , with C− = {𝑧 ∈ C : Re 𝑧 < 0}; Discrete Dynamics in Nature and Society 3 If 𝐴 = (𝐴 𝑖𝑗 ) ∈ R𝑝×𝑞 then (1) 𝐴 ⪰ 0 (in words: 𝐴 is a bounded incidence rate, and 𝐺(𝑧(𝑡)) = 𝐹(𝑧(𝑡))𝑧(𝑡), where nonnegative matrix) means that 𝐴 𝑖𝑗 ≥ 0; ∀𝑖 ∈ 𝑝, ∀𝑗 ∈ 𝑞. 𝐺(𝑧(𝑡)) ∈ R𝑛×(𝑛+𝑚) is a partitioned block matrix function 𝑝×𝑞 An equivalent notation is 𝐴 ∈ R0+ . Similarly, if 𝐴, 𝐵 ∈ R𝑝×𝑞 of the form 𝐺(𝑧(𝑡)) = [𝐺𝑥𝑥 (𝑦(𝑡)) 𝐺𝑥𝑦 (𝑦(𝑡))]. We can also then 𝐴 ⪰ 𝐵 means that 𝐴 − 𝐵 ⪰ 0 and 𝐴 ⪯ 0 means that rename, by obvious reasons, the state vectors 𝑥(𝑡) and 𝑦(𝑡) −(𝐴) ⪰ 0. of the disease and nondisease compartments as the infective (2) 𝐴 ≻ 0 (in words: 𝐴 is a positive matrix) means that and noninfective compartment (or substates) of the epidemic 𝐴(≠ 0) ⪰ 0, that is, 𝐴 ⪰ 0 with at least one positive entry. model, respectively. The following assumptions are made. Similarly, if 𝐴, 𝐵 ∈ R𝑝×𝑞 then 𝐴 ≻ 𝐵 means that 𝐴 − 𝐵 ≻ 0 Assumptions 1. 𝐹𝑖 (0, 𝑦(𝑡)) = 𝑉𝑖 (0, 𝑦(𝑡)) = 0; ∀𝑖 ∈ 𝑛 = and 𝐴 ≺ 0 means that −(𝐴) ≻ 0. {1, 2, . . . , 𝑛}, 𝐹𝑖 (𝑥(𝑡), 𝑦(𝑡)) ≥ 0, 𝑉𝑖 (𝑥(𝑡), 𝑦(𝑡)) ≤ 0 when 𝑥𝑖 = 0, (3) 𝐴 ≻≻ 0 (in words: 𝐴 is a strictly positive matrix) and ∑𝑛𝑖=1 𝑉𝑖 (𝑥(𝑡), 𝑦(𝑡)) ≥ 0 for all 𝑥(𝑡), 𝑦(𝑡) ⪰ 0. means that 𝐴 𝑖𝑗 > 0; ∀𝑖 ∈ 𝑝, ∀𝑗 ∈ 𝑞. An equivalent notation is 𝑝×𝑞 The conditions of Assumptions 1 are invoked for the 𝐴 ∈ R+ . Similarly, if 𝐴, 𝐵 ∈ R𝑝×𝑞 then 𝐴 ≻≻ 𝐵 means that nonnegativity of the state trajectory solution for initial con- 𝐴 − 𝐵 ≻≻ 0 and 𝐴 ≺≺ 0 means that −(𝐴) ≻≻ 0, ditions in the first orthant and for the local stability of the It turns out that 𝐴 ≻≻ 0 󳨐⇒ 𝐴 ≻ 0 󳨐⇒ 𝐴 ⪰ 0. disease-free equilibrium point in the absence of infection, The logic conjunction (“And”) and logic disjunction i.e., under a zero reproduction number. See [1–3] for more (“Or”) of propositions are defined by the symbols “∧” and specific details. In [1–3], the matrices 𝐹 and 𝑉 are real square “∨,” respectively. The logic negation (“No”) is defined by the 𝑛-matrices defined as the following Jacobian matrices around symbol “¬”. For instance, if 𝐴 ∈ R𝑝×𝑞 then ¬𝐴 = (𝐴 𝑖𝑗 ) ≻ 0 the disease-free equilibrium point 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) : means that 𝐴 𝑖𝑗 < 0 for some (𝑖, 𝑗) ∈ 𝑝 × 𝑞. For vectors V ∈ R𝑛 , similar nonnegativity, positivity, and 𝜕𝐹𝑖 (0, 𝑦𝑑𝑓 ) strictly positivity notations are used, namely, V ⪰ 0, V ≻ 0, 𝐹=[ ]; 𝜕𝑥𝑗 V ≻≻ 0 and the corresponding nonpositivity, negativity, and strict negativity counterpart notations. (4) 𝐼𝑛 is the identity 𝑛-matrix, and superscript 𝑇 denotes a 𝜕𝑉𝑖 (0, 𝑦𝑑𝑓 ) 𝑉=[ ] matrix transposition, 𝜕𝑥𝑗 𝜌(𝐴) = |𝜆 max (𝐴)| is the spectral radius of 𝐴 ∈ R𝑝×𝑝 , 𝑝×𝑝 𝐴 ∈ R𝑝×𝑝 is a Metzler matrix, denoted by 𝐴 ∈ 𝑀𝐸 , if The following assumptions on the above Jacobian matrices 𝐴 𝑖𝑗 ≥ 0; ∀𝑖, 𝑗(≠ 𝑖) ∈ 𝑝, are reasonable in the context of epidemic models [1]. The notations 𝐹, 𝑉 stand through the manuscript for the Jacobian 𝐴 ∈ R𝑝×𝑝 is an 𝑀-matrix if 𝐴 𝑖𝑖 ≥ 0 and 𝐴 𝑖𝑗 ≤ 0; ∀𝑖, 𝑗(≠ matrices around the disease-free equilibrium point. The 𝑖) ∈ 𝑝, notations 𝐹𝑒 , 𝑉𝑒 stand for their counterparts around the ‖𝐹(𝑠)‖∞ is the 𝐻∞ - norm of complex rational matrices endemic equilibrium point 𝑥𝑒𝑛𝑑 while the notations 𝐹0 , 𝑉0 𝐹 : C 󳨀→ 𝐶𝑝×𝑞 which are analytic in Re 𝑠 > 0; 𝑠 ∈ C. The stand for any generic equilibrium point 𝑧0 . It is well-known argument “s” stands for the Laplace transform argument. in many typical existing epidemic models that 𝑧𝑒𝑛𝑑 and 𝑧𝑑𝑓 are coincident when the reproduction number is unity. 2. Results for Delay-Free Epidemic Models Assumptions 2. (a) The disease-free equilibrium point 𝑧𝑑𝑓 = A general compartmental disease model of dimension (𝑚 + (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) is unique and the noninfective substate dynamics 𝑛)can be written as [1] ̇ = 𝑔(0, 𝑦(𝑡)) is locally asymptotically stable. 𝑦(𝑡) 𝑥̇ (𝑡) = 𝐹 (𝑥 (𝑡) , 𝑦 (𝑡)) − 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) (b) The minus transition matrix 𝑉 of the infective states (1) of the linearized infective subsystem around the disease-free = (𝐹 − 𝑉) 𝑥 (𝑡) − 𝑓 (𝑥 (𝑡) , 𝑦 (𝑡)) equilibrium point is a nonsingular 𝑀-matrix with 𝑉−1 ≻ 0 while the corresponding transmission matrix fulfills 𝐹 ⪰ 0. 𝑦̇ (𝑡) = 𝑔 (𝑥 (𝑡) , 𝑦 (𝑡)) (2) The above assumptions are very relevant from a physical point of view having in mind that (−𝑉) and 𝐹 are, respectively, for any given 𝑧(0) = (𝑥𝑇 (0), 𝑦(0)𝑇 )𝑇 , where 𝑛 and 𝑚 are the transition and transmission matrices of the linearized the respective dimensions of the disease and nondisease system around the disease-free equilibrium point while (𝐹 − compartments of respective state vectors 𝑥(𝑡) and 𝑦(𝑡), 𝑉) is the matrix of the dynamics of such a system. It turns 𝐹 ∈ R𝑛×𝑛 and (−𝑉) ∈ R𝑛×𝑛 are, respectively, the disease out, in practice, that 𝐹 = 0 if and only if the coefficient transmission and state transition matrices of the linearized transmission rate 𝛽 is zero, that is, there is no disease system around the disease-free equilibrium point, and contagion in practice. Some conditions for the uniqueness of 𝑓 (𝑥 (𝑡) , 𝑦 (𝑡)) = (𝐹 − 𝑉) 𝑥 (𝑡) − 𝐹 (𝑥 (𝑡) , 𝑦 (𝑡)) the disease-free equilibrium point are given in Theorem B.1 (3) of Appendix B based on the implicit function theorem. Some + 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝜃 (𝑧 (𝑡)) 𝐺 (𝑧 (𝑡)) conditions are also given for the uniqueness of the endemic equilibrium point in Theorem B.2 of Appendix B based on describes the higher-order contributions to the infective the Rouché-Frobenius test for compatibility of solutions in compartment dynamics, where 𝑧(𝑡) = (𝑥𝑇 (𝑡), 𝑦𝑇 (𝑡))𝑇 is algebraic systems. This alternative way of proof is taken on the whole model state vector, 𝜃(𝑧(𝑡)) is the scalar positive the basis that distinct endemic equilibrium points can have 4 Discrete Dynamics in Nature and Society distinct disease and nondisease endemic steady-state equi- (iv) The linearized infective substate around the disease-free librium compartments while the disease-free equilibrium equilibrium point is nonnegative for any given initial conditions has always zero infective substates. Both theorems are also 𝑥𝐿 (0) = 𝑥(0) ⪰ 0, 𝑦𝐿 (0) = 𝑦(0) ⪰ 0 and unstable if some of valid for the case of presence of internal delays. Note that the conditions given below hold: Assumption 2(b) can be stated in the following equivalent (1) (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is not a stability matrix. form (see Theorem A.1 of Appendix A): (2) (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 and (𝑉 − 𝐹)−1 either does not exist or if it exists is not positive. Assumption 3(b). The transition matrix (−𝑉) of the infec- (3) (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 and it exists as 𝑉−1 which is not positive, tive state of the linearized system around the disease-free 𝐹 ⪰ 0 and 𝜌(𝐹𝑉−1 ) > 1. equilibrium point is a stability Metzler matrix while the corresponding transmission matrix fulfills 𝐹 ⪰ 0. Proof. The solution of the linearized infective substate around Note that Assumption 3(b) is of a clear interpretation for the disease-free equilibrium point is the linearization around the disease-free equilibrium point while its equivalent form of Assumption 2 is easy to test since 𝑥𝐿 (𝑡) = 𝑒(𝐹−𝑉)𝑡 𝑥𝐿 (0) it is not needed to calculate the allocation of the eigenvalues of 𝑡 (5) the matrix (−𝑉). Note also that 𝐹 = 𝐹(𝑅0 , 𝑦𝑑𝑓 ) ⪰ 0 depends = 𝑒−𝑉𝑡 (𝑥𝐿 (0) + ∫ 𝑒𝑉𝜏 𝐹𝑥𝐿 (𝜏) 𝑑𝜏) on the disease-free equilibrium substate of the noninfective 0 components and on the reproduction number 𝑅0 associated with the disease propagation. If 𝑅0 = 0 then 𝐹 = 𝐹(0, 𝑦𝑑𝑓 ) = It turns out that, for any 𝑥𝐿 (0) ⪰ 0, 𝑥𝐿 (𝑡) ⪰ 0 since 0. Also, it usually occurs that if the epidemic model has only 𝑒(𝐹−𝑉)(𝑡−𝜏) ≻ 0 for any 𝜏(≤ 𝑡), 𝑡 ∈ R0+ and (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 a single constant transmission coefficient rate 𝛽, then 𝑅0 = 0 [14], then, 𝑥𝐿 (𝑡) ⪰ 0; ∀𝑡 ∈ R0+ for any given 𝑥𝐿 (0) ⪰ 0 and 𝐹 = 0 if 𝛽 = 0. This section gives some elementary results if (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 (first term of the solution 𝑥𝐿 (𝑡)) or if on positivity of the solutions and stability for the linearized (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 and 𝐹 ⪰ 0 (second term of the solution 𝑥𝐿 (𝑡)). system around the disease-free equilibrium point in the case Then (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 and (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 and 𝐹 ⪰ 0 are of absence of modelling delays. The results are relevant since both sufficient conditions for 𝑥𝐿 (0) ⪰ 0 󳨐⇒ 𝑥𝐿 (𝑡) ⪰ 0; if the linearized system around the disease- free equilibrium ∀𝑡 ∈ R0+ . The necessity of (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 follows by point fails to be either positive or, respectively, stable then the contradiction. If (𝐹 − 𝑉) ∉ 𝑀𝐸𝑛×𝑛 then there exists at least whole system cannot be either positive or, respectively, stable. one entry 𝑖, 𝑗 ∈ 𝑛 such that (𝑒(𝐹−𝑉)𝑡 )𝑖𝑗 < 0 for some 𝑡 > 0. Two basic auxiliary results to be invoked concerning certain Then, it suffices to take 𝑥𝐿 𝑗 (0) > 0 and 𝑥𝐿 𝑘 (0) = 0; ∀𝑘(≠ technical relations between M-matrices and Metzler matrices 𝑗) ∈ 𝑛 to conclude that 𝑥𝐿 𝑖 (𝑡) = −|(𝑒(𝐹−𝑉)𝑡 )𝑖𝑗 |𝑥𝐿 𝑗 (0) < 0 so and their stability properties are given in Appendix A. The that the infective substate is not positive. Thus, (𝐹 − 𝑉) ∈ following result on stability and positivity of the linearized 𝑀𝐸𝑛×𝑛 is a necessary and sufficient condition of nonnegativity infective substate is derived under some of the conditions of of the solution of the infective linearized substate around Assumptions 1-2: the disease-free equilibrium point for all time and any Theorem 1. The following properties hold. nonnegative initial condition. Property (i) has been proved. (i) The linearization of the epidemic model (1)-(2) around Property (ii) is obvious from the fact that a necessary the disease-free equilibrium point has a nonnegative infective condition for nonnegativity of any solution under arbitrary substate trajectory solution for any initial conditions 𝑥(0) ⪰ 0 nonnegative initial conditions is that the property holds for its and 𝑦(0) ⪰ 0 (i.e., the infective substate is nonnegative for all linearized counterpart system. Property (iii) follows since, on time) if and only if (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 . A sufficient condition is one hand, the given conditions of Metzler matrices guarantee that (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 and 𝐹 ⪰ 0. the nonnegativity of the linearized solution of the infective (ii) A necessary condition for an epidemic model (1)-(2) to substates for any nonnegative initial conditions and all time. have a nonnegative solution trajectory, irrespective of the given And, on the other hand, the convergence to the disease-free particular initial conditions 𝑥(0) ⪰ 0, 𝑦(0) ⪰ 0, is that (𝐹 − equilibrium of such a linearized subsystem follows from the 𝑉) ∈ 𝑀𝐸𝑛×𝑛 . stability of the Metzler matrices under the nonsingularity (iii) The linearized infective substate around the disease- and positivity of their negative 𝑀-matrices counterparts free equilibrium point has a nonnegative solution which is according to Theorem A.1 [(i) to (iii)] and Theorem A.2 (ii) uniformly bounded for all time with 𝑥𝐿 (𝑡) ⪰ 0; ∀𝑡 ∈ R0+ and of Appendix A. Property (iv) is a dual version for instability 𝑥𝐿 (𝑡) 󳨀→ 0 exponentially as 𝑡 󳨀→ ∞ for any given initial of the sufficiency parts of Property (iii). conditions 𝑥𝐿 (0) = 𝑥(0) ⪰ 0, 𝑦𝐿 (0) = 𝑦(0) ⪰ 0 if and only if Note that if (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 for any given 𝐹 ⪰ 0 then any of the two equivalent conditions given below holds: (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 (otherwise, (𝐹 − 𝑉) ∉ 𝑀𝐸𝑛×𝑛 if 𝐹 = 0). (1) (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix. Therefore, [(𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 ; ∀𝐹 ⪰ 0] ⇐⇒ [(−𝑉) ∈ (2) (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is nonsingular (or, equivalently, (𝑉 − 𝐹) 𝑀𝐸𝑛×𝑛 ∧ 𝐹 ⪰ 0]. This equivalence is reasonable, in practice, is a nonsingular 𝑀-matrix) and (𝑉 − 𝐹)−1 ≻ 0. since the nonnegativity of the linearized trajectory solution A joint sufficiency-type condition for any of the above around the disease-free equilibrium point of the infective Conditions (1)-(2) to hold is that (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is nonsingular compartment is also needed for zero reproduction numbers (or, equivalently, 𝑉 is a nonsingular 𝑀-matrix), 𝑉−1 ≻ 0, 𝐹 ⪰ 0 or zero coefficient transmission rates. The joint sufficiency- and 𝜌(𝐹𝑉−1 ) < 1. type condition of Theorem 1 (iii) is also necessary and this Discrete Dynamics in Nature and Society 5 condition is equivalent to the other two Conditions (1)-(2). of 𝐹 resulting in (𝐹 − 𝑉) to lose the stability of (𝐹 − 𝑉) when From Theorem 1 on the linearized infective substate and 𝐹 = 0, i.e., for 𝐹 − 𝑉 = −𝑉. In the following discussions (−𝑉) some further conditions on the nonlinear contribution of the is the transition matrix in-between state of the linearized dynamics in the model (1)-(3), we can obtain some further subsystem of the infected variables around the disease-free results for the nonnegativity of the solution and the stability equilibrium point while 𝐹 is the transmission matrix around of the whole nonlinear model as follows. such an equilibrium point. On the other hand, it turns out that the nonnegativity Theorem 2. The following properties hold. 𝑡 condition ∫0 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏 ⪯ 𝑒(𝐹−𝑉)𝑡 𝑥(0); ∀𝑡 ∈ R0+ of (i) The state trajectory solution of (1)-(3) is nonnegative for Theorem 2 for the whole trajectory solution, subject to non- all time, i.e., 𝑧(𝑡) = (𝑥𝑇 (𝑡), 𝑦𝑇 (𝑡))𝑇 ⪰ 0; ∀𝑡 ∈ R0+ for any given linear dynamics, of the infective substate does not require that initial condition 𝑧(0) = (𝑥𝑇 (0), 𝑦𝑇 (0))𝑇 ⪰ 0 if and only if the (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix or any conditions on the two subsequent constraints hold: nonlinear term 𝑓(𝑥(𝑡), 𝑦(𝑡)). Note that the condition holds 𝑡 directly if −𝑓(𝑥(𝑡), 𝑦(𝑡)) ⪰ 0 since then 0 ⪰ −𝑒(𝐹−𝑉)𝑡 𝑥(0); ∫ 𝑓 (𝑥 (𝜏) , 𝑦 (𝜏)) 𝑑𝜏 ⪯ 𝑒(𝐹−𝑉)𝑡 𝑥 (0) ; ∀𝑡 ∈ R0+ ; ∀𝑥(0) ∈ R0+ 𝑛 𝑡 so that − ∫0 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏 ⪰ 0 ⪰ −𝑒(𝐹−𝑉)𝑡 𝑥(0). 0 (6) The nonnegativity of the trajectory solution can be also easily 𝑡 ∫ 𝑔 (𝑥 (𝜏) , 𝑦 (𝜏)) 𝑑𝜏 ⪰ −𝑦 (0) ; ∀𝑡 ∈ R0+ . tested on the differential system (1)-(3) by ensuring that, for 0 any zero value of a state component, its corresponding time- Furthermore, one has that derivative is nonnegative. So, we have the subsequent result which is alternative to the two first constraints of Theorem 2 𝑡 (i). lim sup (∫ 𝑓 (𝑥 (𝜏) , 𝑦 (𝜏)) 𝑑𝜏 − 𝑒(𝐹−𝑉)𝑡 𝑥 (0)) ⪯ 0; 𝑡󳨀→∞ 0 𝑡 (7) Theorem 3. The following properties hold. lim inf (𝑦 (0) + ∫ 𝑔 (𝑥 (𝜏) , 𝑦 (𝜏)) 𝑑𝜏) ⪰ 0. (i) The state trajectory solution of (1)-(3) is nonnegative for 𝑡󳨀→∞ 0 all time, i.e., 𝑧(𝑡) = (𝑥𝑇 (𝑡), 𝑦𝑇 (𝑡))𝑇 ⪰ 0; ∀𝑡 ∈ R0+ for any given initial condition 𝑧(0) = (𝑥𝑇 (0), 𝑦𝑇 (0))𝑇 ⪰ 0 if and only if, for (ii) Assume, furthermore, that (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability all 𝑖 ∈ 𝑛, 𝑗 ∈ 𝑚 and all 𝑡 ∈ R0+ , the following conditions hold: matrix or that any of the two constraints (1) or (2) of Theorem 1 (iii) holds. Then, ∞ (𝑥𝑖 (𝑡) = 0) 󳨐⇒ (𝑓𝑖 (𝑥 (𝑡) , 𝑦 (𝑡)) ∫ 𝑓 (𝑥 (𝜏) , 𝑦 (𝜏)) 𝑑𝜏 = 0, 0 (8) 𝑛 𝑛 (9) 𝑓 (𝑥 (𝑡) , 𝑦 (𝑡)) 󳨀→ 0 as 𝑡 󳨀→ ∞ ≤ ∑ (𝐹𝑖𝑗 − 𝑉𝑖𝑗 ) 𝑥𝑗 (𝑡) = ∑ (𝐹𝑖𝑗 − 𝑉𝑖𝑗 ) 𝑥𝑗 (𝑡)) 𝑗(=𝑖)=1 ̸ 𝑗=1 𝑛 𝑚 𝑛 provided that 𝑓 : (R0+ × R0+ ) × (R0+ × R0+ ) 󳨀→ R0+ is everywhere uniformly continuous in its definition domain and (𝑦𝑗 (𝑡) = 0) 󳨐⇒ (𝑔𝑗 (𝑥 (𝑡) , 𝑦 (𝑡)) ≥ 0) . 𝑓(𝑥(𝑡), 𝑦(𝑡)) ⪰ 0; ∀𝑡 ∈ R0+ ; and 𝑓(𝑥(𝑡), 𝑦(𝑡)) 󳨀→ 0 as ∞ (ii) Property (i) also implies that 𝑓𝑖 (0, 𝑦𝑑𝑓 ) = 0; ∀𝑖 ∈ 𝑛, 𝑡 󳨀→ ∞ (but nonnecessarily ∫0 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏 = 0) if 𝑛 𝑚 𝑛 𝑔𝑗 (0, 𝑦𝑑𝑓 ) = 0; ∀𝑗 ∈ 𝑚 if a disease-free equilibrium point 𝑓 : (R0+ × R0+ ) × (R0+ × R0+ ) 󳨀→ R0+ is everywhere uniformly continuous in its definition domain. 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) exists. Proof. Property (i) follows directly from (1)-(3). On the other Proof. It is obvious that, for any given 𝑧(0) ⪰ 0, if 𝑧𝑖 (𝑡) = 0 hand, if (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 , and 𝑓(𝑥(𝑡), 𝑦(𝑡)) ⪰ 0; ∀𝑡 ∈ for some 𝑡 ≥ 0, then 𝑧̇𝑖 (𝑡) ≥ 0 from the given conditions and R0+ and it is everywhere uniformly continuous for all time it cannot give 𝜀 ∈ R+ such that 𝑧𝑖 (𝑡 + 𝜀) < 0 for some 𝑖 ∈ 𝑛. for its arguments in the first closed orthant of R𝑛+𝑚 for As a result, 𝑧(𝑡) ⪰ 0; ∀𝑡 ∈ R0+ . The first part of the result ∞ has been proved. The second part is proved by contradiction. all time; then it follows that ∫0 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏 = 0, and First note from the nonnegativity condition of Property (i) 𝑒(𝐹−𝑉)𝑡 𝑥(0) 󳨀→ 0 and 𝑓(𝑥(𝑡), 𝑦(𝑡)) 󳨀→ 0 as 𝑡 󳨀→ ∞. Also, that −𝑓𝑖 (0, 𝑦𝑑𝑓 ) ≥ −(∑𝑛𝑗=1 (𝐹𝑖𝑗 − 𝑉𝑖𝑗 ).0) = 0. If 𝑓𝑖 (0, 𝑦𝑑𝑓 ) = 0; if 𝑓(𝑥(𝑡), 𝑦(𝑡)) is everywhere continuous for all time in R𝑛+𝑚 𝑡 ∀𝑖 ∈ 𝑛 then Property (ii) has been proved for the infective and 0 = lim𝑡󳨀→∞ 𝑒(𝐹−𝑉)𝑡 ⪰ lim𝑡󳨀→∞ |∫0 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏| since substate. Now, assume that there exists some 𝑖 ∈ 𝑛 such that (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 and it is a stability matrix. Then, it follows 𝑓𝑖 (0, 𝑦𝑑𝑓 ) < 0. Since 𝜗(𝑡) = ∑𝑛𝑗=1 (𝐹𝑖𝑗 − 𝑉𝑖𝑗 )𝑥𝑗 (𝑡)]𝑥(𝑡)=0 = 0, from Barbalat’s lemma [18, 24] that 𝑓(𝑥(𝑡), 𝑦(𝑡)) 󳨀→ 0 as it follows that 𝑥𝑖̇ ]𝑥𝑖 =0 > 0 so that 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) is not 𝑡 󳨀→ ∞. Property (ii) has been proved. an equilibrium point; hence a contradiction follows. In the Generally speaking, Theorem 1 (iv.(3)) is also guaranteed same way, if 𝑔𝑗 (0, 𝑦𝑑𝑓 ) > 0 for some 𝑗 ∈ 𝑚 then 𝑧𝑑𝑓 = under sufficient conditions if the involved spectral radii [15, (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) is not an equilibrium point; hence we have again 25, 26] are replaced by matrix norms which are their upper- a contradiction. Property (ii) has been proved. bounds. Also, note that Theorem 2 (ii) has the assumption that (𝐹−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is stable which is needed to guarantee that Theorem 4. Let 𝑔 : R𝑛+𝑚 󳨀→ R𝑚 be everywhere continuously no eigenvalue crosses the imaginary axis under perturbations differentiable with a Jacobian matrix: 6 Discrete Dynamics in Nature and Society = [𝐽𝑔,𝑥 (𝑧0 ) | 𝐽𝑔,𝑦 (𝑧0 )] at any 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 ∈ 𝐽𝑔,𝑧 (𝑧0 ) Furthermore, it is locally asymptotically stable if 𝜌(𝐹𝑉−1 ) < 1 R 𝑛+𝑚 with 𝑥0 ∈ R𝑛 and 𝑦0 ∈ R𝑚 , where 𝑧 = (𝑥𝑇 , 𝑦𝑇 )𝑇 ∈ R𝑛+𝑚 , since then (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix, equivalently 𝑥 ∈ R𝑛 , 𝑦 ∈ R𝑚 . Assume that the 𝑚 × 𝑚 matrix 𝐽𝑔,𝑦 (𝑧0 ) is (𝑉 − 𝐹)−1 ≻ 0 so that the linearized system around the nonsingular at any equilibrium point 𝑧0 of (1)-(3). Then, the disease-free equilibrium point is locally exponentially (then following properties hold asymptotically) stable. On the other hand, it is unstable if (i) There exists an open set 𝑈 of R𝑛 containing 𝑥0 such that 𝜌(𝐹𝑉−1 ) > 1 [Theorem 1 (iii)-(iv)]. The part (a) of Property 𝑇 𝑇 𝑇 there exists a unique continuously differentiable function ℎ : (ii) has been proved. Now let 𝑧0 = 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) be the 𝑈 󳨀→ R𝑚 such that 𝑦0 = ℎ(𝑥0 ) and 𝑓(𝑥, ℎ(𝑥)) = 0; ∀𝑥 ∈ endemic equilibrium point. The following cases can occur 𝑈. Furthermore, any equilibrium point 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 can be (1) If 𝜌(𝐹𝑉−1 ) ≥ 1, then note that expressed being dependent on the infective substate only, i.e., −1 𝑧0 = (𝑥0𝑇 , ℎ𝑇 (𝑥0 ))𝑇 , while satisfying 𝑥𝑒𝑛𝑑 = (𝑉𝑒 − 𝐹𝑒 ) (−𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 ))) (12) 𝑦0 = ℎ (𝑥0 ) ; ⪰ (𝑉 − 𝐹)−1 (−𝑓 (𝑥𝑑𝑓 , ℎ (𝑥𝑑𝑓 ))) ⪰ 0 (10) −1 𝑥0 = (𝐹0 (ℎ (𝑥0 )) − 𝑉0 (𝑥0 )) 𝑓 (𝑥0 , ℎ (𝑥0 )) since 𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ⪯ 0 (being a strict inequality if and only if 𝑥𝑒𝑛𝑑 ≠ 𝑥𝑑𝑓 = 0) and (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is stable and then provided that (𝐹0 (ℎ(𝑥0 )) − 𝑉0 (𝑥0 )) is nonsingular at any nonsingular equilibrium point, where 𝐹0 (ℎ(0)) = 𝐹 and 𝑉0 (0) = 𝑉 if (2) If 𝑅0 = 𝜌(𝐹𝑉−1 ) < 1 then (𝑉−𝐹)−1 ≻ 0, since (𝐹−𝑉) ∈ 𝑧0 = 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 is the disease-free equilibrium point 𝑀𝐸 , and 𝐹𝑒 ≻ 𝐹. Therefore, (𝐼𝑛 + (𝑉 − 𝐹)−1 (𝐹𝑒 − 𝐹)) ≻ 0 and 𝑛×𝑛 and 𝐹0 (ℎ(𝑥𝑒𝑛𝑑 )) = 𝐹𝑒 and 𝑉0 (𝑥𝑒𝑛𝑑 ) = 𝑉𝑒 if 𝑧0 = 𝑧𝑒𝑛𝑑 = 𝜌[(𝑉−𝐹)−1 (𝐹𝑒 −𝐹)] > 0 then (𝐼𝑛 +(𝐼𝑛 −𝑉−1 𝐹)−1 𝑉−1 (𝐹𝑒 −𝐹))−1 𝑇 (𝑥𝑒𝑛𝑑 , ℎ𝑇 (𝑥𝑒𝑛𝑑 ))𝑇 is the endemic equilibrium point. exists, and (ii) Assume, furthermore, that the transition and transmis- −1 −1 sion matrices of the disease-free equilibrium point satisfy that (𝑉𝑒 − 𝐹𝑒 ) = (𝐼𝑛 + (𝑉 − 𝐹)−1 (𝐹𝑒 − 𝐹)) (𝑉 − 𝐹)−1 (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix and 𝐹 ⪰ 0. Then, (a) the disease-free equilibrium point exists and it is defined by −1 −1 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 and it is unique and locally asymptotically = (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) (𝑉 − 𝐹)−1 (13) stable if 𝜌(𝐹𝑉−1 ) < 1. If 𝜌(𝐹𝑉−1 ) > 1 then it is unstable; and ≺ (𝑉 − 𝐹)−1 (b) the endemic equilibrium point exists (while it is unique under the sufficiency-type conditions of Theorem B.2 Note that 𝑥𝑒𝑛𝑑 is unreachable if 𝑥𝑒𝑛𝑑 ≠ 𝑥𝑑𝑓 = 0 since 𝑇 𝑇 𝑇 of Appendix B) and it is defined by 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) = −𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ≺ −𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) = 0, and 𝑇 (𝑥𝑒𝑛𝑑 , ℎ𝑇 (𝑥𝑒𝑛𝑑 ))𝑇 subject to −1 𝑥𝑒𝑛𝑑 = (𝑉𝑒 − 𝐹𝑒 ) (−𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 ))) 𝑦𝑒𝑛𝑑 = ℎ (𝑥𝑒𝑛𝑑 ) ; (11) −1 −1 −1 𝑥𝑒𝑛𝑑 = (𝐹𝑒 − 𝑉𝑒 ) 𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 )) ≺ (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) (14) If it exists then it is reachable if 𝜌(𝐹𝑉−1 ) ≥ 1 and 𝑓(𝑥𝑒𝑛𝑑 , × (𝑉 − 𝐹)−1 (−𝑓 (𝑥𝑑𝑓 , ℎ (𝑥𝑑𝑓 ))) ℎ(𝑥𝑒𝑛𝑑 )) ⪯ 𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) while it is unreachable, in the sense −1 −1 that 𝑧𝑒𝑛𝑑 ∉ R0+ 𝑛+𝑚 , if 𝜌(𝐹𝑉−1 ) < 1 supposing that 𝑓(𝑥𝑒𝑛𝑑 , = (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) 𝑥𝑑𝑓 = 0. ℎ(𝑥𝑒𝑛𝑑 )) ≻ 𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) if 𝑧𝑒𝑛𝑑 ≠ 𝑧𝑑𝑓 . Then, 𝑥𝑒𝑛𝑑 ⪯ 0. But, for the case, 𝑥𝑒𝑛𝑑 = 0, the endemic Proof. Assume that 𝐽𝑔,𝑦 (𝑥0 , 𝑦0 ) ∈ R𝑚 × R𝑚 is nonsingular. equilibrium point is coincident with the disease-free one, that Then, there exits an open set 𝑈 of R𝑛 containing 𝑥0 such that is, 𝑥𝑒𝑛𝑑 = 𝑥𝑑𝑓 = 0 and 𝜌(𝐹𝑉−1 ) = 1, while, for 𝜌(𝐹𝑉−1 ) < 1, there exists a unique continuously differentiable function ℎ : 𝑥𝑒𝑛𝑑 ≺ 0,; hence a contradiction to its reachability follows 𝑈 󳨀→ R𝑚 such that 𝑦0 = ℎ(𝑥0 ) and 𝑓(𝑥, ℎ(𝑥)) = 0; ∀𝑥 ∈ 𝑈. since the equilibrium point is never allocated in the open first Let 𝑧0 be either the disease-free equilibrium point 𝑧0 = 𝑧𝑑𝑓 = orthant of R𝑛 . Property (ii) has been proved. (0𝑇 , 𝑦0𝑇 )𝑇 = (0𝑇 , ℎ𝑇 (0))𝑇 or the endemic one 𝑧0 = 𝑧𝑒𝑛𝑑 = 𝑇 𝑇 𝑇 𝑇 Remark 5. It has not been proved that the endemic equi- (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) = (𝑥𝑒𝑛𝑑 , ℎ𝑇 (𝑥𝑒𝑛𝑑 )). Both of them are unique in a librium point is necessarily unique, independently of the neighbourhood centred at the respective equilibrium point, concrete epidemic model, in the whole state space. A detailed since ℎ(⋅) is unique, while they satisfy (10) and, respectively, proof of the global uniqueness of the disease-free equilibrium (11) if 𝐽𝑔,𝑧 (𝑧𝑑𝑓 ) and, respectively, 𝐽𝑔,𝑧 (𝑧𝑒𝑛𝑑 ) are nonsingular point is given in Theorem B.1 (i) of Appendix B under the since (𝐹0 (𝑥0 ) − 𝑉0 (𝑥0 )) is a nonsingular matrix. Property (i) same conditions as those given in Theorem 4 (ii) since has been proved. In order to prove Property (ii), first note that the infective equilibrium substate is unique and identically (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is nonsingular with 𝑉−1 ≻ 0 (since (−𝑉) is zero what guarantees also the uniqueness of the nondisease stable) with 𝐹 ⪰ 0. Then, the disease-free equilibrium point compartment via the uniqueness of the function ℎ(⋅). On the exists being defined by 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 ; it is unique since other hand, the endemic equilibrium point 𝑧𝑒𝑛𝑑 is unique ℎ(⋅) is unique and given by 𝑥𝑑𝑓 = (𝐹 − 𝑉)−1 𝑓(0, ℎ(0)) = 0. in the whole state space if 𝑥𝑒𝑛𝑑 , i.e., its infective substate, Discrete Dynamics in Nature and Society 7 is unique since from the uniqueness of ℎ(⋅), the uniqueness 𝑆 ̇ (𝑡) = 𝑏1 − (𝑏2 + 𝛽𝐼 (𝑡) + 𝛽𝐴 𝐴 (𝑡) + 𝛽𝐷𝐷 (𝑡)) 𝑆 (𝑡) of the nondisease compartment and then that of the whole endemic equilibrium point are also guaranteed under some + 𝜂𝑅 (𝑡) − 𝑉 (𝑡) algebraic conditions given in Theorem B.2 in Appendix B. 𝐸̇ (𝑡) = − (𝑏2 + 𝛾) 𝐸 (𝑡) It is well-known from the background literature on epi- demic models, subject to a unique disease-free equilibrium + (𝛽𝐼 (𝑡) + 𝛽𝐴 𝐴 (𝑡) + 𝛽𝐷𝐷 (𝑡)) 𝑆 (𝑡) point and a unique endemic equilibrium one, that, typically, if the reproduction number 𝑅0 < 1 then the disease-free 𝐼 ̇ (𝑡) = − (𝑏2 + 𝛼 + 𝜏0 + 𝐾𝜉 ) 𝐼 (𝑡) + 𝛾𝑝𝐸 (𝑡) equilibrium point is asymptotically stable while the endemic (15) one is not reachable since it has negative infective values 𝐴̇ (𝑡) = − (𝑏2 + 𝜏0 ) 𝐴 (𝑡) + 𝛾 (1 − 𝑝) 𝐸 (𝑡) what is incompatible with the positivity of the state trajectory solution. If 𝑅0 > 1 then the endemic equilibrium point is 𝐷̇ (𝑡) = −𝜇𝐷 (𝑡) + 𝑏2 (𝐼 (𝑡) + 𝐴 (𝑡)) + 𝛼𝐼 (𝑡) stable and the disease-free one is unstable and, if 𝑅0 = 1, then the infective variables of both equilibrium points are 𝑅̇ (𝑡) = − (𝑏2 + 𝜂) 𝑅 (𝑡) + 𝜏0 (𝐼 (𝑡) + 𝐴 (𝑡)) + 𝐾𝜉 𝐼 (𝑡) coincident. Theorem 4 describes formally via algebraic tools the reachability/unreachability of the endemic equilibrium + 𝑉 (𝑡) point depending on the value of the reproduction number. 𝑉 (𝑡) = 𝑉0 + 𝐾𝑉𝑆 (𝑡) This fact is illustrated by the subsequent example. Example 6. A particular case of the SEIADR (susceptible- where 𝑉(𝑡) is a vaccination control of constant gain 𝑉0 exposed-symptomatic infectious-asymptomatic infectious- plus a linear feedback term with gain 𝐾𝑉 with information infectious corpses-recovered) proposed in [6, 7] follows of the susceptible subpopulation. There is also an eventual below for the case when the vaccination and treatment treatment control on the infectious subpopulation of gain 𝐾𝜉 . control parameters 𝑉0 , 𝐾𝑉 , and 𝐾𝜉 are constant and there is no The disease-free and endemic Jacobian matrices for the joint impulsive control for the infectious corpses removal control noninfective infective dynamically coupled compartments action: are, respectively, − (𝑏2 + 𝐾𝑉) 0 −𝛽𝑆𝑑𝑓 −𝛽𝐴 𝑆𝑑𝑓 −𝛽𝐷𝑆𝑑𝑓 𝜂 [ ] [ 0 − (𝑏2 + 𝛾) 𝛽𝑆𝑑𝑓 𝛽𝐴 𝑆𝑑𝑓 𝛽𝐷𝑆𝑑𝑓 0 ] [ ] [ ] [ 0 𝛾𝑝 − (𝑏2 + 𝛼 + 𝜏0 + 𝐾𝜉 ) 0 0 0 ] A𝑑𝑓 =[ [ ] ] (16) [ 0 𝛾 (1 − 𝑝) 0 − (𝑏2 + 𝜏0 ) 0 0 ] [ ] [ 0 0 𝑏2 + 𝛼 𝑏2 −𝜇 0 ] [ ] [ 𝐾𝑉 0 𝜏0 + 𝐾𝜉 𝜏0 0 − (𝑏2 + 𝜂)] where the infective compartment has a Jacobian matrix 𝐹 − 𝑉 0 𝑆𝑑𝑓 𝛽𝐴𝑟 𝑆𝑑𝑓 𝛽𝐷𝑟 𝑆𝑑𝑓 uncoupled to the noninfective one with [ ] [0 0 0 0 ] 𝐹 = 𝛽[ [0 0 ] 𝑏2 + 𝛾 0 0 0 [ 0 0 ] ] [ ] [0 0 0 0 ] [ −𝛾𝑝 𝑏2 + 𝛼 + 𝜏0 + 𝐾𝜉 0 0] 𝑉=[ [−𝛾 (1 − 𝑝) ]; [ 0 𝑏2 + 𝜏0 0 ] ] (17) [ 0 − (𝑏2 + 𝛼) −𝑏2 𝜇] and − (𝑏2 + 𝛽𝐼𝑒𝑛𝑑 + 𝛽𝐴 𝐴 𝑒𝑛𝑑 + 𝛽𝐷𝐷𝑒𝑛𝑑 + 𝐾𝑉) 0 −𝛽𝑆𝑒𝑛𝑑 −𝛽𝐴𝑆𝑒𝑛𝑑 −𝛽𝐷𝑆𝑒𝑛𝑑 𝜂 [ ] [ 𝛽𝐼𝑒𝑛𝑑 + 𝛽𝐴 𝐴 𝑒𝑛𝑑 + 𝛽𝐷𝐷𝑒𝑛𝑑 − (𝑏2 + 𝛾) 𝛽𝑆𝑒𝑛𝑑 𝛽𝐴 𝑆𝑒𝑛𝑑 𝛽𝐷𝑆𝑒𝑛𝑑 0 ] [ ] [ ] [ 0 𝛾𝑝 − (𝑏2 + 𝛼 + 𝜏0 + 𝐾𝜉 ) 0 0 0 ] 𝐴 𝑒𝑛𝑑 =[ [ ] ] (18) [ 0 𝛾 (1 − 𝑝) 0 − (𝑏2 + 𝜏0 ) 0 0 ] [ ] [ 0 0 𝑏2 + 𝛼 𝑏2 −𝜇 0 ] [ ] [ 𝐾𝑉 0 𝜏0 + 𝐾𝜉 𝜏0 0 − (𝑏2 + 𝜂)] 8 Discrete Dynamics in Nature and Society where the infective compartment Jacobian matrix is 𝐹𝑒 − 𝑉𝑒 endemic equilibrium point is unstable when unattainable and with asymptotically stable when attainable. 𝑏2 + 𝛾 0 0 0 Proposition 7. Assume the following. [ ] [ −𝛾𝑝 𝑏2 + 𝛼 + 𝜏0 + 𝐾𝜉 0 0] (1) The endemic and the disease-free transition matrices 𝑉𝑒 = 𝑉 = [ [−𝛾 (1 − 𝑝) ] [ 0 𝑏2 + 𝜏0 0 ] ] (−𝑉𝑒 ) and (−𝑉) are identical and independent of the reproduc- tion number, and (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix. [ 0 − (𝑏2 + 𝛼) −𝑏2 𝜇] (2) The disease-free and endemic equilibrium points exist 0 𝑆𝑒𝑛𝑑 𝛽𝐴𝑟 𝑆𝑒𝑛𝑑 𝛽𝐷𝑟 𝑆𝑒𝑛𝑑 and they are unique for any 𝑅0 ∈ [0, +∞]. [ ] (3) The disease-free and the endemic transmission matrices [0 0 0 0 ] are nonnegative, i.e., 𝐹 = 𝐹(𝑅0 ) ⪰ 0, and 𝐹𝑒 = 𝐹𝑒 (𝑅0 ) = 𝐹𝑒 = 𝛽 [ [0 0 ]=𝐹 [ 0 0 ] ] 𝜍(𝑅0 )𝐹 ⪰ 0, where 𝑅0 = 𝑅0 (𝐹) = 𝜌(𝐹𝑉−1 ) is the reproduction [0 0 0 0 ] number, and 𝜍(𝑅0 ) ≥ 0. (19) (4) 𝐹 − 𝐹𝑒 = (𝜆 − 1)𝐹 ⪰ 0 for some 𝜆 ∈ R. 0 𝑆𝑑𝑓 − 𝑆𝑒𝑛𝑑 𝛽𝐴𝑟 (𝑆𝑑𝑓 − 𝑆𝑒𝑛𝑑 ) 𝛽𝐷𝑟 (𝑆𝑑𝑓 − 𝑆𝑒𝑛𝑑 ) (5) 𝐹 ≻ 𝐹𝑒 if 𝑅0 > 1, 𝐹 ≺ 𝐹𝑒 if 𝑅0 < 1, and 𝐹 = 𝐹𝑒 if 𝑅0 = 1. [ ] Then, the endemic equilibrium point is locally asymptoti- [0 0 0 0 ] − 𝛽[ [ ] ] cally stable if 𝑅0 ≥ 1 and unstable if 𝑅0 < 1. [0 0 0 0 ] [0 0 0 0 ] Proof. Note that 0 1 𝛽𝐴𝑟 𝛽𝐷𝑟 𝐹 − 𝑉 = (𝐼𝑛 − 𝐹𝑉−1 ) (−𝑉) (22) [ ] [0 0 0 0 ] = 𝐹 − 𝛽 (𝑅0 − 1) 𝑆𝑒𝑛𝑑 [ [0 0 0 ] Since (−𝑉) is a stability matrix, the disease-free equilibrium [ 0 ]] point is locally asymptotically stable if 𝑅0 = 𝜌(𝐹𝑉−1 ) < 1 and [0 0 0 0 ] unstable if 𝑅0 = 𝜌(𝐹𝑉−1 ) > 1, the critical stability case being with 𝑅0 = 𝜌(𝐹𝑉−1 ) = 𝑆𝑑𝑓 /𝑆𝑒𝑛𝑑 . Note that 𝐹𝑒 ⪰ 𝐹 if 𝑅0 ≤ 1; 𝑅0 = 𝜌(𝐹𝑉−1 ) = 1. Note also that 𝐹 ⪰ 𝐹𝑒 if 𝑅0 ≥ 1; and 𝐹 = 𝐹𝑒 if 𝑅0 = 1 (what implies 𝑥𝑒𝑛𝑑 = 𝐹𝑒 − 𝑉 = (𝐼𝑛 − 𝐹𝑒 𝑉−1 ) (−𝑉) 𝑥𝑑𝑓 ), and (23) = (𝐼𝑛 − 𝐹𝑉−1 + (𝐹 − 𝐹𝑒 ) 𝑉−1 ) (−𝑉) . 𝑓 (0, 𝑦𝑑𝑓 ) = 0; Since 𝐹 − 𝐹𝑒 = (𝜆 − 1)𝐹 ⪰ 0 with 𝜆 = 𝜆(𝑅0 ) ≥ 1 for 𝑅0 ≤ 1 − 𝑓 (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) since 𝐹 ⪰ 𝐹𝑒 , then 𝜌(𝐹 − 𝐹𝑒 ) = (𝜆 − 1)𝜌(𝐹) ⪰ 0 and both (𝛽𝐼𝑒𝑛𝑑 + 𝛽𝐴𝐴 𝑒𝑛𝑑 + 𝛽𝐷𝐷𝑒𝑛𝑑 ) 𝑆𝑒𝑛𝑑 of them are proportional matrices because of their structure, (20) [ ] it follows that the endemic equilibrium point is unstable for [ 0 ] 𝑅0 < 1 (when the disease-free is locally stable ) and locally =[ [ ]≥0 ] [ 0 ] stable for 𝑅0 > 1 (when the disease-free one is unstable). [ 0 ] Note that for Example 6, 𝜆(𝑅0 ) = 2 − 1/𝑅0 in the proof of Proposition 7. Note that assumption 4 of Proposition 7 irrespective that the infective variables be positive or negative is very reasonable if 𝐹 and 𝐹𝑒 are related by a scalar factor since 𝐼𝑒𝑛𝑑 , 𝐴 𝑒𝑛𝑑 , and 𝐷𝑒𝑛𝑑 have the same sign as 𝐸𝑒𝑛𝑑 , if 𝐸𝑒𝑛𝑑 ≠ which is the usual case in epidemic models. The subsequent 0, that is, if the endemic equilibrium point has not coincident examples show that the endemic equilibrium is unstable in components with the disease-free one, that is, if 𝑅0 ≠ 1, as it is the unattainable region. deduced from the epidemic model. Note from the matrix A𝑑𝑓 that if the linearized infective subsystem around the disease- Example 8. Consider the SEIADR epidemic model of free equilibrium is identically zero, then the characteristic Example 6 with the following parameterization: equation of the linearized noninfective subsystem is given by 𝑏1 = 𝑏2 =1/25550 𝑑𝑎𝑦𝑠−1 ; 1/𝜏0 = 15.8 𝑑𝑎𝑦𝑠; 𝜂 = 2𝜇 = 2 𝑝𝑛𝑖 (𝑠) = 𝑠 + (2𝑏2 + 𝐾𝑉 + 𝜂) 𝑠 + 𝑏2 (𝑏2 + 𝜂 + 𝐾𝑉) = 0 (21) 0.1 𝑑𝑎𝑦𝑠−1 ; whose roots are 𝑠1 = −𝑏2 < 0 and 𝑠2 = −(𝑏2 + 𝐾𝑉 + 𝜂) < 𝑝 = 0.15; 1/𝛾 = 15.8 𝑑𝑎𝑦𝑠; 1/𝛼 = 13.3 𝑑𝑎𝑦𝑠; 0. So, such a subsystem is asymptotically stable satisfying 𝑉0 = 𝑏1 ; 𝐾𝑉 = 20𝑏1 ; 𝐾𝜉 = 𝛼; Assumption 2(a). As a result, the whole linearized subsystem is asymptotically stable if 𝑅0 ≤ 1. 𝛽 = 0.11𝑑𝑎𝑦𝑠−1 ; 𝛽𝐴 = 0.05𝑑𝑎𝑦𝑠−1 ; 𝛽𝐷 = 0.15𝑑𝑎𝑦𝑠−1 . It has been proved in Theorem 4 that the disease-free Figure 1 shows the maximum of the real parts of the eigen- equilibrium point is stable if 𝜌(𝐹𝑉−1 ) ≤ 1, the endemic one is values of the Jacobian around the disease-free and endemic unattainable if 𝜌(𝐹𝑉−1 ) < 1, and the disease-free equilibrium equilibrium points versus the reproduction number. In the point is unstable if 𝜌(𝐹𝑉−1 ) > 1 while the endemic one is proposed example, they coincide with the maximum eigen- reachable if 𝜌(𝐹𝑉−1 ) ≥ 1. The next result establishes that the value since it is real. The reproduction number is displayed Discrete Dynamics in Nature and Society 9 ×10−4 14 12 10 8 6 4 2 0 −2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ２0 max[eig(＊？Ｈ＞ )] max[eig(＊＞＠？ )] zero level Figure 1: Maximum Eigenvalues of the Jacobians of the model of Example 6. 0.3 0.25 Re(max(eigenvalue(＊？Ｈ＞ )) 0.2 0.15 0.1 0.05 0 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 ２0 Figure 2: Maximum real part of the eigenvalues versus the reproduction number. in the real axis by keeping constant all the disease and unattainable region under nonnegativity conditions of the control model parameters except the coefficient transmission solution while the disease-free one is the unique asymptot- rates which are modified from their given basic values ically stable attractor if the reproduction number is less than proportionally via the same common direct proportionality one. However, if such a reproduction number exceeds unity, constant. Note that the reproduction number is proportional then the disease-free equilibrium point is unstable while the to the disease coefficient rate if all the remaining parameters endemic one is reachable and asymptotically stable. are kept constant. It is seen in Figure 1 that the value 𝑅0 = 1 is A global stability theorem now follows by combining the the critical reproduction number giving the frontier between preceding results and the analysis of Poincaré indices and the the stability (instability) of the disease-free equilibrium point alternate stability characteristics of limit cycles surrounding together with the unattainability and instability (attainability singular points in any hyperplanes of the state space. It is and stability) of the endemic one. proved, in particular, that the local asymptotic stability of the Example 9. Consider a true mass action (i.e., the nonlinear disease-free equilibrium point for a reproduction number less infective terms of susceptible-infected products are normal- than one (implying also the unattainability of the endemic ized with the total population) variant of the SEIADR epi- equilibrium point) leads to the global asymptotic stability of demic model of Examples 6 and 8 with the parameterization the whole state towards the disease-free equilibrium point. of Example 8. Figure 3 shows a zoom of Figure 2 centred in Note that Proposition 7 concludes that only one of the the stability boundary examined on the maximum eigenvalue equilibrium points is locally stable for each given value of evolution of the Jacobian of the endemic equilibrium point. 𝑅0 . In particular the disease-free one is locally asymptotically It is seen that the endemic equilibrium is unstable in its stable if 𝑅0 ≤ 1 and the endemic one is locally asymptotically 10 Discrete Dynamics in Nature and Society ×10−3 20 Re(max(eigenvalue(＊？Ｈ＞ )) 15 10 5 0 −5 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 ２0 Figure 3: Zoom of the maximum real part of the eigenvalues versus the reproduction number close to the stability region boundary. stable if 𝑅0 ≥ 1. Note also that Proposition 7 agrees with the conditions in the first closed orthant of R𝑛+𝑚 . Also, any such conclusions of Examples 6–9. an oscillation, if it exists, should surround the equilibrium point, since the net global Poincaré index of all the reachable Theorem 10. Under Assumptions 1-2, assume also that (−𝑉) ∈ equilibrium points is still +1, while it should be unstable and 𝑀𝐸𝑛×𝑛 , 𝑉−1 ≻ 0, 𝐹 ⪰ 0, and 𝜌(𝐹𝑉−1 ) < 1. Moreover, assume then asymptotically vanishing to the locally asymptotically 𝑇 that (1) 𝐽𝑔,𝑧 (0 𝑦𝑑𝑓 ) is nonsingular, stable disease-free equilibrium point, since such a unique (2) 𝑔𝑖 (𝑧(𝑡)) ≥ 0 if 𝑦𝑖 (𝑡) = 0 for any 𝑖 ∈ 𝑚 and 𝑡 ∈ R0+ , equilibrium point within the region defined by such a curve, (3) 𝑓𝑖 (𝑧(𝑡)) ≤ 𝑒𝑖𝑇 (𝐹 − 𝑉)𝑥(𝑡) if 𝑥𝑖 (𝑡) = 0 for any 𝑖 ∈ 𝑛 and is locally asymptotically stable. As a result, the disease-free 𝑡 ∈ R0+ (it suffices that 𝑓𝑖 (𝑧(𝑡)) ≤ 0 if 𝑥𝑖 (𝑡) = 0 for any 𝑖 ∈ 𝑛). equilibrium point is globally asymptotically stable for any (i) Then, the total population, i.e., the sum of all the sub- initial condition in the first closed orthant of R𝑛+𝑚 . Property populations, is uniformly bounded for all time. Furthermore, (i) has been proved. Now, assume that the total population is any trajectory solution is nonnegative and uniformly bounded bounded for 𝑅0 > 1 so that the disease-free equilibrium point for all time for any given finite initial condition 𝑥(0) ⪰ 0 and is unstable and the endemic one is attainable. Two cases can 𝑦(0) ⪰ 0 while it is globally convergent at an exponential occur. rate to the disease-free equilibrium point which is the unique Case a. The endemic equilibrium point is globally asymptot- reachable equilibrium point which is a globally asymptotically ically stable. Property (ii) follows directly. stable attractor. (ii) Assume that for 𝑅0 > 1 the total population, i.e., the Case b. The endemic equilibrium point is not globally sum of all the subpopulations, is uniformly bounded for all asymptotically stable. Since the total population is bounded time. Assume also that 𝑓(𝑥(𝑡), 𝑦(𝑡)) and 𝑔(𝑥(𝑡), 𝑦(𝑡)) are by hypothesis and all the subpopulations are nonnegative uniformly continuous in the first closed orthant of R𝑛+𝑚 . Then, for all time, all the subpopulations are bounded for all time all the trajectory solutions are bounded for all time while they and a bounded limit cycle, if it exists, should surround the converge asymptotically to the endemic equilibrium point at an endemic point to attract any trajectories in the first orthant exponential rate which is also the unique asymptotically stable of the state space. This consideration follows by examining attractor. the admissible Poincaré-type combinations of allowed sta- ble/instable combinations of configurations of singular points Proof. The nonnegativity of the solutions and the local and limit cycles. Rewrite compactly (1) to (3) in the form asymptotic stability of the disease-free equilibrium point, ̂ ̂ 𝑥(𝑡) ̇ 𝑧(𝑡) = 𝑓(𝑥(𝑡), 𝑦(𝑡)), where 𝑓(𝑥(𝑡), 𝑦(𝑡)) = [ 𝐹−𝑉 0 0 0 ] [ 𝑦(𝑡) ] + which is unique, follow directly from Theorem 1 (iii) and ∞ Theorem 4 (ii. (a)), since from Theorem 4 (ii. (b)), the [ −𝑓(𝑥(𝑡),𝑦(𝑡)) ̂ ]. Assume that ∫ 𝑓(𝑥(𝜏), 𝑦(𝜏))𝑑𝜏 = ±∞. Then, 𝑔(𝑥(𝑡),𝑦(𝑡)) 0 endemic equilibrium is not reachable if 𝜌(𝐹𝑉−1 ) < 1. 𝑧(𝑡) 󳨀→ ±∞ as 𝑡 󳨀→ ∞ which is not compatible Since the unique disease-free equilibrium point is locally with the nonnegativity of the solution trajectories within asymptotically stable, its Poincaré index [17] is +1 in any the ∞ first orthant and their boundenness. Therefore, −∞ < plane of the phase space containing the evolution of any ̂ ∫0 𝑓(𝑥(𝜏), ̂ 𝑦(𝜏))𝑑𝜏 < +∞. Since 𝑓(𝑥(⋅).𝑦(⋅)) is uniformly two components of the state trajectory solution. It is well- continuous, it follows from Barbalat’s lemma, [24], that known from the existence theorem of oscillations that any ̇ 𝑧(𝑡) 󳨀→ 0 as 𝑡 󳨀→ ∞ so that 𝑧(𝑡) cannot have a limit such a solution, if it exists, should be bounded. This is a oscillation in any of its components. So, a stable limit cycle consequence of the fact that the total population is bounded cannot surround the endemic equilibrium point if unstable for all time and that all the subpopulations are nonnegative or critically stable. Therefore, the endemic equilibrium point for all time what leads to the conclusion that any of them is globally asymptotically stable so that 𝑧(𝑡) 󳨀→ 𝑧𝑒𝑛𝑑 as 𝑡 󳨀→ is uniformly bounded for all time for any finite initial ∞. Property (ii) has been proved. Discrete Dynamics in Nature and Society 11 A very important remark is that the mathematical and 𝜃(̂𝑧(𝑡)) is the scalar positive bounded incidence rate, hypothesis of boundedness of the total population for and 𝐺(̂𝑧(𝑡)) = ∑𝑟𝑖=0 𝐹𝑖 (𝑧(𝑡 − ℎ𝑖 ))𝑧(𝑡 − ℎ𝑖 ) where 𝐺(̂𝑧(𝑡)) is the cases of reproduction numbers exceeding unity of a partitioned block matrix function of the form 𝐺(̂𝑧(𝑡)) = Theorem 10 (ii) is not strong, then fully feasibly, in practice. ∑𝑟𝑖=0 [𝐺𝑥𝑥 (𝑥(𝑡 − ℎ𝑖 )) 𝐺𝑥𝑦 (𝑦(𝑡 − ℎ𝑖 ))]. For commensurate point This consideration is based on the proved boundedness of delays, (24)-(26) remain valid under the additional constraint the total population in the case that the reproduction number ℎ𝑖 = 𝑖ℎ1 for 𝑖 ∈ 𝑟. It turns out that the solution is unique is less than unity implying the global asymptotic stability. In on [−ℎ, ∞) for any such initial conditions from the Cauchy- fact, it can be conjectured from simple inspection that, for Peano theorem. Note that, (24)- (26) can be rewritten as any given initial conditions in the first orthant and any given follows: time instant, the total population cannot exceed its value for 𝑟 the disease-free case at any time because thereof the disease 𝑥̇ (𝑡) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 − ℎ𝑖 ) − 𝑓 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) mortality rates. Even if there is no infection mortality both 𝑖=0 values would be coincident. 𝑟 = (𝐹0 − 𝑉0 ) 𝑥 (𝑡) + ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 − ℎ𝑖 ) 3. Models with Multiple Incommensurate 𝑖=1 (27) Point Time-Delays − 𝑓 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) A compartmental disease model of dimension (𝑚 + 𝑛) with 𝑟 𝑟 constant, in general incommensurate, internal point delays = (𝐹 − 𝑉) 𝑥 (𝑡) + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥 (𝑡 − ℎ𝑖 ) − 𝑥 (𝑡)) 𝑖=1 ℎ𝑖 (𝑖 ∈ 𝑟) satisfying 0 = ℎ0 ≤ ℎ1 ≤ ℎ2 ≤ ⋅ ⋅ ⋅ ≤ ℎ𝑟 = ℎ ≤ ℎ ≤ ∞, what is assumed in the sequel, can be generalized from − 𝑓 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) (1)-(2) as follows: 𝑟 𝑦̇ (𝑡) ≡ 𝑔 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) (28) 𝑥̇ (𝑡) = ∑ (𝐹𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 )) where 𝑖=0 𝑟 𝑟 𝑓 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 − ℎ𝑖 ) − 𝑉𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 ))) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 (24) 𝑖=0 𝑖=0 𝑟 − ℎ𝑖 ) − 𝑓 (𝑥 (𝑡) , 𝑦 (𝑡) , 𝑥 (𝑡 − ℎ1 ) , 𝑦 (𝑡 − ℎ1 ) , . . . , 𝑥 (𝑡 − ∑ (𝐹𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 )) 𝑖=0 − ℎ𝑟 ) , 𝑦 (𝑡 − ℎ𝑟 )) − 𝑉𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 ))) = (𝐹0 − 𝑉0 ) 𝑥 (𝑡) 𝑦̇ (𝑡) ≡ 𝑔 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) = 𝑔 (𝑥 (𝑡) , 𝑦 (𝑡) , 𝑥 (𝑡 − ℎ1 ) , 𝑦 (𝑡 𝑟 (25) − ℎ1 ) , . . . , 𝑥 (𝑡 − ℎ𝑟 ) , 𝑦 (𝑡 − ℎ𝑟 )) + ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 − ℎ𝑖 ) 𝑖=1 subject to any absolutely continuous function of initial con- 𝑟 ditions 𝜑 : [−ℎ, 0] 󳨀→ R𝑛+𝑚 with eventual finite jumps on a (29) − ∑ (𝐹𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 )) subset of [−ℎ, 0] of zero measure with 𝜑(0) = (𝑥𝑇 (0), 𝑦𝑇 (0))𝑇 . 𝑖=0 Any member of such a class of functions is referred to in the sequel as an admissible function of initial conditions, − 𝑉𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 ))) = (𝐹0 − 𝑉0 ) 𝑥 (𝑡) where 𝑛 and 𝑚 are the respective dimensions of the disease 𝑟 and nondisease compartments of respective state vectors 𝑥(𝑡) + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥 (𝑡 − ℎ𝑖 ) − 𝑥 (𝑡)) and 𝑦(𝑡), and the nonlinear effects of (3) are replaced by the 𝑖=1 subsequent delayed counterpart dynamics: 𝑟 − ∑ (𝐹𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 )) 𝑓 (𝑥̂ (𝑡) , 𝑦̂ (𝑡)) ≡ 𝑓 (𝑥 (𝑡) , 𝑦 (𝑡) , 𝑥 (𝑡 − ℎ1 ) , 𝑦 (𝑡 − ℎ1 ) , 𝑖=0 𝑟 . . . , 𝑥 (𝑡 − ℎ𝑟 ) , 𝑦 (𝑡 − ℎ𝑟 )) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥 (𝑡 − ℎ𝑖 ) − 𝑉𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 ))) 𝑖=0 with (26) 𝑟 𝜕𝐹𝑗𝑖 (0, 𝑦𝑑𝑓 ) − ∑ (𝐹𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 )) 𝐹𝑗 = [ ]; 𝑖=0 𝜕𝑥𝑗 (𝑡 − ℎ𝑗 ) − 𝑉𝑖 (𝑥 (𝑡 − ℎ𝑖 ) , 𝑦 (𝑡 − ℎ𝑖 ))) = 𝜃 (̂𝑧 (𝑡)) 𝐺 (̂𝑧 (𝑡)) 𝜕𝑉𝑗𝑖 (0, 𝑦𝑑𝑓 ) ̂ 𝑦(𝑡)) where 𝑓(𝑥(𝑡), ̂ describes the dynamics of higher-order 𝑉𝑗 = [ ]; 𝜕𝑥𝑗 (𝑡 − ℎ𝑗 ) contributions to the dynamics, 𝑧̂(𝑡) = (𝑥̂𝑇 (𝑡), 𝑦̂𝑇 (𝑡)) = (𝑧𝑇 (𝑡), 𝑧𝑇 (𝑡 − ℎ1 ), . . . , 𝑧𝑇 (𝑡 − ℎ𝑟 )), where 𝑧(𝑡) = (𝑦𝑇 (𝑡), 𝑥𝑇 (𝑡))𝑇 𝑗 ∈ 𝑟 ∪ {0} 12 Discrete Dynamics in Nature and Society 𝜕𝐹𝑒𝑗𝑖 (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) general differential epidemic model subject to time-delays 𝐹𝑒𝑗 = [ ]; becomes: 𝜕𝑥𝑗 (𝑡 − ℎ𝑗 ) 𝑟 𝜕𝑉𝑒𝑗𝑖 (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) 𝑆 ̇ (𝑡) = −𝛽 ∑ 𝑆 (𝑡 − ℎ𝑖 ) 𝐼 (𝑡 − ℎ𝑖 ) ; 𝑉𝑒𝑗 = [ ]; 𝑖=0 𝜕𝑥𝑗 (𝑡 − ℎ𝑗 ) 𝑟 𝑟 (34) 𝐼 ̇ (𝑡) = 𝛽 ∑ 𝑆 (𝑡 − ℎ𝑖 ) 𝐼 (𝑡 − ℎ𝑖 ) − 𝛾 ∑ 𝐼 (𝑡 − ℎ𝑖 ) 𝑗 ∈ 𝑟 ∪ {0} , 𝑖=0 𝑖=0 (30) which corresponds to that in the integral form: 𝑟 ∞ respectively, at the disease-free equilibrium point 𝑧𝑑𝑓 = ̃ (𝜏) 𝑆 ̇ (𝑡 − ℎ𝑖 − 𝜏) 𝑑𝜏 𝑆 ̇ (𝑡) = ∑ 𝑆 (𝑡 − ℎ𝑖 ) ∫ 𝐴 (35) (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 𝑇 ) and at the endemic one 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 𝑇 𝑇 , 𝑦𝑒𝑛𝑑 ) and 𝑖=0 0 the redefinitions: The physical interpretation relies on the arrival of several infectious groups at different time instants or to the combined 𝑟 contributions of several infective strains at different time 𝐹 = ∑ 𝐹𝑖 , 𝑖=0 instants both with the same coefficient rates. (31) Sufficient conditions for the stability of the linearized 𝑟 system independent of the delay sizes around the disease-free 𝑉 = ∑ 𝑉𝑖 equilibrium point can be obtained for the cases when either 𝑖=0 𝑉0 or 𝑉 is a stability matrix. For this purpose, we first write the equations of the linearized system in two ways depending The extended versions of Assumptions 1 and 2 are kept for the on the nonsingularity of 𝑉0 or that of 𝑉as follows. delayed epidemic models as follows. Lemma 12. The following properties hold. Assumptions 4. ∑𝑟𝑗=0 𝐹𝑖𝑗 (0, 𝑦(𝑡 − ℎ𝑗 )) = ∑𝑟𝑗=0 𝑉𝑖𝑗 (0, 𝑦(𝑡 − ℎ𝑗 )) = (i) The infective linearized subsystem obtained from (24)- 0; ∀𝑖 ∈ 𝑛 = {1, 2, . . . , 𝑛}, 𝐹𝑖𝑗 (𝑥(𝑡 − ℎ𝑗 ), 𝑦(𝑡 − ℎ𝑗 )) ≥ 0, (25) around the disease-free equilibrium point is given by the 𝑉𝑖𝑗 (𝑥(𝑡 − ℎ𝑗 ), 𝑦(𝑡 − ℎ𝑗 )) ≤ 0 when 𝑥𝑖 (𝑡 − ℎ𝑗 ) = 0, and subsequent dynamics: ∑𝑛𝑖=1 ∑𝑟𝑗=0 𝑉𝑖𝑗 (𝑥(𝑡), 𝑦(𝑡)) ≥ 0 for all 𝑥(𝑡), 𝑦(𝑡) ⪰ 0. 𝑟 𝑥𝐿̇ (𝑡) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝑡 − ℎ𝑖 ) (36) Assumptions 5. (a) (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 , 𝑉0−1 ≻ 0, 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟. 𝑖=0 (b) The disease-free equilibrium point 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) is for given initial conditions 𝑥𝐿 (𝑡) = 𝑥(𝑡) = 𝜑(𝑡); ∀𝑡 ∈ [−ℎ, 0], unique and the noninfective subsystem 𝑦(𝑡) ̇ = 𝑔(0, 𝑦(𝑡)) is where 𝜑 : [−ℎ, 0] 󳨀→ R𝑛+𝑚 is an absolutely continuous locally asymptotically stable. function with eventual finite jumps on a subset of [−ℎ, 0] of zero measure with 𝜑(0) = (𝑥𝑇 (0), 𝑦𝑇 (0))𝑇 . Example 11. A simple motivation of the model (24)-(26) (ii) If 𝑉0 is nonsingular then (36) can be equivalently subject to delays covers and generalizes classical models expressed as in the literature which have a physical interpretation. A well-known classical Kermack-McKendrick-based epidemic 𝑥𝐿̇ (𝑡) = −𝑉0 (𝐼𝑛 − 𝑉0−1 𝐹0 ) 𝑥𝐿 (𝑡) model in integral form, revisited in [27], is the subsequent 𝑟 one: + ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝑡 − ℎ𝑖 ) ∞ 𝑖=1 ̃ (𝜏) 𝑆 ̇ (𝑡 − 𝜏) 𝑑𝜏 𝑆 ̇ (𝑡) = 𝑆 (𝑡) ∫ 𝐴 (32) 𝑟 (37) 0 = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝑡) 𝑖=0 where 𝑆(𝑡) denotes the spatial density of susceptibles, i.e., the number of individuals per area at time 𝑡 where 𝐴(𝜏) ̃ 𝑟 is the expected infectivity of an individual which become + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥𝐿 (𝑡 − ℎ𝑖 ) − 𝑥𝐿 (𝑡)) 𝑖=1 infected 𝜏 unity of times ago. That model can be interpreted in differential form as the SI- epidemic model: (iii) If 𝑉 is nonsingular then (36) can be equivalently expressed as 𝑆 ̇ (𝑡) = −𝛽𝑆 (𝑡) 𝐼 (𝑡) ; 𝑥𝐿̇ (𝑡) (33) 𝐼 ̇ (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡) − 𝛾𝐼 (𝑡) 𝑟 𝑟 −1 𝑟 = − (∑ 𝑉𝑖 ) (𝐼𝑛 − (∑ 𝑉𝑖 ) (∑ (𝐹𝑖 ))) 𝑥𝐿 (𝑡) 𝑡 𝑖=0 𝑖=0 𝑖=0 by defining 𝐼(𝑡) = −𝛽−1 ∫−∞ 𝐴(𝑡̃ − 𝜏)𝑆(𝜏)𝑑𝜏, ̇ subject to the 𝑟 ̃ −𝛾𝜏 special infectivity 𝐴(𝜏) = 𝛽𝑒 , where 𝛽 and 𝛾 are the + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥𝐿 (𝑡 − ℎ𝑖 ) − 𝑥𝐿 (𝑡)) coefficient transmission rate and the infection rate. A more 𝑖=1 Discrete Dynamics in Nature and Society 13 𝑟 (iii) The linearized infective substate around the disease-free = (∑ (𝐹𝑖 − 𝑉𝑖 )) 𝑥𝐿 (𝑡) equilibrium point is positive and globally exponentially stable 𝑖=0 independent of the delays for any admissible function of initial 𝑟 conditions 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0] if and only if the following + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥𝐿 (𝑡 − ℎ𝑖 ) − 𝑥𝐿 (𝑡)) conditions jointly hold: ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) is a stability matrix, 𝑖=1 (𝐹0 − 𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 (equivalently, (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 and 𝐹0 ⪰ 0), (38) 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟, and the matrix (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 )𝑒−𝑠ℎ𝑖 ) has no (iv) If 𝑉0 and 𝑉 are nonsingular then the linearized system nonzero imaginary eigenvalues. around the disease-free equilibrium point is described indis- (iv) The whole linearized model around the unique tinctly by both (37) and (38) and also by disease-free equilibrium point has a nonnegative state 𝑟 trajectory solution for any given admissible nonnegative 𝑥𝐿̇ (𝑡) = −𝑉0 (𝐼𝑛 + 𝑉0−1 (∑ 𝑉𝑖 )) function of initial conditions which converges asymptotically 𝑖=1 to the unique disease-free equilibrium point. 𝑟 −1 𝑟 Proof. Note that the solution of the linearized infective ⋅ (𝐼𝑛 − (𝑉0 (𝐼𝑛 + 𝑉0−1 (∑ 𝑉𝑖 )) (∑ 𝐹𝑖 ))) (39) substate around the disease-free equilibrium point is 𝑖=1 𝑖=0 𝑟 𝑥𝐿 (𝑡) = 𝑒(𝐹0 −𝑉0 )𝑡 𝑥𝐿 (0) ⋅ 𝑥𝐿 (𝑡) + ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑥𝐿 (𝑡 − ℎ𝑖 ) − 𝑥𝐿 (𝑡)) 𝑟 𝑡 𝑖=1 + ∑ ∫ 𝑒(𝐹0 −𝑉0 )(𝑡−𝜏) (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝜏 − ℎ𝑖 ) 𝑑𝜏 Proof. Eq. (36) is a linearized version of (24) around the 𝑖=1 0 disease-free equilibrium point. Eqs. (37) and (38) are direct = 𝑒(𝐹0 −𝑉0 )𝑡 𝑥𝐿 (0) from (36) if 𝑉0 and 𝑉 are, respectively, nonsingular. Further- more, if both matrices are nonsingular then 𝑉−1 = (𝐼𝑛 + 𝑟 𝑡−ℎ𝑖 𝑉0−1 (∑𝑟𝑖=1 𝑉𝑖 ))−1 𝑉0−1 and (𝐼𝑛 +𝑉0−1 (∑𝑟𝑖=1 𝑉𝑖 ))−1 exists what leads + ∑∫ 𝑒(𝐹0 −𝑉0 )(𝑡−ℎ𝑖 −𝜏) (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝜏) 𝑑𝜏 𝑖=1 −ℎ𝑖 (41) to (39). Some basic results, given in Theorem 1 on positivity and = 𝑒(𝐹0 −𝑉0 )𝑡 𝑥𝐿 (0) stability for the delay-free linearized infective subsystem 𝑟 0 around the disease-free equilibrium point, for the case of + ∑∫ 𝑒(𝐹0 −𝑉0 )(𝑡−ℎ𝑖 −𝜏) (𝐹𝑖 − 𝑉𝑖 ) 𝜑 (𝜏) 𝑑𝜏 multiple point delays follows. 𝑖=1 −ℎ𝑖 Theorem 13. The following properties hold under Assump- 𝑟 𝑡−ℎ𝑖 tions 4-5. + ∑∫ 𝑒(𝐹0 −𝑉0 )(𝑡−ℎ𝑖 −𝜏) (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝜏) 𝑑𝜏 (i) The linearized epidemic model (24)-(25) around the 𝑖=1 0 disease-free equilibrium point has a nonnegative solution of the The sufficiency part is direct from (41) since if (𝐹0 − 𝑉0 ) ∈ infective substate independent of the delays for any bounded 𝑀𝐸𝑛×𝑛 and 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟 then 𝑥𝐿 (𝑡) ⪰ 0 for any 𝑡 > 0 admissible function of initial conditions 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0] and any given 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0]. The necessity is if and only if (𝐹0 −𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 (equivalently, (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 and proved by contradiction in two separate steps. Assume that 𝐹0 ⪰ 0) and 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟. These conditions are also jointly 𝐴 0 = (𝐴 0𝑖𝑗 ) = (𝐹0 − 𝑉0 ) ∉ 𝑀𝐸𝑛×𝑛 . Then, there is some pair necessary for the epidemic model (24)-(25) to have a non- (𝑖, 𝑘(≠ 𝑖)) ∈ 𝑛 × 𝑛 such that 𝐴 0𝑖𝑘 < 0. Now, take a function of negative solution trajectory independent of the delays for any initial conditions satisfying 𝜑𝑘 (0) > 0, 𝜑𝑖 (0) = 0; ∀𝑖(≠ 𝑘) ∈ 𝑛, admissible function of initial conditions 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0]. and 𝜑(𝑡) = 0 for 𝑡 ∈ [−ℎ, 0). Thus, 𝑥𝐿̇ 𝑖 (0) = −|𝐴 0𝑖𝑘 |𝜑𝑘 (0) < 0 (ii) The linearized infective substate around the disease- free equilibrium point has a nonnegative solution trajectory and 𝑥𝐿 𝑖 (0) = 0 so that 𝑥𝐿 𝑖 (0+ ) < 0 and the system is not and it is globally exponentially stable independent of the positive. On the other hand, assume that there is some 𝑖 ∈ 𝑟 delays for any admissible function of initial conditions such that 𝑉𝑖 ≻ 𝐹𝑖 so that there is some pair (𝑗, 𝑘) ∈ 𝑛 × 𝑛 such 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0] if the following conditions jointly hold: that 𝑉𝑖𝑗𝑘 > 𝐹𝑖𝑗𝑘 . Take initial conditions as follows: 𝜑(𝑡) = 0 (𝐹0 − 𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix (equivalently, it exists for 𝑡 ∈ [−ℎ, −ℎ𝑖 ) ∪ (−ℎ𝑖 , 0], 𝜑ℓ (−ℎ𝑖 ) = 0; ∀𝑘, ℓ(≠ 𝑘) ∈ 𝑛 (𝑉0 − 𝐹0 )−1 ≻ 0; equivalently (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 , it exists 𝑉0−1 ≻ 0, and 𝜑𝑘 (−ℎ𝑖 ) > 0. Then, 𝑥𝐿̇ 𝑗 (0) = −|𝐹𝑖𝑗𝑘 − 𝑉𝑖𝑗𝑘 |𝜑𝑘 (−ℎ𝑖 ) < 0 𝐹0 ⪰ 0, and 𝜌0∞ = 𝜌(𝐹0 𝑉0−1 ) < 1), 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟, and and 𝑥𝐿 𝑗 (0) = 0 so that 𝑥𝐿 𝑗 (0+ ) < 0 and the system is not 󵄩󵄩󵄩 𝑟 󵄩󵄩󵄩 positive. Therefore, 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟 and (𝐹0 − 𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 󵄩 −1 󵄩 𝜌0∞ = 󵄩󵄩󵄩(𝑠𝐼𝑛 + 𝑉0 − 𝐹0 ) (∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑒−𝑠ℎ𝑖 )󵄩󵄩󵄩 are necessary for positivity. The first part of Property (i) for 󵄩󵄩 󵄩󵄩 󵄩 𝑖=1 󵄩∞ the linearized system around the disease-free equilibrium 󵄩󵄩 󵄩󵄩 point has been proved. The fact that the conditions are also 󵄩󵄩 𝑟 󵄩󵄩 (40) −1 necessary for the positivity of the whole nonlinear model = sup 󵄩󵄩(i𝜔𝐼𝑛 + 𝑉0 − 𝐹0 ) (∑ (𝐹𝑖 − 𝑉𝑖 ))󵄩󵄩󵄩 󵄩 𝜔∈R0+ 󵄩 󵄩󵄩 𝑖=1 󵄩󵄩 󵄩2 are obvious in the guidelines of Theorem 1 (i) for the delay- free case since the positivity of the whole model requires that <1 of its linearized version around the disease-free equilibrium 14 Discrete Dynamics in Nature and Society point. This proves the second part of Property (i). Property ∀𝑖 ∈ 𝑟 for any bounded admissible function of initial conditions (ii) contains conditions to guarantee the positivity (borrowed 𝜑(𝑡) ⪰ 0; ∀𝑡 ∈ [−ℎ, 0] if and only if ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) is a stability from Property (i)) and, furthermore, the global asymptotic matrix, (𝐹0 − 𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 , 𝐹𝑖 ⪰ 𝑉𝑖 ; ∀𝑖 ∈ 𝑟, and stability is guaranteed as follows. Since the transfer matrix 𝑟 (𝑠𝐼𝑛 + 𝑉0 − 𝐹0 )−1 is stable, since (𝐹0 − 𝑉0 ) is a stability matrix, det (𝜄𝜔𝐼𝑛 + ∑ (𝑉𝑖 − 𝐹𝑖 ) 𝑧𝑖 ) ≠ 0; that is det(𝑠𝐼𝑛 + 𝑉0 − 𝐹0 ) ≠ 0 for Re 𝑠 ≥ 0, then condition (40) (43) 𝑖=0 is equivalent to ∀𝜔 ∈ R0+ , ∀𝑧 ∈ 𝐶 (0, 1) . 𝑟 det (𝜄𝜔𝐼𝑛 + ∑ (𝑉𝑖 − 𝐹𝑖 ) 𝑒−𝜄𝜔ℎ𝑖 ) = det (𝜄𝜔𝐼𝑛 + 𝑉0 If the set 𝑆 = {(𝜔, 𝑧) ∈ R0+ × 𝜕𝐶(0, 1)} ≠ ⌀ then the 𝑖=0 infective substate around the disease-free equilibrium point is not exponentially stable for the sets of commensurate delays − 𝐹0 ) det (𝐼𝑛 𝑆𝐻 = {ℎ𝑖 = − ln 𝑧/𝜔 : (𝜔, 𝑧) ∈ 𝑆}. (42) Taking into account the expression (36) to describe the 𝑟 −1 −𝜄𝜔ℎ𝑖 linearized infective dynamics around the disease-free equi- − (𝜄𝜔𝐼𝑛 + (𝑉0 − 𝐹0 ) ) (∑ (𝑉𝑖 − 𝐹𝑖 ) 𝑒 )) librium point, it is possible to give an alternative sufficiency- 𝑖=0 type condition for global stability of the infective subsystem ≠ 0; ∀𝜔 ∈ R0+ . as the subsequent result reflects. Thus, constraint (42) is a sufficient condition for global Corollary 15. The linearized infective substate around the asymptotic stability. Since the linearized system is time- disease-free equilibrium point is exponentially stable indepen- invariant and globally asymptotically stable and since time- dent of the set of 𝑟 incommensurate delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟 if invariant time-delay systems with constant point delays can 󵄩󵄩 𝑟 󵄩󵄩 also have a finite number of characteristic zeros to the right of 󵄩󵄩 󵄩󵄩 𝜌00 = 2 󵄩󵄩∑ (𝐹𝑖 − 𝑉𝑖 )󵄩󵄩󵄩 󵄩 any vertical line in the complex plane of abscissa greater than 󵄩󵄩𝑖=1 󵄩󵄩 󵄩 󵄩2 −∞, then the solution is a function of time of exponential (44) 󵄩󵄩 −1 󵄩󵄩󵄩 negative order so that one concludes that the proved global 󵄩󵄩 𝑟 󵄩 asymptotic stability is also exponential. Property (ii) has been ⋅ sup 󵄩󵄩(i𝜔𝐼𝑛 − (∑ (𝐹𝑖 − 𝑉𝑖 ))) 󵄩󵄩󵄩󵄩 < 1 󵄩 󵄩 𝜔∈R+ 󵄩 󵄩 󵄩󵄩 proved. On the other hand, note that if the jointly condition 󵄩 𝑖=1 󵄩2 that ∑𝑟𝑖=0 (𝐹𝑖 −𝑉𝑖 ) is a stability matrix and the matrix (∑𝑟𝑖=0 (𝐹𝑖 − provided that ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) is a stability matrix, which holds if 𝑉𝑖 )𝑒−𝑠ℎ𝑖 ) has no nonzero imaginary eigenvalues holds then the global asymptotic stability at exponential rate is also guaran- and only if 𝜌00 = 𝜌[(∑𝑟𝑖=0 𝐹𝑖 )(∑𝑟𝑖=0 𝑉𝑖 )−1 ] < 1. teed since (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 )𝑒−𝑠ℎ𝑖 ) has no imaginary eigenvalues Proof. One gets the characteristic equation from (37) as and it is a stability matrix in the delay-free case. Such a jointly follows: condition is also necessary [28] for global asymptotic stability from the continuity of the characteristic zeros with respect to 𝑟 𝑟 the parameters. Since the linearized system is time-invariant det (𝑠𝐼𝑛 − (∑ (𝐹𝑖 − 𝑉𝑖 )) − ∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑒−ℎ𝑖 𝑠 − 1)) and globally asymptotically stable and since time-invariant 𝑖=0 𝑖=1 (45) time-delay systems with constant point delays can also have = 0 for 𝑠 ∈ C a finite number of characteristic zeros to the right of any vertical line in the complex plane of abscissa greater than Since (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 )) is a stability matrix then det(𝑠𝐼𝑛 − −∞, then the solution is a function of time of exponential (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ))) ≠ 0 for 𝑠 ∈ C0+ so that negative order so that one concludes that the proved global asymptotic stability is also exponential. Property (iii) has 𝑟 been proved. Since Assumptions 4-5 hold, then both the det (𝑠𝐼𝑛 − (∑ (𝐹𝑖 − 𝑉𝑖 ))) det (𝐼𝑛 linearized infective substate and the noninfective one around 𝑖=0 the unique disease-free equilibrium point have a nonnegative −1 𝑟 state trajectory solution which converges exponentially to − (𝑠𝐼𝑛 − (∑ (𝐹𝑖 − 𝑉𝑖 ))) (46) the disease-free equilibrium point for any given admissible 𝑖=0 nonnegative function of initial conditions. Hence, Property (iv) is proved. 𝑟 ⋅ (∑ (𝐹𝑖 − 𝑉𝑖 ) (𝑒−ℎ𝑖 𝑠 − 1))) ≠ 0; ∀𝑠 ∈ C0+ Taking into account (42) in the proof of Theorem 13, one 𝑖=1 gets the subsequent result. provided that ‖(𝑠𝐼𝑛 − (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 )))−1 (∑𝑟𝑖=1 (𝐹𝑖 − 𝑉𝑖 )(𝑒−ℎ𝑖 𝑠 − Corollary 14. The linearized infective substate around the 1))‖∞ < 1 which is guaranteed if 𝜌00 < 1. Since ∑𝑟𝑖=0 (𝐹𝑖 −𝑉𝑖 ) is disease-free equilibrium point is positive and exponentially a stability matrix, the above test is fulfilled at the origin of the stable independent of the set of 𝑟 commensurate delays ℎ𝑖 = 𝑖ℎ1 ; complex plane. Note also that ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) = ∑𝑟𝑖=0 (−𝑉𝑖 )[𝐼𝑛 − Discrete Dynamics in Nature and Society 15 ∑𝑟𝑖=0 (𝑉𝑖 )−1 ∑𝑟𝑖=0 𝐹𝑖 ] is a stability matrix if and only if 𝜌00 < 1 where 𝑅00 = max(𝜌00 , 𝜌00 ), 𝑅0∞ = max(𝜌0∞ , 𝜌0∞ ). Note since ∑𝑟𝑖=0 (−𝑉𝑖 ) is a stability matrix. Therefore, a sufficient that 𝜌00 and 𝜌0∞ are the respective reproduction numbers of condition to achieve the global (exponential) asymptotic the AILDFS and the AILDDFS which are delay-free auxiliary stability test independent of the delays is constraint (44) since linearized systems and the amounts 𝜌00 and 𝜌0∞ express sup𝜔∈R0+ |𝑒−iℎ𝑖 𝜔 − 1| ≤ 2; ∀𝑖 ∈ 𝑟. the tolerance to stability independent of the delays of the linearized systems around the disease-free equilibrium point It follows from Corollary 15 and the “If Part” of compatible with guaranteeing the stability independent of the Theorem 13 (iii) that since ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) is a stability matrix, delay sizes provided that those amounts are less than unity. constraint (44) guarantees that (∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 )𝑒−𝑠ℎ𝑖 ) has no Define 𝑅0 = min(𝑅00 , 𝑅0∞ ) and note that 𝑅𝑚0 ≤ 𝑅0 (0, ℎ1 , nonzero imaginary eigenvalues. Otherwise, the infective part ℎ2 , ..., ℎ𝑟 ) ≤ 𝑅0 ≤ 𝑅𝑀0 ; ∀ℎ𝑖 ∈ R0+ , ∀𝑖 ∈ 𝑟. of the linearized system around the disease-free equilibrium Thus, the asymptotic stability independent of the delays point could not be globally asymptotically stable. of the linearized system around the disease-free equilibrium point holds if 𝑅𝑀0 ≤ 1 and it is then guaranteed if 𝑅0 ≤ 1. Remark 16. The infective linearized system around the However, the above constraint has only a partial information disease-free equilibrium point has two important particular about the stability and attainability of the endemic equilib- cases which define auxiliary linearized systems which are rium point for each particular combination of delays. For, useful for the local stability analysis independent of the instance, we can intuitively discuss the following cases. delays, namely, (1) For a set of delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟, it holds that 𝑅0 (0, ℎ1 , ℎ2 , (a) auxiliary infective linearized delay-free system . . . , ℎ𝑟 ) ≤ 1 < 𝑅0 ≤ 𝑅𝑀0 . We get the conclusion that the (AILDFS) describing the particular case ℎ𝑖 = 0; ∀𝑖 ∈ 𝑟: disease-free equilibrium point is locally asymptotically stable 𝑟 and the endemic is unattainable for this value of 𝑅0 (⋅). Since 𝑥𝐿̇ (𝑡) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝑡) (47) 𝑅𝑀0 ≥ 𝑅0 > 1, we can say that there are combinations 𝑖=0 of delays, all of them with the same amounts of delayed 𝑛 dynamics, for which the disease is endemic but it also holds for 𝑡 > 0 for admissible initial conditions 𝜑 : [−ℎ, 0] 󳨀→ R0+ that there exist some particular delays for which the disease with 𝜑(0) = 𝑥(0) = 𝑥0 and ℎ = ℎ𝑟 ; asymptotically extinguishes. On the other hand, since 𝑅𝑀0 > (b) auxiliary infective linearized delayed dynamics-free system (AILDDFS) describing the particular case 𝐹𝑖 − 𝑉𝑖 = 0; 1, there are other sets of delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟 for the same matrices ∀𝑖 ∈ 𝑟: of dynamics such that the disease-free equilibrium point is unstable and the endemic one is stable and attainable. 𝑥𝐿̇ (𝑡) = (𝐹0 − 𝑉0 ) 𝑥𝐿 (𝑡) (48) (2) For a set of delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟, it holds that 𝑅0 (0, ℎ1 , ℎ2 , 𝑛 . . . , ℎ𝑟 ) ≤ 𝑅0 < 1 < 𝑅𝑀0 . We get the conclusion that for 𝑡 > 0 for admissible initial conditions 𝜑 : [−ℎ, 0] 󳨀→ R0+ the disease-free equilibrium point is locally asymptotically with 𝜑(0) = 𝑥(0) = 𝑥0 and a given maximum delay ℎ = ℎ𝑟 ≤ stable and the endemic one is unattainable. However, the +∞. This particular system can also be interpreted for the reproduction number range for which this holds exceeds the case of point initial conditions 𝜑(𝑡) = 0 for 𝑡 ∈ [−∞, 0) and lower-bound given by 𝑅0 . Furthermore, there are other sets 𝜑(0) = 𝑥(0) = 𝑥0 as defined under any amounts of delayed of delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟, under the same amounts of delayed dynamics associated with infinity delays ℎ𝑖 = +∞; ∀𝑖 ∈ 𝑟. dynamics, such that the disease-free equilibrium point is Note that the disease reproduction number 𝑅0 (0, ℎ1 , unstable and the endemic one is stable and attainable. ℎ2 , ..., ℎ𝑟 ) is, in general, delay-dependent with particular (3) In the inequalities of the above cases (a) and (b), if 𝑅0 values 𝑅00 = 𝑅0 (0), with 0 ∈ R𝑛+1 , which is the reproduction is replaced with 𝑅00 or 𝑅0∞ , we get similar conclusions. number of the AILDFS, and 𝑅0∞ = 𝑅0 (0, inf), where inf ∈ R𝑛 has all components equal to +∞, which is the repro- The following three results discuss the stability and duction number of the AILDDFS. Define the maximum positivity independent of the delays of the linearized system and minimum reproduction numbers for any given set of around the disease-free equilibrium point if either 𝑅00 (i.e., incommensurate delays: the reproduction number of the AILDFS) or 𝑅0∞ (i.e., the reproduction number of the AILDDFS) or both amounts are 𝑅𝑀0 = max 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ; 0≤ℎ𝑖 ≤∞ less than unity. In the first case, the complementary sufficient- (49) type stability condition (44) of worst-case tolerance against 𝑅𝑚0 = min 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) delayed dynamics is also invoked. In the second case the 0≤ℎ𝑖 ≤∞ complementary worst-case tolerance condition (40) replaces Note also that to (44). 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ∈ [𝑅𝑚0 , 𝑅𝑀0 ] ; Corollary 17. If Assumption 5 holds then the following prop- erties hold. ∀ℎ𝑖 ∈ R0+ , ∀𝑖 ∈ 𝑟, (50) (i) If constraint (44) holds, then there exists a real number 𝑅00 ∈ [𝑅𝑚0 , 𝑅𝑀0 ] , 𝜀 = 𝜀(ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ∈ R0+ dependent, in general, on the delays ℎ𝑖 ≥ 0 for 𝑖 ∈ 𝑟, such that the reproduction number 𝑅0∞ ∈ [𝑅𝑚0 , 𝑅𝑀0 ] . is given by 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) = (1 + 𝜀)𝑅00 − 𝜀, so that 16 Discrete Dynamics in Nature and Society (𝑅00 ≤ 1) 󳨐⇒ (𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ≤ 1), and the implication of initial conditions, if and only if 𝜌0∞ < 1. If 𝜌0∞ < 1 then the also holds under strict inequalities, for any given set of delays ℎ𝑖 ; endemic equilibrium point of the AILDDFS is unattainable. If ∀𝑖 ∈ 𝑟, and then 𝑅𝑚0 ≤ 𝑅𝑀0 ≤ 1. Then, the infective substate of 𝜌0∞ = 1 then the disease-free and the endemic equilibrium the linearized system around the disease-free equilibrium point points coincide. If 𝜌0∞ > 1 then the disease-free equilibrium is globally asymptotically stable independent of the delays, with point is unstable. a nonnegative state trajectory solution for any given admissible Proof. It follows under a similar reasoning as that in the proof nonnegative function of initial conditions, if 𝑅00 < 1. of Corollary 17 by using Lemma 12 (iii). (ii) The AILDFS is globally asymptotically stable around the disease-free equilibrium point, with a nonnegative state tra- Corollary 19. If Assumption 5 holds, 𝑉 is nonsingular and jectory solution for any given admissible nonnegative function (40) and (44) jointly hold, then 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) = (1 + of initial conditions, if and only if 𝜌00 ≤ 1. If 𝜌00 < 1 then, 𝜀2 )𝜌 − 𝜀2 for any given set of delays ℎ𝑖 ≥ 0 for 𝑖 ∈ 𝑟 and some furthermore, the endemic equilibrium point is unattainable. 𝜀2 = 𝜀2 (ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ∈ R0+ , where 𝑅0 = max(𝑅00 , 𝑅0∞ ) < 1, If 𝜌00 = 1 then the disease-free and the endemic equilibrium and then the infective substate of the linearized system around points coincide. If 𝜌00 > 1 then the disease-free equilibrium the disease-free equilibrium point has a globally asymptotically point is unstable. stable nonnegative state trajectory solution independent of the delays for any given nonnegative admissible function of initial Proof. The positivity and stability are guaranteed for the lin- conditions. earized infective substate from Theorem 13 (ii) and Lemma 12 (ii). Now assume that 𝜌00 ≤ 1 and (44) holds, that is, if 𝜌00 < 1, Proof. It follows under a similar reasoning as that in the proof both combined guarantee that 𝑅00 < 1. Note that the critical of Corollary 17 by taking into account Corollary 18. stability of the infective linearized system around the disease- free equilibrium point is given by 𝜌00 = 1. It is now proved Remark 20. If Assumption 5 holds, and 𝑉 = ∑𝑟𝑖=0 𝑉𝑖 and that (𝑅00 ≤ 1) 󳨐⇒ (𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ≤ 1) for any set 𝑉󸀠 = ∑𝑟𝑖=1 𝑉𝑖 = 𝑉 − 𝑉0 are nonsingular then 𝜌1 = of incommensurate delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟 if (44) holds. Assume, max(𝜌(𝐹0 𝑉0−1 ), 𝜌((∑𝑟𝑖=0 𝐹𝑖 )(∑𝑟𝑖=0 𝑉𝑖 )−1 )) = 𝜌 ≤ 1 and then on the contrary, that (𝑅00 ≤ 1) 󳨐⇒ (𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) > the condition 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ≤ 𝜌1 = 𝜌 for some set of 1) for some set of delays ℎ𝑖 ; ∀𝑖 ∈ 𝑟, so that 𝑅𝑀0 > 1. delays ℎ𝑖 ≥ 0; ∀𝑖 ∈ 𝑟 if 𝜌 ≤ 1 if both (40) and (44) hold Then, the linearized infective substate around the disease-free included in Corollary 19. Thus, the linearized description equilibrium point is unstable for at least one set of delays, of Lemma 12 (iv) does not add any new information to a contradiction to Theorem 13 (ii) from the joint conditions reformulate alternative conditions to those of Corollary 19. 𝜌00 < 1 and (44). Hence, (𝑅00 ≤ 1) 󳨐⇒ (𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ≤ Remark 21. Note that, since Theorem 13 (ii) gives sufficient 1). Since both 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) and 𝑅00 are positive real (while not necessary) stability conditions independent of constants which depend on the model parameterization then the delays, it cannot be proved in Corollary 17 that (𝑅00 > 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) = (1 + 𝜀) 𝑅00 − 𝜀 1) 󳨐⇒ (𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) > 1) for some set of delays (51) ℎ𝑖 ≥ 0; ∀𝑖 ∈ 𝑟 to conclude instability. That is, 𝑅00 ≤ 1 = (𝑅00 − 1) 𝜀 + 𝑅00 guarantees stability since 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ≤ 1 but the converse is not true, in general. In fact, if 𝑅00 > 1 then for some 𝜀 = 𝜀(ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ∈ R0+ for each set of delays 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) > 𝑅00 since 𝜀(𝑅00 − 1) > 0 so that 𝜌00 ℎ𝑖 ≥ 0; ∀𝑖 ∈ 𝑟. The nonnegativity of any solution under is a lower-bound of the maximum reproduction number for any nonnegative admissible initial conditions follows from any set of admissible delays. Similar considerations can be Assumption 5. Property (i) has been proved. Property (ii) said related to Corollaries 18 and 19 which do not provide follows since if Assumption 5 holds, then the AILDFS is necessary conditions either, in general, having in mind globally asymptotically stable if and only if 𝜌00 ≤ 1 since 𝜌00 Theorem 13. So, 𝜌0∞ is also a lower-bound of the maximum is the reproduction number of the AILDFS and any trajectory reproduction number for any set of admissible delays. solution is nonnegative under nonnegative initial conditions. The other assertions in Property (ii) follow from Theorem 4 Remark 22. A case of interest might be when 𝐹𝑖 = 𝛼𝑖 𝐹0 ; (ii) since the AILDFS is a delay-free system. 𝑖 ∈ 𝑟 ∪ {0} with 𝛼0 = 1. Then, 𝜌((∑𝑟𝑖=0 𝐹𝑖 )(∑𝑟𝑖=0 𝑉𝑖 )−1 ) = (1 + ∑𝑟𝑖=1 𝛼𝑖 )𝜌(𝐹0 (∑𝑟𝑖=0 𝑉𝑖 )−1 ) in Corollaries 18 and 19. If 0 < Corollary 18. Under Assumption 5, if 𝑉 = ∑𝑟𝑖=0 𝑉𝑖 is 𝛼𝑖+1 ≤ 𝛼𝑖 (respectively, 𝛼𝑖+1 > 𝛼𝑖 ) for 𝑖 ∈ 𝑟 − 1 ∪ {0} nonsingular and (40) holds then the following properties hold. then the delays lose their influence, or, at least, such an (i) 𝑅0 (0, ℎ1 , ℎ2 , . . . , ℎ𝑟 ) = (1+𝜀1 )𝑅0∞ −𝜀1 for any given set of influence does not increase (respectively, increases) in the delays ℎ𝑖 ≥ 0 for 𝑖 ∈ 𝑟 and some 𝜀1 = 𝜀1 (ℎ1 , ℎ2 , . . . , ℎ𝑟 ) ∈ R0+ . transmission of the disease as the delay sizes increase. So, Thus, the infective substate of the linearized system around the 𝜌((∑𝑟𝑖=0 𝐹𝑖 )(∑𝑟𝑖=0 𝑉𝑖 )−1 ) ≤ (𝑟 + 1)𝜌(𝐹0 (∑𝑟𝑖=0 𝑉𝑖 )−1 ), respectively, disease-free equilibrium point is globally asymptotically stable 𝜌((∑𝑟𝑖=0 𝐹𝑖 )(∑𝑟𝑖=0 𝑉𝑖 )−1 ) ≥ (𝑟 + 1)𝜌(𝐹0 (∑𝑟𝑖=0 𝑉𝑖 )−1 ). independent of the delays with a nonnegative state trajectory solution for any given admissible nonnegative function of initial Example 23. Consider the subsequent linearized model conditions if 𝜌0∞ ≤ 1 and (40) holds, that is, if 𝜌0∞ < 1. around the disease-free equilibrium point: (ii) the AILDDFS is globally asymptotically stable around 1 the disease-free equilibrium point, with a nonnegative state tra- 𝑥𝐿̇ (𝑡) = ∑ (𝐹𝑖 − 𝑉𝑖 ) 𝑥𝐿 (𝑡 − ℎ𝑖 ) (52) jectory solution for any given admissible nonnegative function 𝑖=0 Discrete Dynamics in Nature and Society 17 for any delay ℎ ≥ 0. The nonnegativity of the solutions of Theorem 24. The following properties hold: the linearized system holds with (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 and 𝐹0 ⪰ 0 (i) Assume that (𝐹0 − 𝑉0 ) and ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) are stability (AILDDFS-positivity, that is, (𝐹0 − 𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 ) and 𝐹1 ⪰ 𝑉1 matrices, and that (these three joint conditions implying AILDFS-positivity, that is, ∑1𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) ∈ 𝑀𝐸𝑛×𝑛 ). If furthermore, the nonlinear 𝑟 𝑇 −1 contribution to the infective dynamics of the whole epidemic 𝜌 [(𝐼𝑛 , 𝐼𝑛 , ⏞⏞⏞⋅⏞⏞ ⋅ ⋅⏞⏞, 𝐼𝑛 ) (i𝜔𝐼𝑛 − 𝐹0 + 𝑉0 ) ̂ 𝑦(𝑡)) model fulfills −𝑓(𝑥(𝑡), ̂ ⪰ 0; ∀𝑡 ∈ R0+ , independent of (54) ℎ, then the whole solution trajectory is nonnegative for all time independent of ℎ. The relevant amounts to guarantee the ⋅ (𝐹1 − 𝑉1 , 𝐹2 − 𝑉2 , . . . , 𝐹𝑟 − 𝑉𝑟 )] < 1; ∀𝜔 ∈ R0+ . asymptotic stability of the auxiliary delay-free systems around the disease-free equilibrium point and to keep it under a certain tolerance to the delayed dynamics amount (specified Then, the infective linearized subsystem around the disease-free by the matrix (𝐹1 − 𝑉1 )) independently of the delay ℎ, via equilibrium point is globally asymptotically stable independent Theorem 13 and Corollaries 15 and 17, are of the delays. (ii) If 𝐹𝑖 = 𝐹0 ≻ 0; ∀𝑖 ∈ 𝑟, (−𝑉0 ) and (−𝑉) = − ∑𝑟𝑖=0 𝑉𝑖 are 𝜌0∞ = 𝜌 (𝐹0 𝑉0−1 ) ; stability matrices, 𝜌(𝐹0 𝑉−1 ) < 1/(𝑟 + 1), and 󵄩 󵄩 𝜌0∞ = sup 󵄩󵄩󵄩󵄩(i𝜔𝐼𝑛 + 𝑉0 − 𝐹0 ) (𝐹1 − 𝑉1 )󵄩󵄩󵄩󵄩2 ; −1 𝜔∈R 0+ 𝑟 𝑇 −1 𝜌 [(𝐼𝑛 , 𝐼𝑛 , ⏞⏞⋅⏞⏞⏞ ⋅ ⋅⏞⏞, 𝐼𝑛 ) (i𝜔𝐼𝑛 − 𝐹0 + 𝑉0 ) −1 𝜌00 = 𝜌 [(𝐹0 + 𝐹1 ) (𝑉0 + 𝑉1 ) ] ; (55) 󵄩 󵄩 󵄩 −1 󵄩 𝜌00 = 2 󵄩󵄩󵄩𝐹1 − 𝑉1 󵄩󵄩󵄩2 sup 󵄩󵄩󵄩󵄩(i𝜔𝐼𝑛 − (𝐹1 − 𝑉1 )) 󵄩󵄩󵄩󵄩2 ; (53) ⋅ (𝐹0 − 𝑉1 , 𝐹0 − 𝑉2 , . . . , 𝐹0 − 𝑉𝑟 )] < 1; ∀𝜔 ∈ R0+ . 𝜔∈R + 𝑅0 = min (𝑅00 , 𝑅0∞ ) ; Then the infective linearized subsystem is globally asymptoti- cally stable independent of the delays. 𝑅00 = max (𝜌00 , 𝜌00 ) ; Proof. If (𝐹0 − 𝑉0 ) and ∑𝑟𝑖=0 (𝐹𝑖 − 𝑉𝑖 ) are stability matrices 𝑅0∞ = max (𝜌0∞ , 𝜌0∞ ) . then the linearized subsystem is globally asymptotically stable Note that for infinity and zero delays and the property also holds (1) 𝜌0∞ ≤ 1 if and only if the AILDDFS is locally independent of the sizes of the delays [18] if (54) holds. asymptotically stable. Since the endemic equilibrium point Property (i) has been proved. Property (ii) follows in the is not attainable if 𝜌0∞ < 1 and it is confluent with the same way when 𝐹𝑖 = 𝐹0 ≻ 0; ∀𝑖 ∈ 𝑟 since 𝜌(𝐹0 𝑉−1 ) < disease-free one if 𝜌0∞ = 1, the condition also implies that 1/(𝑟 + 1) guarantees that (−𝑉) is a stability matrix if (−𝑉0 ) the AILDDFS is globally asymptotically stable. is a stability matrix and (55) is identical to (54) if 𝐹𝑖 = 𝐹0 ≻ 0; (2) 𝜌0∞ > 1 implies that the disease-free equilibrium ∀𝑖 ∈ 𝑟. point is unstable and the endemic one is stable and attainable. (3) 𝜌00 ≤ 1 if and only if the AILDFS is locally Remark 25. Condition 2 of Theorem 24 holds if (−𝑉0 ) is asymptotically stable. Since the endemic equilibrium point is a nonsingular stability matrix (it suffices, for instance, that not attainable if 𝜌00 < 1 and it is confluent with the disease- (−𝑉0 ) ∈ 𝑀𝐸𝑛×𝑛 and it exists 𝑉0−1 ≻ 0, the second condition free one if 𝜌00 = 1, the condition also implies that the AILDFS holds if and only if the Metzler matrix (−𝑉0 ) is a stability is globally asymptotically stable. matrix) and 𝜌(𝑉0−1 (∑𝑟𝑖=1 𝑉𝑖 )) < 1 since such a condition (4) 𝜌00 > 1 implies that the disease-free equilibrium point implies that (−𝑉) is a stability matrix provided that (−𝑉0 ) is is unstable and the endemic one is stable and attainable. also a stability matrix. (5) If 𝑅0∞ < 1, or if 𝑅00 < 1, then all the disease-free A related result to Theorem 4 for the case of presence of equilibrium points (which satisfy the respective conditions delays adopts the subsequent form. 𝜌0∞ < 1 or 𝜌00 ≤ 1) for any value of ℎ ≥ 0 are globally asymptotically stable. In particular, 𝐹0 −𝑉0 and 𝐹1 −𝑉1 are such Theorem 26. Let 𝑔 : R𝑛+𝑚 󳨀→ R𝑚 be everywhere contin- that either 𝜌00 ≤ 1 or 𝜌0∞ ≤ 1 so that the corresponding aux- uously differentiable with a Jacobian matrix: 𝐽𝑔,𝑧 (𝑧0 ) = iliary delay-free system around the disease-free equilibrium [𝐽𝑔,𝑥 (𝑧0 ) | 𝐽𝑔,𝑦 (𝑧0 )] at any 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 ∈ R𝑛+𝑚 with 𝑥0 ∈ point is globally asymptotically stable. Under those respective conditions, all the resulting endemic equilibrium points are R𝑛 and 𝑦0 ∈ R𝑚 , where 𝑧 = (𝑥𝑇 , 𝑦𝑇 )𝑇 ∈ R𝑛+𝑚 , 𝑥 ∈ R𝑛 , unattainable. They are attainable and stable if 𝑅0∞ ≥ 1 or if 𝑦 ∈ R𝑚 . Assume that the 𝑚 × 𝑚 matrix 𝐽𝑔,𝑦 (𝑧0 ) is nonsingular 𝑅00 ≥ 1. at any equilibrium point 𝑧0 of (24)-(26). Then, the following The following result relies on sufficiency-type conditions properties hold. of global asymptotic stability independent of the delays (i) There exists an open set 𝑈 of R𝑛 containing 𝑥0 such of the linearized infective substate around the disease-free that there exists a unique continuously differentiable function equilibrium point including the case when the transmission ℎ : 𝑈 󳨀→ R𝑚 such that 𝑦0 = ℎ(𝑥0 ) and 𝑓(𝑥, ℎ(𝑥)) = 0; matrices for the various delays are all identical. ∀𝑥 ∈ 𝑈. Also, any equilibrium point 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 can be 18 Discrete Dynamics in Nature and Society expressed being dependent only on the infective substate, i.e., since 𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ⪯ 0 (being a strict inequality if and only 𝑧0 = (𝑥0𝑇 , ℎ𝑇 (𝑥0 ))𝑇 , while satisfying if 𝑥𝑒𝑛𝑑 ≠ 𝑥𝑑𝑓 = 0) and (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is stable, then it is nonsingular. 𝑦0 = ℎ (𝑥0 ) ; (b) If 𝑅00 < 1 then (𝑉 − 𝐹)−1 ≻ 0, since (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 , −1 and 𝐹𝑒 ≻ 𝐹. Therefore, (𝐼𝑛 + (𝑉 − 𝐹)−1 (𝐹𝑒 − 𝐹)) ≻ 0 and 𝑟 (56) 𝜌[(𝑉−𝐹)−1 (𝐹𝑒 −𝐹)] > 0 then (𝐼𝑛 +(𝐼𝑛 −𝑉−1 𝐹)−1 𝑉−1 (𝐹𝑒 −𝐹))−1 𝑥0 = [∑ (𝐹𝑖 (ℎ (𝑥0 )) − 𝑉𝑖 (𝑥0 ))] 𝑓 (𝑥0 , ℎ (𝑥0 )) 𝑖=0 exists, and −1 −1 −1 provided that [∑𝑟𝑖=0 (𝐹𝑖 (ℎ(𝑥0 )) − 𝑉𝑖 (𝑥0 ))] is nonsingular at (𝑉𝑒 − 𝐹𝑒 ) = (𝐼𝑛 + (𝑉 − 𝐹)−1 (𝐹𝑒 − 𝐹)) (𝑉 − 𝐹)−1 any equilibrium point 𝑧0 = (𝑥0𝑇 , ℎ𝑇 (𝑥0 ))𝑇 , where 𝐹𝑖 (ℎ(0)) = 𝐹𝑖 −1 −1 and 𝑉𝑖 (𝑧0 ) = 𝑉𝑖 ; ∀𝑖 ∈ 𝑟 ∪ {0} if 𝑧0 = 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 is = (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) (𝑉 − 𝐹)−1 (59) the disease-free equilibrium point and 𝐹𝑖 (ℎ(𝑥𝑒𝑛𝑑 )) = 𝐹𝑖𝑒 and 𝑇 𝑉𝑖 (𝑥𝑒𝑛𝑑 ) = 𝑉𝑖𝑒 ; ∀𝑖 ∈ 𝑟 ∪ {0} if 𝑧0 = 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 , ℎ𝑇 (𝑥𝑒𝑛𝑑 ))𝑇 is ≺ (𝑉 − 𝐹)−1 the endemic equilibrium point. (ii) Assume, furthermore, that (− ∑𝑟𝑖=0 𝑉𝑖 ) ∈ 𝑀𝐸𝑛×𝑛 is Note that 𝑥𝑒𝑛𝑑 is unreachable if 𝑥𝑒𝑛𝑑 ≠ 𝑥𝑑𝑓 = 0 since nonsingular with 𝐹0 ⪰ 0. Then, (a) the disease-free equilibrium −𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ≻ −𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) = 0, and point exists being defined by 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 , it is unique −1 and locally asymptotically stable independent of the delays if 𝑥𝑒𝑛𝑑 = (𝑉𝑒 − 𝐹𝑒 ) (−𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 ))) 𝑅00 < 1 and −1 −1 (b) the endemic equilibrium point exists being defined by ≺ (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) 𝑇 𝑇 𝑇 𝑇 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) = (𝑥𝑒𝑛𝑑 , ℎ𝑇 (𝑥𝑒𝑛𝑑 ))𝑇 while it satisfies (60) × (𝑉 − 𝐹)−1 (−𝑓 (𝑥𝑑𝑓 , ℎ (𝑥𝑑𝑓 ))) 𝑦𝑒𝑛𝑑 = ℎ (𝑥𝑒𝑛𝑑 ) ; (57) −1 −1 −1 𝑥𝑒𝑛𝑑 = (𝐹𝑒 − 𝑉𝑒 ) 𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 )) . = (𝐼𝑛 + (𝐼𝑛 − 𝑉−1 𝐹) 𝑉−1 (𝐹𝑒 − 𝐹)) 𝑥𝑑𝑓 = 0. It is reachable independent of the delays if 𝑅00 ≥ 1 and Then, 𝑥𝑒𝑛𝑑 ⪯ 0. But, for the case, 𝑥𝑒𝑛𝑑 = 0, the endemic 𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ⪯ 𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) while it is unreachable, equilibrium point is coincident with the disease-free one, that 𝑛+𝑚 is, 𝑥𝑒𝑛𝑑 = 𝑥𝑑𝑓 = 0 and 𝑅00 = 1 while, for 𝑅00 < 1, 𝑥𝑒𝑛𝑑 ≺ in the sense that 𝑧𝑒𝑛𝑑 ∉ R0+ , if 𝑅00 < 1 supposing that 𝑓(𝑥𝑒𝑛𝑑 , ℎ(𝑥𝑒𝑛𝑑 )) ≻ 𝑓(𝑥𝑑𝑓 , ℎ(𝑥𝑑𝑓 )) if 𝑧𝑒𝑛𝑑 ≠ 𝑧𝑑𝑓 . 0, hence a contradiction to its reachability follows since the infective components of the endemic equilibrium point are Proof. Since 𝐽𝑔,𝑧 (𝑥0 , 𝑦0 ) is nonsingular, there exists an open never allocated in the open first orthant of R𝑛 . Property (ii) set 𝑈 of R𝑛 containing 𝑥0 such that there exists a unique has been proved. continuously differentiable function ℎ : 𝑈 󳨀→ R𝑚 such that 𝑦0 = ℎ(𝑥0 ) and 𝑓(𝑥, ℎ(𝑥)) = 0; ∀𝑥 ∈ 𝑈. Let 𝑧0 be either the The above result can be reformulated directly under the disease-free equilibrium point 𝑧0 = 𝑧𝑑𝑓 = (0𝑇 , 𝑦0𝑇 )𝑇 or the replacements 𝑅00 󳨀→ 𝑅0∞ and 𝑅00 󳨀→ 𝑅0 . Note that if 𝑅00 ≤ endemic one 𝑧0 = 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 𝑇 𝑇 𝑇 , 𝑦𝑒𝑛𝑑 ) . Property (i) follows if 1 or 𝑅0∞ ≤ 1 or 𝑅0 ≤ 1, one concludes that the disease-free 𝑟 −1 is guaranteed to be locally asymptotically stable independent [∑𝑖=0 (𝐹𝑖 (ℎ(𝑥0 )) − 𝑉𝑖 (𝑥0 ))] exists for any equilibrium point. of the delays. It can also be guaranteed that the endemic To prove Property (ii), note that (−𝑉) ∈ 𝑀𝐸𝑛×𝑛 is equilibrium point is attainable and stable independent of the nonsingular with 𝐹 ⪰ 0. Then, the disease-free equilibrium delays if 𝑅00 ≥ 1, or 𝑅0∞ ≥ 1, or 𝑅0 ≥ 1. However, point exists being defined by 𝑧𝑑𝑓 = (0𝑇 , ℎ𝑇 (0))𝑇 , it is unique combinations of delays with identical amounts of associated since h(⋅) is unique, given by 𝑥𝑑𝑓 = (𝐹 − 𝑉)−1 𝑓(0, ℎ(0)) = delayed dynamics can exist with reproduction numbers less 0 and it is locally asymptotically stable if 𝑅00 < 1 since than 𝑅00 , respectively, 𝑅0∞ , or, respectively, 𝑅0 so that the then (𝐹 − 𝑉) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix, equivalently disease-free equilibrium point is unstable and the endemic (𝑉 − 𝐹)−1 ≻ 0 so that the linearized system around the one stable and attainable. From Theorem 13, and based on disease-free equilibrium point is locally exponentially (then the previous results of Theorem 10 for the delay-free case, the asymptotically ) stable. On the other hand, it is unstable if following result holds on positivity and global asymptotically 𝑅00 > 1 [Theorem 1 (iii)-(iv)]. The part (a) of Property (ii) stability of the system subject to delays. 𝑇 𝑇 𝑇 has been proved. Now let 𝑧0 = 𝑧𝑒𝑛𝑑 = (𝑥𝑒𝑛𝑑 , 𝑦𝑒𝑛𝑑 ) be the endemic equilibrium point. Note that it is also unique, since Theorem 27. Consider the epidemic model (27)-(29) under 𝑓 and ℎ are everywhere continuous. Note that the following Assumptions 4-5 and, furthermore, assume that cases can occur. (1) any of the sets of conditions of either Theorem 13 (ii), or (a) If 𝑅00 ≥ 1, note that Theorem 13 (iii), or Corollary 18(ii), or Corollary 19 hold, (2) there are a unique disease-free equilibrium point and a −1 𝑥𝑒𝑛𝑑 = (𝑉𝑒 − 𝐹𝑒 ) (−𝑓 (𝑥𝑒𝑛𝑑 , ℎ (𝑥𝑒𝑛𝑑 ))) unique endemic equilibrium point, (58) (3) 𝑔𝑖 (̂𝑧(𝑡)) ≥ 0 in (28) if 𝑦𝑖 (𝑡) = 0 for any 𝑖 ∈ 𝑚 and ⪰ (𝑉 − 𝐹)−1 (−𝑓 (𝑥𝑑𝑓 , ℎ (𝑥𝑑𝑓 ))) ⪰ 0 𝑡 ∈ R0+ , Discrete Dynamics in Nature and Society 19 (4) 𝑓𝑖 (̂𝑧(𝑡)) ≤ 𝑒𝑖𝑇 (𝐹 − 𝑉)𝑥(𝑡) if 𝑥𝑖 (𝑡 − ℎ𝑗 ) = 0 for any 𝑖 ∈ Since 𝑓(̂ 𝑥(⋅). ̂ 𝑦(⋅)) ̂ is uniformly continuous, it follows from 𝑛; ∀𝑗 ∈ 𝑟 ∪ {0} and 𝑡 ∈ R0+ (it suffices that 𝑓𝑖 (̂𝑧(𝑡)) ≤ 0 if Barbalat’s lemma that 𝑧(𝑡)̇ 󳨀→ 0 as 𝑡 󳨀→ ∞ so that 𝑧(𝑡) 𝑥𝑖 (𝑡 − ℎ𝑗 ) = 0), cannot have a limit oscillation in any of its components. So, a (5) the epidemic model is subject to any bounded absolutely stable limit cycle cannot surround the endemic equilibrium continuous function of initial conditions 𝜑 : [−ℎ, 0] 󳨀→ point. Therefore, 𝑧(𝑡) 󳨀→ 𝑧𝑒𝑛𝑑 as 𝑡 󳨀→ ∞ for any set of 𝑛+𝑚 R0+ with eventual finite jumps on a subset of [−ℎ, 0] of zero initial conditions satisfying condition (5). Property (ii) has measure, where 𝜑(0) = (𝑥𝑇 (0), 𝑦𝑇 (0))𝑇 , such that the resulting been proved. 𝑛+𝑚 function of initial conditions 𝜑 : [−ℎ, 0] 󳨀→ R0+ . (i) Then, any trajectory solution is nonnegative and uni- Remarks 28. Note that the equilibrium points do not depend formly bounded independent of the delays while it is globally on the delays but only on the delay-free and delayed matrices. convergent to the disease-free equilibrium point if 𝑅0 < 1, However, the reproduction numbers are dependent, in gen- which is also a globally asymptotically stable attractor. eral, on each particular set of delays although the threshold If furthermore, the following condition holds: 𝑅0 is independent of the delays. (6) if 𝑅0 > 1 then the total population of (27)-(29), (2) Note also that Theorem 27 concludes the stability i.e., the sum of all the subpopulations, is uniformly bounded (respectively, instability) of the disease-free equilibrium point independent of the delays for all time. for each set of delays if 𝑅0 < 1 (respectively, 𝑅0 > 1). However, ̂ 𝑦(𝑡)) (ii) Then, provided that 𝑓(𝑥(𝑡), ̂ ̂ 𝑦(𝑡)) and 𝑔(𝑥(𝑡), ̂ are it does not conclude that, for a particular set of delays, the uniformly continuous in the first closed orthant of R𝑛+𝑚 , all the endemic equilibrium point is unattainable/ unstable if 𝑅0 < 1. trajectory solutions are bounded and nonnegative for all time (3) From Remark 21, Theorem 13, Corollaries 18 and 19, while they converge asymptotically to the endemic equilibrium it follows that 𝑅00 and 𝑅0∞ are the reproduction numbers point which is also the unique asymptotically stable attractor. for the auxiliary delay-free systems AILDFS and AILDDFS, respectively, and also lower bounds of the reproduction Proof. If 𝑅0 < 1 then conditions (1)-(5) with Assumptions number for the whole set of delays. They also guarantee global 4-5 guarantee that the linearized model around the disease- asymptotic stability independent of the delays towards the free equilibrium point is locally asymptotically stable and disease-free equilibrium under sufficiency-type conditions of all the trajectories within the open first orthant of R𝑛+𝑚 tolerance to the delayed dynamics given, respectively, by (44) remain within it for all time. Since it is the unique attainable and (40). equilibrium the local asymptotic stability is also global (that is, for any bounded function of initial conditions within the first orthant) and asymptotic. Since all the subpopulations Appendix of both the infective and noninfective compartments are bounded and nonnegative for all time, the total population is A. Some Technical Results on also nonnegative and bounded through time. Property (i) has Positivity and Stability of Matrices and been proved. On the other hand, if condition (6) holds and Meztler and 𝑀-Matrices 𝑅0 > 1 then the disease-free equilibrium point is unstable and the endemic one is attainable. Assume the two subsequent Some technical results on positivity and stability of matrices cases potentially to occur. and their perturbed counterparts are given. Those results are then used concerning the stability of the linearized system Case a. The endemic equilibrium point is globally asymptoti- around the disease-free equilibrium point. cally stable independent of the delays. Property (ii) is proved directly. Theorem A.1 (see [14, 15]). Assume that 𝐺 ∈ R𝑛×𝑛 . Then, the following properties hold. Case b. The endemic equilibrium point is unstable indepen- (i) 𝐺 is an 𝑀-matrix if and only if (−𝐺) ∈ 𝑀𝐸𝑛×𝑛 . dent of the delays. Since the total population is bounded by (ii) 𝐺 is nonsingular with 𝐺−1 ≻ 0 if and only if 𝐺 is an hypothesis and all the subpopulations are nonnegative for all 𝑀-matrix. time, all the subpopulations are bounded for all time and a (iii) 𝐺 is nonsingular with 𝐺−1 ≻ 0 if and only if (−𝐺) ∈ bounded limit cycle should surround the endemic point to 𝑛×𝑛 𝑀𝐸 is a stability matrix. attract any trajectories in the first orthant from the admis- (iv) There exists 𝐺𝐵 (∈ R𝑛×𝑛 ) ≻ 0 with maximal eigenvalue sible Poincaré-type combinations of allowed stable/instable 𝑟 such that 𝐺 = 𝑐𝐼𝑛 − 𝐺𝐵 , where 𝑐 ≥ 𝑟. combinations of configurations of singular points and limit cycles. Rewrite compactly (27) to (29) in the form 𝑧(𝑡) ̇ = Theorem A.2. Assume that 𝐺 ∈ R𝑛×𝑛 . Then, the following ̂ 𝑥(𝑡), 𝑓( ̂ 𝑦(𝑡)) ̂ by defining 𝑓̂ : R0+ × R𝑛 × R𝑚 󳨀→ R𝑛+𝑚 properties hold. from the linearized infective dynamics and the functions (i) Assume that 𝐺 is decomposed as 𝐺 = 𝐺1 + 𝐺2 with 𝐺1 𝑓 : R0+ × R𝑛+𝑚 󳨀→ R𝑛 and 𝑔 : R0+ × R𝑛+𝑚 󳨀→ R𝑚 . being nonsingular and 𝐺2 = 𝐺 − 𝐺1 . Then, 𝐺 is nonsingular ∞ Assume that ∫0 𝑓( ̂ 𝑥(𝜏), ̂ ̂ 𝑦(𝜏))𝑑𝜏 = ±∞. Then 𝑧(𝑡) 󳨀→ ±∞ if ‖𝐺2 ‖ < 𝜀 for some sufficiently small (in general, norm- as 𝑡 󳨀→ ∞ which is not compatible with the nonnegativity dependent) 𝜀 ∈ R+ . of the solution trajectories within the first orthant and their ∞ (ii) Assume that 𝐺 ∈ R𝑛×𝑛 is an 𝑀-matrix that is boundenness. Therefore, −∞ < ∫0 𝑓( ̂ 𝑥(𝜏), ̂ ̂ 𝑦(𝜏))𝑑𝜏 < +∞. decomposed as 𝐺 = 𝐺1 + 𝐺2 with 𝐺1 being an 𝑀-matrix with 20 Discrete Dynamics in Nature and Society 𝐺1−1 ≻ 0 and 𝐺2 = 𝐺−𝐺1 ≺ 0. Then, 𝐺 ∈ R𝑛×𝑛 is a nonsingular B. Two Technical Results Which Guarantee antistable matrix with 𝐺−1 ≻ 0 if and only if 𝜌(𝐺2 𝐺1−1 ) < 1. the Uniqueness of the Equilibrium Points Equivalently, (−𝐺) ∈ 𝑀𝐸𝑛×𝑛 , while it is a stability matrix if and only if 𝜌(𝐺2 𝐺1−1 ) < 1. Theorem B.1. The following properties hold. (i) Let 𝑔 : R𝑛+𝑚 󳨀→ R𝑚 be a continuously differentiable Proof. Since (𝐺1 + 𝐺2 ) is a regular splitting of 𝐺 such that 𝐺1 function such that 𝑔(0, 𝑦0 ) = 0 for 0 ∈ R𝑛 and some 𝑦0 ∈ is nonsingular, one has from Banach’s Perturbation Lemma R𝑚 . If the Jacobian matrix 𝐽𝑔,𝑦 (0, 𝑦0 ) = ((𝜕𝑔𝑖 /𝜕𝑦𝑗 )(0, 𝑦0 )) [25] that is nonsingular then the disease-free equilibrium point 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) , 𝑦𝑑𝑓 = 𝑦0 , is unique. 𝐺 = 𝐺1 + 𝐺2 = 𝐺1 (𝐼𝑛 + 𝐺1−1 𝐺2 ) (A.1) (ii) Let 𝑔 : R𝑛+𝑚 󳨀→ R𝑚 be continuous with 𝑔(0, 𝑦0 ) = 0 for 0 ∈ R𝑛 and some 𝑦0 ∈ R𝑚 . If there exist open neighborhoods Thus, 𝐺−1 exists and 𝐴 ⊂ R𝑛 of 0 ∈ R𝑛 and 𝐵 ⊂ R𝑚 of 𝑦0 , such that for all 𝑥 ∈ 𝐴, 𝑔(𝑥, .) : 𝐴 󳨀→ R𝑚 is locally one-to-one then the disease-free 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝐺 󵄩󵄩 = 󵄩󵄩(𝐺1 + 𝐺2 )−1 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩(𝐼𝑛 + 𝐺1−1 𝐺2 ) 𝐺1−1 󵄩󵄩󵄩󵄩 −1 󵄩 󵄩 󵄩 󵄩 󵄩 equilibrium point 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) , 𝑦𝑑𝑓 = 𝑦0 , is unique. 󵄩 󵄩󵄩 −1 󵄩󵄩 (A.2) 󵄩 󵄩 󵄩󵄩 −1 󵄩 󵄩 󵄩󵄩𝐺𝑑 󵄩󵄩 Proof. Let 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) , for 𝑦𝑑𝑓 = 𝑦0 ∈ R𝑚 , 𝑦𝑑𝑓 = ≤ 󵄩󵄩󵄩󵄩𝐺1−1 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩(𝐼𝑛 + 𝐺1−1 𝐺2 ) 󵄩󵄩󵄩 ≤ 󵄩 󵄩 󵄩 󵄩 1 − 󵄩󵄩󵄩󵄩𝐺1−1 𝐺2 󵄩󵄩󵄩󵄩 𝑦01 (≠ 𝑦0 ) ∈ R𝑚 be two distinct disease-free equilibrium points and 𝑔 : R𝑛+𝑚 󳨀→ R𝑚 a continuously differentiable for any matrix norm ‖ ⋅ ‖, provided that ‖𝐺1−1 𝐺2 ‖ < 1, that function such that 𝑔(0, 𝑦0 ) = 0 for 0 ∈ R𝑛 and some 𝑦0 ∈ is, if ‖𝐺2 ‖ < 𝜀 = 1/‖𝐺1−1 ‖ = 𝐾(𝐺1 )/‖𝐺1 ‖, where 𝐾(𝐺1 ) = R𝑚 . If the Jacobian matrix 𝐽𝑔,𝑦 (0, 𝑦0 ) = ((𝜕𝑔𝑖 /𝜕𝑦𝑗 )(0, 𝑦0 )) is ‖𝐺1 ‖‖𝐺1−1 ‖ is the condition number of 𝐺1 with respect to the nonsingular then, from the implicit function theorem, there matrix norm ‖ ‖. Property (i) has been proved. exists an open set 𝑈 of R𝑛 with 0 ∈ 𝑈 such that there exists a To prove Property (ii), note that (𝐺1 + 𝐺2 ) is a regular unique continuously differentiable function ℎ : 𝑈 ⊂ R𝑛 󳨀→ splitting of the 𝑀-matrix 𝐺 since 𝐺1 is a nonsingular 𝑀- R𝑚 such that 𝑦0 = ℎ(0), 𝑓(𝑥, 𝑔(𝑥)) = 0; ∀𝑥 ∈ 𝑈, and matrix with 𝐺1−1 ≻ 0. Then, (−𝐺1 ) ∈ 𝑀𝐸𝑛×𝑛 is a stability matrix (𝜕ℎ/𝜕𝑥𝑗 )(𝑥) = −[𝐽𝑔,𝑦 (𝑥, ℎ(𝑥))]−1 [(𝜕𝑔/𝜕𝑥𝑗 )(𝑥, ℎ(𝑥))]; ∀𝑥 ∈ 𝑈. (Theorem A.1 (iii)) and 𝐺1 is anti-stable. Thus, 𝐺𝑎2 = −𝐺2 ≻ 0 Now, take a similar reasoning for 𝑦𝑑𝑓 = 𝑦01 , again assuming and 𝐺1−1 𝐺𝑎2 ≻ 0 and it follows that that 𝐽𝑔,𝑦 (0, 𝑦01 ) = ((𝜕𝑔𝑖 /𝜕𝑦𝑗 )(0, 𝑦01 )) is nonsingular, what concludes that 𝑦01 = ℎ1 (0) with ℎ1 : 𝑈1 󳨀→ 𝑅𝑚 but since 𝐺 = 𝐺1 + 𝐺2 = 𝐺1 − 𝐺𝑎2 = 𝐺1 (𝐼𝑛 − 𝐺1−1 𝐺𝑎2 ) (A.3) both ℎ : 𝑈 󳨀→ 𝑅𝑚 and ℎ1 : 𝑈1 󳨀→ 𝑅𝑚 are unique in 𝑈 and 𝑈1 which both are open and contain 0 ∈ R𝑛 so that they have a nonempty intersection, it follows that and 𝐺−1 exists and it is given by 𝐺−1 = (𝐼𝑛 − 𝐺1−1 𝐺𝑎2 )−1 𝐺1−1 . ℎ ≡ ℎ1 in 𝑈 ∩ 𝑈1 so that 𝑦01 = ℎ1 (0) ≠ 𝑦0 = ℎ(0) is a Since 𝜌(𝐺1−1 𝐺𝑎2 ) = 𝜌(𝐺1−1 𝐺2 ) < 1, (𝐼𝑛 − 𝐺1−1 𝐺𝑎2 )−1 = contradiction. Therefore, 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) is unique. Property ∑∞ −1 𝑛 𝑛=0 (𝐺1 𝐺𝑎2 ) ≻ 0 from Von Neumann’s series expansion (i) has been proved. Now, let again 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓 𝑇 𝑇 ) be a and since (𝐺1 𝐺𝑎2 )𝑛 ≻ 0 for all 𝑛 ≥ 0 and 𝐺1−1 ≻ 0, it follows −1 disease-free equilibrium point with 𝑦𝑑𝑓 = 𝑦0 . The proof of that 𝐺−1 = (∑∞ −1 𝑛 −1 𝑛=0 (𝐺1 𝐺𝑎2 ) )𝐺1 ≻ 0. Property (ii) is supported by the implicit function theorem On the other hand, since 𝐺𝑎2 ≻ 0 and 𝐺1−1 ≻ 0 for continuous (nonnecessarily continuously differentiable) both have a simple, in general, nonstrictly dominant Perron functions. If there exist open neighborhoods 𝐴 ⊂ R𝑛 of root. But the inverse of the Perron root of 𝐺1 is also a real 0 ∈ R𝑛 and 𝐵 ⊂ R𝑚 of 𝑦0 ∈ 𝐵, such that for all 𝑥 ∈ 𝐴, positive eigenvalue of 𝐺1−1 with the minimum value of all its 𝑔(𝑥, ⋅) : 𝐴 󳨀→ R𝑚 is locally one-to-one, then there exist eigenvalues. Therefore, since 𝐺1 and 𝐺1−1 are both antistable open neighborhoods 𝐴 0 ∈ R𝑛 of 0 ∈ R𝑛 , and 𝐵0 ∈ R𝑚 of and the eigenvalues of a matrix are continuous function with 𝑦0 ∈ R𝑚 such that for all 𝑥 ∈ 𝐴 0 , 𝑦 = ℎ(𝑥) ∈ 𝐵0 ⊂ R𝑚 is respect to all its entries, one concludes that the condition the unique local solution of 𝑔(𝑥, 𝑦) = 0, defined by the pairs 𝜌(𝐺1−1 𝐺2 ) < 1 guarantees that 𝐺−1 ≻ 0 and 𝐺 are both (𝑥, 𝑦 = ℎ(𝑥)) ∈ 𝐴 0 × 𝐵0 , where ℎ is a continuous function antistable and then (−𝐺) is a stability Metzler matrix. The from 𝐴 0 into 𝐵0 . Since any disease-free equilibrium point is above condition can also be expressed as 𝜌(𝐺2 𝐺1−1 ) < 1 since of the form 𝑧𝑑𝑓 = (0𝑇 , 𝑦𝑑𝑓𝑇 = ℎ𝑇 (0))𝑇 and ℎ is a continuous the order of the matrix product does not affect the spectral function from 𝐴 0 , which contains 0 ∈ R𝑛 , into 𝐵0 , such an radius since the eigenvalues of a product of two matrices do equilibrium point is unique; that is, two solutions (0𝑇 , 𝑦0𝑇 )𝑇 not depend on the product order [25]. The sufficient part has and 𝑧𝑑𝑓 = (0𝑇 , 𝑦1𝑇 )𝑇 with 𝑦0 , 𝑦1 (≠ 𝑦0 ) cannot exist. Property been proved. The necessity of 𝜌(𝐺2 𝐺1−1 ) < 1 for 𝐺 to be an (ii) has been proved. antistable 𝑀-matrix follows since if 𝜌(𝐺2 𝐺1−1 ) = 1 then, from the continuity of all the eigenvalues with respect to the matrix Theorem B.2. Assume that all the entries of the Jacobian entries, one of the eigenvalues of 𝐺 crosses the imaginary axis matrix around any equilibrium point are additive functions of the complex plane. It has been proved that 𝐺 ∈ R𝑛×𝑛 is and that all the equilibrium points have the same associate a nonsingular antistable matrix with 𝐺−1 ≻ 0 if and only transition matrix (−𝑉0 ). Let 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 = (𝑥0𝑇 , ℎ𝑇 (𝑥0 ))𝑇 if 𝜌(𝐺2 𝐺1−1 ) < 1. Equivalently, (−𝐺) ∈ 𝑀𝐸𝑛×𝑛 , while it is a be an endemic equilibrium point. Then, the following properties stability matrix, if and only if 𝜌(𝐺2 𝐺1−1 ) < 1. hold. Discrete Dynamics in Nature and Society 21 (i) Let 𝑧0 be reachable. Then, it is the unique endemic where Δ𝑥0 = 𝑥1 − 𝑥0 and Δ𝑦0 = 𝑦1 − 𝑦0 . If the reachable equilibrium point if and only if second two constraints are replaced into the first equation, one gets that, from Rouché-Froebenius theorem of Linear [∀Δ𝑥0 (≠ 0) ∈ R𝑛 , ∀Δ𝑦0 ∈ R𝑚 : (Δ𝑥0 ≻ −𝑥0 ) Algebra, such an algebraic equation is solvable (being either compatible determinate with one solution or indeterminate ∧ (ℎ (Δ𝑥0 ) ≻ −𝑦0 ) ∧ (𝑓 (𝑥0 + Δ𝑥0 , 𝑦0 + Δ𝑦0 ) with infinitely many ones) in Δ𝑥0 ≠ 0 if and only if ≺ 0) ∧ (rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) (B.1) rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) + 𝐹 (ℎ (Δ𝑥0 )) + 𝐹 (ℎ (Δ𝑥0 )) − 2𝑉0 , 2𝑓 (Δ𝑥0 , ℎ (Δ𝑥0 ))]) − 2𝑉0 , 2𝑓 (Δ𝑥0 , ℎ (Δ𝑥0 ))] = rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) + 𝐹 (ℎ (Δ𝑥0 )) (B.7) = rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) − 2𝑉0 ]] 󳨐⇒ [ℎ (𝑥0 ) ≠ ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )] + 𝐹 (ℎ (Δ𝑥0 )) − 2𝑉0 ]] (ii) Let 𝑧0 be either reachable or unattainable. Then, it is the unique endemic equilibrium point if and only if This implies the existence of at least another endemic equi- 𝑛 𝑚 librium point, which is not coincident with the disease-free [∀Δ𝑥0 (≠ 0) ∈ R , ∀Δ𝑦0 ∈ R : one since 𝑥1 = Δ𝑥0 ≠ 0 if ℎ(𝑦1 ) = ℎ(𝑥0 + Δ𝑥0 ) = ℎ(𝑥0 ) + (rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) + 𝐹 (ℎ (Δ𝑥0 )) ℎ(Δ𝑥0 ) = 𝑦0 +ℎ(Δ𝑥0 ). Such a point does not exist if either rank condition fails (i.e., the algebraic system is incompatible) or if − 2𝑉0 , 2𝑓 (Δ𝑥0 , ℎ (Δ𝑥0 ))]) it holds but ℎ(𝑥0 ) ≠ ℎ(𝑥0 +Δ𝑥0 )−ℎ(Δ𝑥0 ). There are additional (B.2) conditions on positivity needed for the point to be reachable = rank [𝐹 (ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )) in the case that it exists. This leads directly to the two claimed properties. + 𝐹 (ℎ (Δ𝑥0 )) − 2𝑉0 ]] 󳨐⇒ [ℎ (𝑥0 ) ≠ ℎ (𝑥0 + Δ𝑥0 ) − ℎ (Δ𝑥0 )] Data Availability Proof. Assume that there exist two distinct equilibrium Previously reported data used to support this study are cited points with infective and noninfective variables 𝑥(⋅) and 𝑦(⋅) , at relevant places within the text referred to the references in namely, 𝑧0 = (𝑥0𝑇 , 𝑦0𝑇 )𝑇 and 𝑧1 = (𝑥1𝑇 , 𝑦1𝑇 )𝑇 = 𝑧0 + Δ𝑧0 so that the manuscript. 𝐽𝑔,𝑧 (𝑧1 ) 𝑧1 = 𝐽𝑔,𝑧 (𝑧0 + Δ𝑧0 ) (𝑧0 + Δ𝑧0 ) (B.3) Conflicts of Interest = 𝐽𝑔,𝑧 (𝑧0 ) 𝑧0 = 0 The authors declare that there is no conflict of interests Since all the entries of the Jacobian matrices are additive, one regarding the publication of this paper. gets 𝐽𝑔,𝑧 (𝑧1 ) 𝑧1 − 𝐽𝑔,𝑧 (𝑧0 ) 𝑧0 Acknowledgments The authors are grateful to the Spanish Government = 𝐽𝑔,𝑧 (𝑧0 ) Δ𝑧0 + 𝐽𝑔,𝑧 (Δ𝑧0 ) 𝑧0 + 𝐽𝑔,𝑧 (Δ𝑧0 ) Δ𝑧0 (B.4) for Grants DPI2015-64766-R, RTI2018-094336-B-I00, and =0 DPI2016-77271-R (MINECO/FEDER, UE). then References 𝐽𝑔,𝑧 (𝑧0 ) Δ𝑧0 + 𝐽𝑔,𝑧 (Δ𝑧0 ) Δ𝑧0 = −𝐽𝑔,𝑧 (Δ𝑧0 ) 𝑧0 . (B.5) [1] Z. Shuai and P. 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