ON THE GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DISTRIBUTED DELAYS. Yukihiko Nakata. Yoichi Enatsu. Yoshiaki Muroya

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1 Manuscript submitted to AIMS Journals Volume X, Number X, XX 2X Website: ttp://aimsciences.org pp. X XX ON THE GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DISTRIBUTED DELAYS Yukiiko Nakata Basque Center for Applied Matematics Bizkaia Tecnology Park, Building 5 E-4816 Derio, Spain Yoici Enatsu Department of Pure and Applied Matematics Waseda University Okubo, Sinjuku-ku, Tokyo, , Japan Yosiaki Muroya Department of Matematics Waseda University Okubo, Sinjuku-ku, Tokyo, , Japan Abstract. In tis paper, we establis te global asymptotic stability of an endemic equilibrium for an SIRS epidemic model wit distributed time delays. It is sown tat te global stability olds for any rate of immunity loss, if te basic reproduction number is greater tan 1 and less tan or equals to a critical value. Oterwise, tere is a maximal rate of immunity loss wic guarantees te global stability. By using an extension of a Lyapunov functional establised by [C.C. McCluskey, Complete global stability for an SIR epidemic model wit delay-distributed or discrete, Nonlinear Anal. RWA. 11 ( ], we provide a partial answer to an open problem weter te endemic equilibrium is globally stable, wenever it exists, or not. 1. Introduction and main result. Currently, one of te important issues to us is controlling te transmission dynamics of infectious diseases. Epidemic models by a system of differential equations ave been studied in te literature (see [1]-[13] and te references terein. By analyzing suc matematical models, we obtain furter knowledge regarding te disease dynamics and tis will give us a useful information to determine weter te disease dies out or not in te ost population. One of te basic matematical models in epidemiology is called SIR (Susceptible- Infected-Recovered epidemic model (see Hetcote [4]. Based on te model, for vector-borne diseases, Beretta and Takeuci [1] introduced distributed delays and considered te delay effect on global stability of equilibria of te system. Teir 2 Matematics Subject Classification. Primary: 34K2 and 34K25; Secondary: 92D3. Key words and prases. SIRS epidemic model, global asymptotic stability, distributed delays, Lyapunov functional. Te first autor was partially supported by te Grant MTM of te MICINN, Spanis Ministry of Science and Innovation. Te second autor was partially supported by JSPS Fellows, No of Japan Society for te Promotion of Science. Te tird autor was partially supported by Scientific Researc (c, No of Japan Society for te Promotion of Science. 1

2 2 YUKIHIKO NAKATA, YOICHI ENATSU AND YOSHIAKI MUROYA model is given by te following system of delay differential equations: ds(t B S(t βs(t f(τi(t τdτ, di(t βs(t f(τi(t τdτ ( + γi(t, dr(t γi(t R(t. (1.1 S(t, I(t and R(t denote te population of susceptible, infective and recovered individuals at time t, respectively. It is assumed tat all newborns are susceptible. Te positive constant B represents te total number of newborns in te ost population per unit time and te positive constant represents te deat rate of te population. Te positive constant β is te transmission coefficient and is a superior limit of incubation times. Te incubation period distribution f(τ, wic denotes te fraction of vector population in wic te time taken to become infectious is τ, is assumed to be continuous on [, ] satisfying f(τdτ 1 and f(τ for τ. Te positive constant γ represents te recovery rate of infectives. System (1.1 always as a disease-free equilibrium E (S,,, S B/ corresponding to te extinction of te disease. Beretta and Takeuci [1] sowed tat E is globally asymptotically stable for any > wen system (1.1 does Bβ (+γ not ave any equilibria oter tan E. If > 1, ten tere exists a unique endemic equilibrium E (S, I, R, were every component is strictly positive. Tey also sowed tat E is locally asymptotically stable for any >. However, on te global stability analysis, tey required tat sould be small enoug to sow te global stability of te endemic equilibrium. Te global stability of E for any > remained unsolved for a long time. For tis problem, McCluskey [6] introduced te following Lyapunov functional: were U S (t g ( S(t S U (t 1 βi U S(t + 1 βs U I(t + U + (t, (1.2, U I (t g ( I(t I, U + (t g(x x 1 ln x g(1 for x >, α(τ τ ( I(t τ α(τg f(σdσ. I dτ, (1.3 Ten tey proved tat E is globally asymptotically stable for any > wenever it exists. In particular, U + (t in (1.2 plays an important role to obtain tat te global stability does not depend on te duration of te incubation period >. On te oter and, system (1.1 is insufficient to caracterize te disease transmission dynamics wen we take account of an immunity loss of te diseases. In tis situation, recovered individual become susceptible again wen te temporary immunity fades away and ten, individual moves cyclically among tree compartments. Suc a population dynamics is described by SIRS epidemic models (see Mena-Lorcat and Hetcote [7]. Considering suc an immunity loss, we study te following SIRS epidemic model wit distributed delays based on system (1.1 (cf.

3 GLOBAL STABILITY OF A DELAYED SIRS EPIDEMIC MODEL 3 [11, 12, 13]: ds(t B S(t βs(t f(τi(t τdτ + δr(t, di(t βs(t f(τi(t τdτ ( + γi(t, dr(t γi(t ( + δr(t. (1.4 Te state variables and parameters except δ ave te same meaning as tose in system (1.1. δ denotes te rate at wic recovered individuals lose immunity and return to te susceptible class. Zen et al. [13] also studied similar system and added a disease-related deat rate to te second equation of (1.4. Te initial conditions for system (1.4 take te form { S(θ φ1 (θ, I(θ φ 2 (θ, R(θ φ 3 (θ, φ i (θ, θ [, ], φ i ( >, φ i C + (1.5, i 1, 2, 3, were C denotes te Banac space C([, ], R of continuous functions mapping te interval [, ] into R equipped wit te sup-norm ψ sup θ ψ(θ. Te nonnegative cone of C is defined as C + C([, ], R +. Ten, system (1.4 as a unique solution (S(t, I(t, R(t satisfying S(t >, I(t > and R(t > for all t > (see [12, 13]. System (1.4 also as te disease-free equilibrium E (S,,. By Zen et al. [13], it is proved tat E is globally asymptotically stable for any > wen tere is no equilibrium oter tan E (see also Teorem 2.1 below. Similar to system (1.1, te basic reproduction number of system (1.4 is given by R Bβ ( + γ (1.6 and if R > 1, ten system (1.4 as a unique endemic equilibrium E (S, I, R, were S + γ β, I + δ + δ + γ S (1 1R, R γ + δ + γ S (1 1R. (1.7 By Zen et al. [13], it is sown tat E is locally asymptotically stable for any > and teir analysis allows considering te disease-related deat rate. However, on te global stability analysis, tey also required tat te lengt of te incubation period > and te rate of immunity loss δ sould be small enoug, since teir approac is a modified Lyapunov functional from tat of Takeuci [1] for system (1.1. In contrast, for te corresponding SIRS epidemic model witout delay, te endemic equilibrium is globally asymptotically stable wenever it exists (see [2, 7]. Terefore, stability results in [2, 6, 7] motivate us to expect tat if te endemic equilibrium of system (1.4 exists, it is globally asymptotically stable for any > as well as system (1.1. Tis is still an open problem now, as te stability problem for system (1.1 ad been unsolved. Concerning SIRS epidemic models wit distributed delay wic as a nonlinear incidence rate, Muroya et al. [8, 9] and Enatsu et al. [3] establised conditions for te global stability of te endemic equilibrium. Muroya et al. [8, 9] developed a monotone iterative sceme for te model and Enatsu et al. [3] constructed a Lyapunov functional using te nonlinearity of te incidence form.

4 4 YUKIHIKO NAKATA, YOICHI ENATSU AND YOSHIAKI MUROYA In tis paper, a partial answer to te open problem is given by constructing a new Lyapunov functional wic combines te Lyapunov functional (1.2 used in McCluskey [6] wit a quadratic term for te state variable R(t. For a limit system of (1.4, we consider te following functional: U(t U δ (t + γβ(s 2 I (R(t R 2, 2 wit (1.7. Te limit system (see (3.1 below is derived by using te relations I(t N(t S(t R(t and lim t + N(t B/ to I(t in te tird equation of (1.4, were N(t S(t + I(t + R(t is te total population. We establis te following result for te global stability of te endemic equilibrium E of system (1.4. Teorem 1.1. Let R > 1. If S δr, (1.8 ten te endemic equilibrium E of system (1.4 is globally asymptotically stable for any >. In particular, (1.8 is equivalent to δ < +, for 1 < R 1 + γ, δ δ : 1, for R > 1 + γ. (1.9 R 1+ γ For system (1.4, tis is te first global stability result wic does not depend on >. In addition, since te monotone iterative metod in Xu and Ma [11] for a class of delayed SIRS models used te advantage of te nonlinearity on te incidence function and Lyapunov functional tecnique in Zen et al. [13] relies on te positivity of te disease-related deat rate, teir approaces are not directly applicable for system (1.4. Teorem 1.1 is a partial answer to te problem, weter te endemic equilibrium is globally asymptotically stable wenever it exists or not. In particular, we sow tat te rate of immunity loss δ does not give any influence for te global stability wen 1 < R 1 + γ. Terefore, biologically, if te basic reproduction number lies in te interval determined by te deat rate and te recovery rate, ten te disease transmission will eventually equilibrate to te endemic steady state for any rate of immunity loss and any duration of te incubation period. In te oter cases, we establis te maximal rate of immunity loss wic guarantees te global stability of te endemic steady state. Since (1.8 is always true if δ, we extend te result given by McCluskey [6] (see also Figure 1. Te organization of tis paper is as follows. In Section 2, we give basic results for system (1.4 by Zang and Teng [12] and Zen et al. [13]. One can see tat te basic reproduction number R determines weter te disease dies out or uniformly persists in te population. In Section 3, we study te global stability of te endemic equilibrium and give te proof of Teorem 1.1. At first, we introduce a limit system of (1.4 and ten construct a Lyapunov functional for te system. Te Lyapunov functional, used in te proof of Teorem 1.1, combines te functional given by McCluskey [6] for system (1.1 and a quadratic term for R(t. In Section 4, some examples are given. A numerical simulation may support our conjecture tat te endemic equilibrium is globally asymptotically stable even if (1.8 does not old.

5 GLOBAL STABILITY OF A DELAYED SIRS EPIDEMIC MODEL 5 Figure 1. Stability region of te disease-free equilibrium and te endemic equilibrium in te parameter space (R, δ. Te curve denotes te boundary of te global stability region for te endemic equilibrium. LAS and GAS denote locally asymptotically stable and globally asymptotically stable, respectively. 2. Known results. Lemma 2.1. For any solution (S(t, I(t, R(t of system (1.4 wit initial conditions (1.5, it olds tat lim t + (S(t + I(t + R(t B/. Proof. From system (1.4, it follows tat dn(t B (S(t + I(t + R(t B N(t, wic implies tat lim t + N(t B/. Hence, tis completes te proof. Te following results from Zen et al. [13] and Zang and Teng [12] will be used in te proof of Teorem 1.1, indicating tat te disease dies out for R < 1 and persists in te population for R > 1. Teorem 2.1. (Zen et al. [13, Teorem 3.1]. (i Let R < 1. Ten te disease-free equilibrium E of system (1.4 is globally asymptotically stable for any >. (ii Let R > 1. Ten te disease-free equilibrium E is unstable and te endemic equilibrium E of system (1.4 is locally asymptotically stable for any >. Lemma 2.2. (Zang and Teng [12, Propositions 3.2 and 3.3]. Let R > 1. Ten, for any solution of system (1.4 wit initial condition (1.5, it olds tat lim inf S(t v 1 : t + B + βb/, lim inf I(t v 2 : qi exp ( ( + γ( + ρ, lim inf R(t v 3 : γv 2 t + t + + δ, were < q < 1 and ρ > satisfy S < S : B k (1 exp ( kρ, k + βqi.

6 6 YUKIHIKO NAKATA, YOICHI ENATSU AND YOSHIAKI MUROYA 3. Global stability of te endemic equilibrium E for R > 1. In tis section, we study te global stability of E. From Lemma 2.1, system (1.4 as te following limit system: ds(t B S(t βs(t f(τi(t τdτ + δr(t, di(t βs(t f(τi(t τdτ ( + γi(t, (3.1 dr(t γ ( B S(t R(t ( + δr(t. For system (3.1, we obtain te following result: Teorem 3.1. Let R > 1. If (1.8 olds, ten te endemic equilibrium E of system (3.1 is globally asymptotically stable for any >. In particular, (1.8 is equivalent to (1.9. Proof. We construct te following Lyapunov functional wic combines (1.2 by McCluskey [6] for system (1.1 wit a quadratic term for R(t: U(t 1 βi U S(t + 1 βs U δ I(t + U + (t + γβ(s 2 I U R(t, were U S (t, U I (t and U + (t are defined in (1.3 and U R (t 1 2 (R(t R 2. We now sow tat du(t. We use te following notations if necessary. First, we calculate du S(t du S (t x t S(t S, y t I(t I, y t,τ I(t τ I, z t R(t R.. Substituting B S + βs I δr gives as follows: 1 ( (1 S S B S(t βs(t f(τi(t τdτ + δr(t S(t ( ( (1 S S(t S(t S 1 + (1 δr S S R(t S(t R 1 ( +βi f(τ (1 S 1 S(t I(t τ S(t S I dτ (1 1xt (x t 1 + (1 δr 1xt (z t 1 +βi We secondly calculate du I (t du I (t S f(τ (1 1xt (1 x t y t,τ dτ. (3.2. Substituting + γ βs, we obtain 1 ( (1 I I βs(t f(τi(t τdτ βs I(t I(t ( βs f(τ (1 I S(t I(t τ I(t S I I(t I dτ (1 1yt (x t y t,τ y t dτ. (3.3 βs f(τ

7 Next, we calculate du+(t tat GLOBAL STABILITY OF A DELAYED SIRS EPIDEMIC MODEL 7 du + (t. Since U + (t f(τ t t τ I(s g( I dsdτ olds, we ave f(τ(g(y t g(y t,τ dτ. (3.4 By applying tecniques on deformation of equation in McCluskey [6], we use te following relation: (1 1xt (1 x t y t,τ + (1 1yt (x t y t,τ y t + (g(y t g(y t,τ y t,τ x ty t,τ y t + (y t ln y t y t,τ + ln y t,τ x t y ( ( t 1 xt y t,τ g g. x t Ten, combining (3.2, (3.3 and (3.4, we obtain y t ( d 1 βs U I(t + U + (t βi U S(t + 1 (1 βi 1xt (x t 1 + δr βs I ( ( ( 1 xt y t,τ f(τ g + g x t y t (1 1xt (z t 1 dτ. (3.5 Since B S +(+γi δr and γi (+δr imply tat B S + +γ+δ γ R, calculating du R(t gives du R (t { ( } B (R(t R γ S(t R(t ( + δr(t (R(t R { γ(s(t S ( + γ + δ(r(t R } γr S (z t 1 (1 x t ( + γ + δ(r 2 (z t 1 2. (3.6 Under te condition (1.8, we ave (1 βi 1xt (x t 1 + (1 δr βs I 1xt (z t 1 + δr βs I (z t 1 (1 x t } (2 βi x t 1xt + δr βs I (z t 1 {(1 1xt + (1 x t 1 βs I {S + δr (z t 1} (2 x t 1xt 1 βs I (S δr (2 x t 1xt.

8 8 YUKIHIKO NAKATA, YOICHI ENATSU AND YOSHIAKI MUROYA Terefore, combining (3.5 and (3.6, it follows tat du(t (1 βi 1xt (x t 1 + (1 δr βs I 1xt (z t 1 + δr βs I (z t 1 (1 x t δ( + γ + δ(r 2 ( ( ( 1 γβ(s 2 I (z t 1 2 xt y t,τ f(τ g + g dτ δ( + γ + δ(r 2 γβ(s 2 I (z t 1 2 ( f(τ g x t ( 1 x t + g y t ( xt y t,τ y t dτ. Let M be te largest invariant set of { du(t } tat is invariant wit respect to system (3.1. We sow tat M consists of only te endemic equilibrium E. Recalling tat g( 1 x t + g( xtyt,τ y t, if and only if x t 1 and x t y t,τ /y t 1, follows, if and only if du(t S(t S, I(t I(t τ for almost all τ [, ] and R(t R. (3.7 Since M is invariant, we ave ds(t in M. Ten, from te first equation of (3.1 and (3.7, it olds tat ds(t B S βs f(τi(t τdτ + δr B S βs I(t + δr βs (I I(t, wic implies tat I(t I. Terefore, eac element of M satisfies S(t S, I(t I and R(t R for all t. From Teorem 2.1 and LaSalle s invariance principle (see Kuang [5, Corollary 5.2], E of system (3.1 is globally asymptotically stable. Finally, we sow tat te condition (1.8 is equivalent to te condition (1.9. From (1.6 and (1.7, it follows R S S and ( δr S δγ S 1 1 R + δ + γ S Tus, we obtain δr S ( + δ + γ δγ (R 1 + δ + γ ( ( + γ δγ R 1 γ + δ + γ + γ + δ + γ { δ ( δγ + δ + γ (R 1. R 1 + γ Terefore, one can see tat (1.8 is equivalent to te condition (1.9. Tis completes te proof. Proof of Teorem 1.1. By Teorem 3.1, we obtain Teorem }.

9 GLOBAL STABILITY OF A DELAYED SIRS EPIDEMIC MODEL 9 Parameter Value Dimension β Number of individual 1 Year 1 B 16 Number of individual Year Year 1 γ 13 Year 1 δ.35 Year 1 τ.65 Year Table 1. Parameters and teir values in Section 4. Finally, we consider te following corresponding discrete delay model of (1.4: ds(t B S(t βs(ti(t τ + δr(t, di(t βs(ti(t τ ( + γi(t, (3.8 dr(t γi(t ( + δr(t. It is easy to see tat (3.8 also as te same endemic equilibrium E of (1.4 if and only if R > 1. Te condition (1.8, wic is equivalent to (1.9, becomes te sufficient condition for te global stability of te endemic equilibrium of (3.8 similar to te case of SIR epidemic model wit distributed delays (cf. McCluskey [6, Section 5]. 4. Numerical examples. In tis section, using parameter values given in Table 1, we present numerical examples to illustrate our global stability condition. Example 4.1. Set te parameter values B, and γ fixed in Table 1 but wit β.117. Ten, we obtain 1 < R γ From te first case of te condition (1.9 in Teorem 1.1, te endemic equilibrium E of system (1.4 is globally asymptotically stable for any δ. Example 4.2. Set te parameter values B, and γ fixed in Table 1 but wit β.22. Ten, we obtain R > 1 + γ From te second case of te condition (1.9 in Teorem 1.1, te endemic equilibrium E of system (1.4 is globally asymptotically stable for δ δ.17. Finally, we consider a case were te condition (1.9 does not old. We fix te parameter values as in Table 1 but wit β.22. Ten, we ave R > 1 + /γ and δ.35 > δ.17. Tus te condition (1.9 does not old. In Figure 2, we present te grap trajectory of S(t, I(t and R(t of a discrete delay model (3.8 wit tree different initial conditions and one can see tat te endemic equilibrium seems to be globally stable. We leave as an open problem if te endemic equilibrium is globally stable for δ > δ and R > 1 + /γ. REFERENCES [1] E. Beretta and Y. Takeuci, Global stability of an SIR epidemic model wit time delays, J. Mat. Biol. 33 ( [2] J. Cen, An SIRS epidemic model, Appl. Mat. J. Cinese Univ. 19 (

10 1 YUKIHIKO NAKATA, YOICHI ENATSU AND YOSHIAKI MUROYA R t S t 4 2 I t O t Figure 2. Beaviour of S(t, I(t and R(t for R > 1 + γ and δ.35 > δ [3] Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIRS epidemic models wit a class of nonlinear incidence rates and distributed delays, Acta Mat. Sci. To appear. [4] H.W. Hetcote, Te matematics of infectious diseases, SIAM review 42 ( [5] Y. Kuang, Delay Differential Equations wit Applications in Population Dynamics. Academic Press, San Diego, [6] C.C. McCluskey, Complete global stability for an SIR epidemic model wit delay-distributed or discrete, Nonlinear Anal. RWA. 11 ( [7] J. Mena-Lorcat and H.W. Hetcote, Dynamic models of infectious diseases as regulators of population sizes, J. Mat. Biol. 3 7 ( [8] Y. Muroya, Y. Enatsu and Y. Nakata, Global stability of a delayed SIRS epidemic model wit a non-monotonic incidence rate, J. Mat. Anal. Appl. 377 ( [9] Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative tecniques to SIRS epidemic models wit nonlinear incidence rates and distributed delays, Nonlinear Anal. RWA. 12 ( [1] Y. Takeuci, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model wit finite incubation times, Nonlinear Anal. 42 ( [11] R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model wit a nonlinear incidence rate, Caos, Solitons and Fractals. 41 ( [12] T. Zang and Z. Teng, Global beavior and permanence of SIRS epidemic model wit time delay, Nonlinear Anal. RWA. 9 ( [13] J. Zen, Z. Ma and M. Han, Global stability of an SIRS epidemic model wit delays, Acta Mat. Sci. 26 ( Received July 21; revised August 21. address: nakata@bcamat.org address: yo1.gc-rw.docomo@akane.waseda.jp address: ymuroya@waseda.jp