A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) presented an SEIR model using an infinite delay to account for varying infectivity. The analysis in that paper did not resolve the global dynamics for R0 \u3e 1. Here, we show that the endemic equilibrium is globally stable for R0 \u3e 1. The proof uses a Lyapunov functional that includes an integral over all previous states

McCluskey, C. Connell

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Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay

Wilfrid Laurier University

Wilfrid Laurier University Scholars Commons @ Laurier Mathematics Faculty Publications Mathematics 7-2009 Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay C. Connell McCluskey Wilfrid Laurier University, cmccluskey@wlu.ca Follow this and additional works at: https://scholars.wlu.ca/math_faculty Recommended Citation McCluskey, C. Connell, "Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay" (2009). Mathematics Faculty Publications. 11. https://scholars.wlu.ca/math_faculty/11 This Article is brought to you for free and open access by the Mathematics at Scholars Commons @ Laurier. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact scholarscommons@wlu.ca. MATHEMATICAL BIOSCIENCES doi:10.3934/mbe.2009.6.603AND ENGINEERINGVolume 6, Number 3, July 2009 pp. 603–610GLOBAL STABILITY FOR AN SEIR EPIDEMIOLOGICALMODEL WITH VARYING INFECTIVITY AND INFINITE DELAYC. Connell McCluskeyDepartment of Mathematics, Wilfrid Laurier UniversityWaterloo, Ontario, CanadaN2L 3C5(Communicated by Abba Gumel)Abstract. A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) pre-sented an SEIR model using an infinite delay to account for varying infectivity.The analysis in that paper did not resolve the global dynamics for R0 > 1.Here, we show that the endemic equilibrium is globally stable for R0 > 1. Theproof uses a Lyapunov functional that includes an integral over all previousstates.1. Introduction. A recent paper [16] presented an SEIR model for an infectiousdisease that included infection-age structure to allow for varying infectivity. Theincidence is of mass action type, but because of the varying infectivity, has the formβS(t)∫∞0k(a)i(t, a)da. Nevertheless, the authors gave a thorough analysis leavingout only the elusive global stability of the endemic equilibrium.That issue is resolved in this paper using a Lyapunov functional related to thetype of Lyapunov function used for ordinary differential equation (ODE) ecologicalmodels [3, 4] in the 1980s and used more recently for ODE epidemiological models[6, 10, 11, 12, 13, 14, 15]. In [5], an ODE model of arbitrary dimension that includesvarying infectivity is studied using the same type of Lyapunov function. For each ofthese models, the Lyapunov function is a sum of terms of the form f(y) = y−1−lny,where y is a variable of the system. The model studied in this paper has infinitedelay, and so it is necessary to include in the Lyapunov functional a term thatintegrates over all previous states.We now provide a brief outline of the paper. In Section 2 we describe the equa-tions that are to be studied. Section 3 includes results by Ro¨st and Wu from [16],providing the context in which this paper is to be read. Many of these results arethen used in Section 4 where the global stability of the endemic equilibrium is shown— the key result of this paper.2. Model equations. A population is divided into classes: susceptible, exposed,infectious, and recovered, denoted by S, E, I, and R, respectively. The infectiousclass is structured by age of infection (i.e. time since entry into class I). Thedensity of individuals with infection-age a at time t is given by i(t, a) with I(t) =∫∞0i(t, a)da.2000 Mathematics Subject Classification. 34K20, 92D30.Key words and phrases. infinite delay, global stability, Lyapunov functional.The author is supported by an NSERC Discovery Grant.603604 C. CONNELL MCCLUSKEYConstant recruitment into S is given by Λ. Incidence is of mass action typewith baseline coefficient β. The relative infectivity of individuals of infection-age ais k(a), where k is an integrable function taking values in the interval [0, 1]. Thenatural death rate is d, the disease-related death rate is r, the average latency periodis 1/µ and the average period of infectivity is 1/r.The original model equations [16] aredS(t)dt= Λ− βS(t)∫ ∞0k(a)i(t, a)da− dS(t)dE(t)dt= βS(t)∫ ∞0k(a)i(t, a)da− (µ+ d)E(t)dI(t)dt= µE(t)− (d+ δ + r)I(t)dR(t)dt= rI(t) − dR(t)(1)and (∂∂t+∂∂a)i(t, a) = −(d+ δ + r)i(t, a) (2)with the boundary conditioni(t, 0) = µE(t).Solving (2) givesi(t, a) = µe−(d+δ+r)aE(t− a).This allows equation (1) to be rewritten asdS(t)dt= Λ− µβS(t)∫ ∞0k(a)e−(d+δ+r)aE(t− a)da− dS(t)dE(t)dt= µβS(t)∫ ∞0k(a)e−(d+δ+r)aE(t− a)da− (µ+ d)E(t),(3)where the equations for dIdtand dRdtare omitted because they decouple.In order to specify the initial conditions for (3), we introduce the following no-tation. Given a non-negative function E defined on the interval (−∞, T ], for anyt ≤ T we define the function Et : R≤0 → R≥0 by Et(θ) = E(t+ θ) for θ ≤ 0.For equation (1), the initial condition would specify S(0), E(0), R(0) ≥ 0 andi(0, ·) : R≥0 → R≥0. For equation (3), an equation with infinite delay, the initialcondition must specify S(0) ≥ 0 and E0 : R≤0 → R≥0.Due to the infinite delay, it is necessary to determine an appropriate phase space.For any ∆ ∈ (0, d+ δ + r), letC∆ ={ϕ : R≤0 → R such that ϕ(θ)e∆θ is bounded and uniformly continuous}andY∆ = {ϕ ∈ C∆ : ϕ(θ) ≥ 0 for all θ ≤ 0} .Define the norm on C∆ and Y∆ by‖ϕ‖ = supθ≤0∣∣ϕ(θ)e∆θ∣∣ .It follows immediately that ϕ(0) ≤ ‖ϕ‖.Fixing ∆ ∈ (0, d+δ+r), we take the phase space for equation (3) to be R≥0×Y∆.Any initial condition (S(0), E0) ∈ R≥0×Y∆ gives a solution (S(t), Et) that remainsGLOBAL STABILITY WITH INFINITE DELAY 605in the phase space for all time. Furthermore, if (S(t), E(t)) is bounded for t ≥ 0,then the positive orbit Γ+ = {(S(t), Et) : t ≥ 0} has compact closure in R≥0 × Y∆.Relevant developments of infinite delay equations, including determining thephase space, can be found in [1, 8, 9] and references found therein.3. Previous results. In their paper, the authors of [16] give a thorough analysisof equation (3). They find the equilibria, calculate the basic reproduction numberR0 and show that the system is point dissipative. The disease-free equilibrium isshown to be globally stable for R0 < 1. For R0 > 1 the disease-free equilibriumis unstable, there is a unique endemic equilibrium, which is locally asymptoticallystable, and the system is permanent. They also do a final size calculation.All that remains to complete the analysis is to determine the global behaviourfor R0 > 1. This is done in Section 4 of this paper, where it is shown that theendemic equilibrium is globally stable for R0 > 1. In preparation for that, we nowgive results from [16].Theorem 3.1. Equation (3) is point dissipative. That is, there exists M > 0 suchthat for each solution of (3) there is a T > 0 such that S(t) ≤ M and ‖Et‖ ≤ Mfor all t ≥ T .Note that ‖Et‖ ≤M implies E(t) ≤M .The basic reproduction number [2] for the model isR0 =βΛµd(µ+ d)∫ ∞0k(a)e−(d+δ+r)ada.For all values of the parameters, there is a disease-free equilibrium P0 = (S0, 0)where S0 = Λ/d. For R0 ≤ 1, P0 is the only equilibrium. For R0 > 1, there is aunique endemic equilibrium P ∗ = (S∗, E∗) whereS∗ =S0R0=ΛdR0and E∗ =Λµ+ d(1−1R0).Note that while we write an equilibrium of (3) as a point (S¯, E¯) ∈ R2, more formally,an equilibrium point is a point (S˜, E˜) ∈ R≥0 × Y∆ satisfying S˜ = S¯ and E˜(θ) = E¯for all θ ≤ 0. The equilibrium solution is given by (S(t), Et) = (S˜, E˜) ∈ R≥0 × Y∆for each t. Related to this is an equilibrium of (1) for which S(t), E(t), I(t) and R(t)are constant functions and for which i(t, a) = i¯(a) = µE¯e−(d+δ+r)a is independentof time t.Theorem 3.2. If R0 < 1, then all solutions converge to the disease-free equilibrium,which is locally asymptotically stable.As with many finite dimensional models, ifR0 is larger than one, then the disease-free equilibrium attracts disease-free states and repels states for which disease ispresent. Let a˜ = inf{a :∫∞ak(σ)dσ = 0}. For a system with a truly infinite delay,we have a˜ = ∞, whereas, for a system with a bounded distributed delay, we have0 < a˜ <∞.For a state (S˜, E˜) ∈ R≥0 × Y∆, we say that disease is present if E˜(−a) > 0 forsome a ∈ [0, a˜). Recall that elements of Y∆ are continuous. Thus, if E˜ is positive atsome point, then E˜ is positive on an interval about that point. If disease is presentfor (S˜, E˜), then the solution of (3) with initial condition (S˜, E˜) will satisfy E(t) > 0for some t > 0. If E˜ does not satisfy the given condition (i.e. E˜(−a) = 0 for all606 C. CONNELL MCCLUSKEYa ∈ [0, a˜)), then the solution of (3) will have E(t) identically zero for t ≥ 0, and willconverge to P0. For a solution for which disease is present for the initial condition,we say the disease is initially present.Theorem 3.3. Suppose R0 > 1. Then the disease-free equilibrium is unstable andthe endemic equilibrium is locally asymptotically stable. Furthermore, the system ispersistent; that is, there exists η > 0 such that for any solution for which the diseaseis initially present, we havelim inft→∞S(t) ≥ η and lim inft→∞E(t) ≥ η.Remark 1. In [16], it is implicitly understood that a˜ = ∞ meaning that thesystem has a true infinite delay. However, for a bounded distributed delay, whichgives a˜ <∞, the proofs in [16] still hold, as do the new results of this paper.4. Global stability for R0 > 1. Let X(t) = (S(t), Et) be a solution of equation(3) for which disease is initially present. It is shown in the proof of Theorem 6.1of [16] that the semi-flow induced by equation (3) has properties that imply theexistence of a global compact attractor (see Theorem 3.4.6 of [7]). Combined withTheorem 3.1 and Theorem 3.3, it follows that the ω-limit set Ω of X is non-empty,compact, and invariant. It follows that Ω is the union of orbits of equation (3).That is, if (S˜, E˜) ∈ R≥0× Y∆ is an omega limit point of X , then there is a solutionthrough (S˜, E˜) such that every point on the solution is in Ω.Lemma 4.1. Suppose R0 > 1 and Z(t) = (φ(t), ϕt) is a solution to equation (3)that lies in Ω. Then η ≤ φ(t) ≤M and η ≤ ϕ(t) ≤M for all t ∈ R.Proof. Fix ǫ > 0 and T ∈ R, and let Z˜ = Z(T ) = (φ(T ), ϕT ). Then Z˜ ∈ Ω is anomega limit point of X . Thus, there exists a sequence {tn} that increases to infinitysuch that X(tn)→ Z˜.Then S(tn)→ φ(T ). By Theorem 3.1 and Theorem 3.3, we have η− ǫ ≤ S(tn) ≤M for large n, and so the same inequalities apply to φ(T ). Also, 0 ≤ |E(tn)−ϕ(T )| ≤‖Etn − ϕT ‖, which goes to 0 as n → ∞. Thus, since η − ǫ ≤ E(tn) ≤ M for largeenough n, the same is true for ϕ(T ).Because the choice of T was arbitrary, as was the choice of ǫ > 0, the desiredresult follows for all t ∈ R.Theorem 4.2. Suppose R0 > 1 and Z(t) = (φ(t), ϕt) is a solution to equation (3)that lies in Ω. Then Z converges to the endemic equilibrium; that is,limt→∞(φ(t), ϕ(t)) = (S∗, E∗) .Proof. We begin by normalizing. Let s(t) = φ(t)/S∗, x(t) = ϕ(t)/E∗ and xt =ϕt/E∗. Thends(t)dt=ΛS∗− µβE∗s(t)∫ ∞0k(a)e−(d+δ+r)ax(t− a)da− ds(t)dx(t)dt= µβS∗s(t)∫ ∞0k(a)e−(d+δ+r)ax(t − a)da− (µ+ d)x(t).(4)GLOBAL STABILITY WITH INFINITE DELAY 607The endemic equilibrium for (4) is p∗ = (s∗, x∗) = (1, 1). Thus, by evaluating bothsides of (4) at p∗, we have0 =ΛS∗− µβE∗∫ ∞0k(a)e−(d+δ+r)ada− d0 = µβS∗∫ ∞0k(a)e−(d+δ+r)ada− (µ+ d).(5)Letf(y) = y − 1− ln y,and letUs(t) = f(s(t))Ux(t) = αxf(x(t))U+(t) =∫ ∞0α(a)f(x(t − a))da,whereαx =E∗S∗and α(a) = µβE∗∫ ∞ak(σ)e−(d+δ+r)σdσ.We will study the behaviour of the Lyapunov functionalU(t) = Us + Ux + U+.We note that αx is positive, as is α(a) for each a ∈ [0, a˜). The function f hasdomain R>0 and range R≥0. We also note that f has only one extreme value,which is the global minimum: f(1) = 0. Thus, U(t) ≥ 0 with equality if and onlyif s(t) = x(t) = 1 and x(t − a) = 1 for almost all a ∈ [0, a˜). Lemma 4.1 implies Uis well-defined; that is, U+ is finite for all t.For clarity, we calculate the derivatives of each of Us, Ux and U+ separately andthen combine them to get dUdt. Also, instances of s(t) and x(t) will be written as sand x, respectively.dUsdt=(1−1s)dsdt=s− 1s(ΛS∗− µβE∗s∫ ∞0k(a)e−(d+δ+r)ax(t− a)da− ds).Subtracting the right-hand side of the first equation of (5) givesdUsdt=s− 1s(µβE∗∫ ∞0k(a)e−(d+δ+r)a (1− sx(t− a)) da+ d (1− s))= −d(s− 1)2s+ µβE∗∫ ∞0k(a)e−(d+δ+r)a(1− sx(t− a)−1s+ x(t− a))da.(6)In calculating dUxdt, we use the second equation of (5) to replace (µ + d) with theintegral, obtainingdUxdt= αx(1−1x)(µβS∗s∫ ∞0k(a)e−(d+δ+r)ax(t− a)da− (µ+ d)x)=E∗S∗(1−1x)µβS∗∫ ∞0k(a)e−(d+δ+r)a (sx(t− a)− x) da= µβE∗∫ ∞0k(a)e−(d+δ+r)a(sx(t− a)− x−sx(t− a)x+ 1)da.(7)608 C. CONNELL MCCLUSKEYWe now calculate the derivative of U+(t).dU+dt=ddt∫ ∞0α(a)f(x(t − a))da=∫ ∞0α(a)ddtf(x(t− a))da= −∫ ∞0α(a)ddaf(x(t− a))da.Using integration by parts, we getdU+dt= −α(a)f(x(t − a))|∞a=0 +∫ ∞0dda(α(a)) f(x(t− a))da.By Lemma 4.1, since the solution Z(t) is in the omega limit set Ω, we have ηE∗≤x(t) ≤ ME∗for all t ∈ R. Thus, f(x(t − a)) is bounded above and below. Then,noting that 0 ≤ α(a) = µβE∗∫∞ak(σ)e−(d+δ+r)σdσ ≤ µβE∗∫∞ae−(d+δ+r)σdσ =µβE∗(d+δ+r)e−(d+δ+r)a → 0, it follows that lima→∞ α(a)f(x(t − a)) = 0. Also, at a = 0we get α(a)f(x(t − a)) = α(0)f(x(t)), and sodU+dt= α(0)f(x(t)) +∫ ∞0dda(α(a)) f(x(t− a))da.Filling in for α(0), evaluating the derivative ddaα(a) = −µβE∗k(a)e−(d+δ+r)a, andthen combining the two resulting integrals givesdU+dt= µβE∗∫ ∞0k(a)e−(d+δ+r)a(f(x(t))− f(x(t− a)))da= µβE∗∫ ∞0k(a)e−(d+δ+r)a(x− lnx− x(t− a) + ln x(t− a))da.(8)Adding equations (6), (7), and (8), we obtaindUdt= −d(s− 1)2s− µβE∗∫ ∞0k(a)e−(d+δ+r)aC(a)da,whereC(a) = −2 +1s+sx(t− a)x+ lnx− lnx(t − a)=(1s− 1 + ln s)+(sx(t− a)x− 1− (ln s+ lnx(t − a)− lnx))= f(1s)+ f(sx(t − a)x)≥ 0.Thus, dUdt≤ 0 with equality if and only if s(t) = 1 and x(t− a)/x(t) = 1 for almostall a ∈ [0, a˜). It follows that U(t) is a non-increasing function that is bounded belowby zero, and therefore limt→∞ U(t) exists.Next, we show that limt→∞ s(t) = 1. To do this, we first note thatdUdt≤ −g(t) ≤0 where g(t) = d (s(t)−1)2s(t) . Suppose that s(t) does not converge to 1. Then thereexist ǫ > 0 and a sequence {tn} that increases to infinity such that g(tn) ≥ ǫ for eachn. Note that the bounds on Z given by Lemma 4.1 imply that the derivative dsdtisbounded, and so there exists τ > 0 such that g(t) ≥ ǫ2 for t ∈ In = (tn − τ, tn + τ).Then, we have dUdt≤ − ǫ2 for all t ∈ ∪In, which is a set of infinite measure. Hence,GLOBAL STABILITY WITH INFINITE DELAY 609U decreases to −∞, which contradicts the fact that U is bounded below. Thus, s(t)must converge to 1.Finally, we show that limt→∞ x(t) = 1. To do this, let y(t) = s(t)+αxx(t). Thendydt=dsdt+ αxdxdt=ΛS∗− ds− αx(µ+ d)x=ΛS∗+ µs− (µ+ d)ySince s(t) converges to 1, this is an asymptotically autonomous ordinary differentialequation for which solutions of the limiting equation go to a hyperbolic equilib-rium. Thus, limt→∞ y(t) =1µ+d(ΛS∗+ µ). Using (5), it follows that limt→∞ x(t) =limt→∞1αx(y(t)− s(t)) = 1.Since limt→∞(s(t), x(t)) = (1, 1), it follows that limt→∞(φ(t), ϕ(t)) = (S∗, E∗),completing the proof.Theorem 4.3. If R0 > 1, then all solutions of equation (3) for which the diseaseis initially present converge to the endemic equilibrium; that is,limt→∞(S(t), E(t)) = (S∗, E∗) .Proof. Let Z(t) be a solution in Ω, the omega limit set of X . By Theorem 4.2, Z(t)converges to the endemic equilibrium P ∗. Since Ω is closed, we have P ∗ ∈ Ω and soX gets arbitrarily close to P ∗. By Theorem 3.3, P ∗ is locally asymptotically stableand therefore X converges to P ∗.We note that the results here include systems with bounded distributed delay.Acknowledgments. The author thanks Jianhong Wu for discussion that assistedwith the main result.REFERENCES[1] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differentialequations, Funkcial. Ekvac., 31 (1988), 331–347.[2] O. Diekmann, J. A. P. Heesterbeek and J. 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Naito, “Functional Differential Equations with Infinite Delay,”Springer-Verlag, Germany, 1991.[10] A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models withdelays, Math. Biosci., 209 (2007), 51–75.[11] Andrei. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemicmodels, Math. Med. and Biol., 21 (2004), 75–83.610 C. CONNELL MCCLUSKEY[12] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR andSEIR epidemiological models with nonlinear incidence, Math. Biosci. and Eng., 1 (2004),57–60.[13] C. Connell McCluskey, Lyapunov functions for tuberculosis models with fast and slow pro-gression, Math. Biosci. and Eng., 3 (2006), 603–614.[14] C. Connell McCluskey, Global stability for a class of mass action systems allowing for latencyin tuberculosis, J. Math. Anal. Appl., 338 (2008), 518–535.[15] J. Pru¨ss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamicsof prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225–235.[16] G. Ro¨st and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,Math. Biosci. and Eng., 5 (2008), 389–402.Received September 15, 2008; Accepted December 23, 2008.E-mail address: ccmcc8@gmail.com

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Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay

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