Assuming a general distribution for the sojourn time in the in- fectious
class, we consider an SIS type epidemic model formulated as a scalar integral
equation. We prove that the endemic equilibrium of the model is globally
asymptotically stable whenever it exists, solving the conjecture of Hethcote
and van den Driessche (1995) for the case of nonfatal diseases

Nakata, Yukihiko

Rost, Gergely

English

arXiv

Nakata Yukihiko

Röst Gergely

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Differential and Integral Equations Volume 31, Numbers 3-4 (2018), 161–172GLOBAL STABILITY OF AN SIS EPIDEMIC MODELWITH A FINITE INFECTIOUS PERIODYukihiko NakataDepartment of Mathematics, Shimane University1060 Nishikawatsu-cho, Matsue, Shimane, 690-8504, JapanGergely RöstBolyai Institute, University of SzegedH-6720 Szeged, Aradi vértanúk tere 1., Hungary(Submitted by: Glenn Webb)Abstract. Assuming a general distribution for the sojourn time in theinfectious class, we consider an SIS type epidemic model formulated asa scalar integral equation. We prove that the endemic equilibrium ofthe model is globally asymptotically stable whenever it exists, solvingthe conjecture of Hethcote and van den Driessche (1995) for the case ofnonfatal diseases.1. IntroductionIn the paper [7], Hethcote and van den Driessche formulate an SIS typeepidemic model and analyze its qualitative properties. The model has a dis-ease related death rate, which affects the size of the total population. Thesojourn time of the infectious states is assumed to follow a general proba-bility density function, though the authors elaborate the case of a constantinfectious period. In Theorem 5.1 in the paper [7], the authors show that theendemic equilibrium is asymptotically stable when the disease related deathrate is zero and subsequently mention global stability of the endemic equilib-rium as an open problem. See also [8] for a study of a similar SIS epidemicmodel, where the assumed population demography is different from [7].The problem of global stability does not seem to be fully solved up tonow. In the recent paper [9] the authors analyze monotonicity of the semiflowinduced by the model proposed in [7]. It is shown that the solution semiflow ismonotone with respect to the initial function when a key parameter, namelythe basic reproduction number R0, is greater than two. Then the author ofAMS Subject Classifications: 37N25, 45D05.Accepted for publication: April 2017.161162 Yukihiko Nakata and Gergely Röst[10] proves that many solutions, but not all solutions, converge to the stableendemic equilibrium if R0 > 2. In those papers, to derive those results, aconstant infectious period is assumed in the model. Note that in [9, 10] thelong time behavior of the solutions for 1 < R0 ≤ 2 is not studied even forthe case of constant delay.The aim of this paper is to prove that the endemic equilibrium is indeedglobally asymptotically stable for the epidemic model proposed by Hethcoteand van den Driessche in [7, 8] for the case of nonfatal diseases. We provethat all positive solutions tend to the endemic equilibrium if R0 > 1, thusthe endemic equilibrium is always globally asymptotically stable. Our proofallows any probability function which the sojourn time of the infectious statefollows.This paper is organized as follows. In Section 2, we introduce an SISepidemic model, corresponding to the model considered in [7, 8] for the caseof nonfatal diseases. The model is formulated as a scalar integral equationdescribing time evolution of the infectious population. In Section 3, we proveglobal stability of the disease free equilibrium. In Section 4, we show thatthe endemic equilibrium is asymptotically stable analyzing the correspondingcharacteristic equation. Here, an elementary proof for persistence of thesolution is also given. We then prove global attractivity of the endemicequilibrium. In Section 5, we discuss our results.2. Endemic modelFollowing [7], we introduce the SIS epidemic model of which we analyze itsglobal dynamics. Denote by F(a) the probability that infective individualsare infective up to infection-age a (time elapsed since infection). From thebiological interpretation, F : [0, h]→ [0, 1] is nonincreasing withF(0) = 1, F(h) = 0,where h < ∞ is the maximum infectious period. Without loss of generalitywe assume that F(a) > 0 for 0 < a < h.Let S(t) and I(t) be the proportions of the susceptible population and theinfective population at time t, respectively, so that1 = S(t) + I(t), (2.1)holds for any t. The density of infective individuals at time t with respect toinfection-age a is given asi(t, a) = βS(t− a)I(t− a)F(a)e−µa, (2.2)Global stability of an endemic equilibrium 163where µ is the natural mortality rate and β is the constant transmissioncoefficient. Note that the density of newly infective individuals appearing inthe host population at time t isi(t, 0) = βS(t)I(t).Integrating both sides of (2.2) we obtain the following renewal equationI(t) = βˆ h0S(t− a)I(t− a)F(a)e−µada, (2.3)see also the integral equation (2.2) in [7].We scale the time so that the maximum infectious period becomes 1 andthen subsequently we recall that (2.1) holds for t ≥ −1 (so I(t) ≤ 1 fort ≥ −1). Then we get a scalar renewal equation for the proportion of infectivepopulation:I(t) = βˆ 10(1− I(t− a)) I(t− a)F(a)e−µada, (2.4)from (2.3). Since F is a monotone function, F is a bounded variation functionsuch that−ˆ 10dF(a) = F(0)−F(1) = 1,holds. The proportion of individuals who obtain susceptibility per unit timeat time t is given as−βˆ 10S(t− a)I(t− a)e−µadF(a),where the integral is Riemann-Stieljes integral. Differentiating the equation(2.4) one also obtains the following delay differential equationddtI(t) = β(1−I(t))I(t)+βˆ 10(1−I(t−a))I(t−a)e−µadF(a)−µI(t). (2.5)Equations (2.4) and (2.5) withF(a) ={1, 0 ≤ a < 10, a = 1,(2.6)can be found in [7, 8] for the case of no disease induced death rate.Denote by C([−1, 0],R) the Banach space of continuous functions mappingthe interval [−1, 0] into R equipped with the sup-norm. The initial function164 Yukihiko Nakata and Gergely Röstis chosen from the subset of continuous functions:Y := {φ ∈ C ([−1, 0] , [0, 1]) |φ(0) = G(φ)} ,where G : C ([−1, 0] , [0, 1])→ R is defined asG(φ) := βˆ 10(1− φ(−a))φ(−a)F(a)e−µada.The initial function for (2.4), and for (2.5), is given asI(θ) = ψ(θ), θ ∈ [−1, 0] , (2.7)with ψ ∈ Y . We introduce a standard notation It : [−1, 0]→ R+ defined viathe relationIt(θ) = I(t+ θ) for θ ∈ [−1, 0].It is straightforward to see that the set Y is forward invariant under thesemiflow induced by (2.4) with the initial condition (2.7) i.e., It ∈ Y, t > 0.3. Global stability of the disease free equilibriumThe basic reproduction number is easily computed asR0 = βˆ 10F(a)e−µada.We here prove global stability of the disease free equilibrium when R0 ≤ 1holds, by using similar ideas as in the proof of Theorem 4.2 in [7] for the casethat F is given by (2.6).Theorem 1. Let us assume that R0 ≤ 1 holds. Then the disease free equi-librium is globally asymptotically stable in Y .Proof. Consider a sequence {Mn}∞n=0 determined byMn = R0Mn−1(1−Mn−1),with M0 = 14 . It holds that Mn monotonically decreases and thatlimn→∞Mn = 0,if R0 ≤ 1 holds. From the equation (2.4), we haveI(t) ≤ 14R0 ≤ 14, t ∈ [0, 1] .One can see that I(t) ≤ Mn, t ∈ [n, n + 1], thus, we get limt→∞ I(t) = 0.Stability of the disease free equilibrium follows from the estimation above.Global stability of an endemic equilibrium 1654. Global stability of the endemic equilibriumAn endemic equilibrium emerges if R0 > 1 holds. In Theorem 5.1 in [7] theendemic equilibrium is shown to be asymptotically stable when F is given by(2.6), see also Theorem 4 in [8]. In the following proposition we show thatthe endemic equilibrium is asymptotically stable for any F .Let us write pˆ to denote the constant function in Y satisfying pˆ(θ) = p forθ ∈ [−1, 0]. We then define Y+ := Y \{0ˆ}. In the following, to generate apositive solution, we assume ψ ∈ Y+.Proposition 2. Let us assume that R0 > 1 holds. There exists a uniqueendemic equilibrium Iˆ∗ ∈ Y+, whereI∗ = 1− 1R0. (4.1)The endemic equilibrium is asymptotically stable.Proof. Let R0 > 1 hold. The endemic equilibrium satisfies the followingequation:1 = (1− I)βˆ 10F(a)e−µada = R0 (1− I) ,so we get (4.1).We apply the principle of linearized stability established in [1] for Volterrafunctional equations. The Fréchet derivative of G : C ([−1, 0] , [0, 1]) → Revaluated at the endemic equilibrium Iˆ∗ ∈ Y+ can be computed asDG(Iˆ∗)φ = (1− 2I∗)βˆ 10φ(−a)F(a)e−µada.Thus, the characteristic equation is1 = (1− 2I∗)βˆ 10e−λaF(a)e−µada, λ ∈ C. (4.2)DenoteF˜(a) = F(a)e−µa.Using the partial integrationˆ 10e−λaF˜(a)da = 1λ(1 +ˆ 10e−λadF˜(a)),one can deduce a priori bounds for the roots of the characteristic equation.We now show that the real part of all roots of (4.2) is negative. First, it canbe easily seen that the characteristic equation (4.2) does not have a root with166 Yukihiko Nakata and Gergely Röstpositive real part for small I∗ > 0. Suppose that there exists a root λ = iωwith ω > 0. We get the following two equations1 = (1− 2I∗)βˆ 10F˜(a) cos (ωa) da. (4.3a)0 =ˆ 10F˜(a) sin (ωa) da. (4.3b)Integrating by parts we have0 =ˆ 10F˜(a) sin(ωa)da = 1ω([− F˜(a) cos(ωa)]10+ˆ 10cos(ωa)dF˜(a))=1ω(1 +ˆ 10cos(ωa)dF˜(a)),thus, ˆ 10cos (ωa) dF˜(a) = −1,follows. Noting that(ˆ 10cos (ωa) dF˜(a))2+( ˆ 10sin (ωa) dF˜(a))2 ≤ 1,holds, we see ˆ 10sin (ωa) dF˜(a) = 0,holds. Since one has thatˆ 10F˜(a) cos (ωa) da = 1ω([F˜(a) sin (ωa)]10−ˆ 10sin (ωa) dF˜(a))= − 1ωˆ 10sin (ωa) dF˜(a),we finally obtain ˆ 10F˜(a) cos (ωa) da = 0.Therefore, we get a contradiction to the equality in (4.3a). Consequently,in the complex plane the characteristic roots do not cross the imaginaryaxis. Next, we show persistence of the solution when R0 > 1.Global stability of an endemic equilibrium 167Proposition 3. Let R0 > 1 holds. Thenlim inft→∞ I(t) > 0.Proof. We can choose a q ∈ (0, 1) such that qR0 > 1 holds. Assume thatthere is a solution with ψ(0) > 0 such that limt→∞ I(t) = 0. Then there is aT such that for t > T , I(t) < 12 and 1− I(t) > q hold. There is a sequencetn →∞ (n→∞),such thatI(tn) = min{I(t) : t ∈ [tn − 1, tn]},and limn→∞ I(tn) = 0. Then, for tn > T + 1, we haveI(tn) = βˆ 10I(tn − a)(1− I(tn − a))F(a)e−µada,and from the monotonicity of the quadratic map x 7→ x(1− x) on [0, 12 ], weobtainI(tn) ≥ βˆ 10I(tn)(1− I(tn))F(a)e−µada = R0I(tn)(1− I(tn)).Since we have 1 − I(tn) > q, dividing by I(tn), we find the contradiction1 > qR0. We now introduce the notationsI = lim inft→∞ I(t), I = lim supt→∞I(t).From Proposition 3, one can easily see that 0 < I ≤ I ≤ 1. The prooffor the global attractivity of the endemic equilibrium is divided into twocases: 1 < R0 ≤ 2 and 2 < R0. Using the monotonicity of the quadraticmap x 7→ x(1 − x) on the two intervals [0, 12] and [12 , 1] for the two casesrespectively, we show that I = I = I∗ holds.First, let us assume that 1 < R0 ≤ 2 holds. We define the subspaceY1 :={φ ∈ Y : 0 ≤ φ(θ) ≤ 12, θ ∈ [−1, 0]},and first prove that Y1 attracts all positive solutions.Lemma 4. Let us assume that 1 < R0 ≤ 2. Then I(t) ≤ 12 , t ≥ 0. thus,It ∈ Y1, t > 1 for any ψ ∈ Y+.168 Yukihiko Nakata and Gergely RöstProof. Since (1− x)x ≤ 14 for x ∈ [0, 1], we can estimate the equation (2.4)as I(t) ≤ 14R0 ≤ 12 . Then solution It, t > 1 satisfies0 ≤ It(θ) = I(t+ θ) ≤ 12for θ ∈ [−1, 0]. It is easy to see that I∗ = 1 − 1R0 ≤ 12 , when 1 < R0 ≤ 2 holds. To showglobal attractivity, we use the monotonicity of the quadratic function:0 ≤ x ≤ y ≤ 12→ (1− x)x ≤ (1− y)y.Proposition 5. Let us assume that 1 < R0 ≤ 2. Thenlimt→∞ I(t) = I∗ for any ψ ∈ Y+.Proof. From Proposition 3 and Lemma 4, it holds that 0 < I ≤ I ≤ 12 . Itthen holds thatI ≥ G(Iˆ) = R0I (1− I) ,whileI ≤ G(Iˆ) = R0I(1− I) .Therefore, we get the following inequalityR0 (1− I) ≤ 1 ≤ R0(1− I) ,which implies I = I = I∗ = 1− 1R0 . Next, we assume that R0 > 2 holds. It will be shown that all positivesolutions are attracted to the setY2 ={φ ∈ Y : 12≤ φ(θ) ≤ 1, θ ∈ [−1, 0]}.In Iggidr et al. [9], it is shown that Y2 is an invariant set and that thesemiflow is monotone in Y2 when F is the step function (2.6).Lemma 6. Let us assume that R0 > 2 holds. For any ψ ∈ Y+ there exists τsuch that It ∈ Y2, t > τ.Proof. First let us show that there exists no solution that is bounded by 12from above. To derive a contradiction, suppose that there exists a solutionsuch that I(t) ≤ 12 for any t ≥ 0. Then0 < I ≤ I ≤ 12, (4.4)Global stability of an endemic equilibrium 169holds. From (2.4) one sees that I ≥ G(Iˆ) = R0I(1 − I), which impliesI ≥ I∗ = 1 − 1R0 > 12 . Thus, we obtain a contradiction to (4.4), so thesolution can not stay in Y1 and leaves the set Y1 eventually. Now considera solution that oscillates about 12 . We show that the solution never returnsto the set Y1 once it crosses 12 . To see this we compute the differentiation ofI(t) when I(t) = 12 holds. From (2.5), we getI ′(t) = β (1− I(t)) I(t) + βˆ 10(1− I(t− a)) I(t− a)e−µadF(a)− µI(t)=14β + βˆ 10(1− I(t− a)) I(t− a)e−µadF(a)− 12µ.Since for any nonnegative solutions it holds thatˆ 10(1− I(t− a)) I(t− a)e−µadF(a) ≥ 14ˆ 10e−µadF(a),it followsI ′(t) ≥ 14β +14βˆ 10e−µadF(a)− 12µ=12µ{12βµ(1 +ˆ 10e−µadF(a))− 1}.Sinceβµ(1 +ˆ 10e−µadF(a))= βˆ 10e−µaF(a)da = R0,holds, we obtainI ′(t) ≥ 12µ(12R0 − 1)> 0,which implies that if the solution once crosses 12 then it is bounded belowfrom 12 . Proposition 7. Let us assume that R0 > 2 holds. Thenlimt→∞ I(t) = I∗ for any ψ ∈ Y+.Proof. From Lemma 6 it is sufficient to consider a solution in Y2. Consider asequence {pn}∞n=1 such that I(pn)→ I as n→∞. We apply the fluctuationlemma to the equation (2.5) to get0 = β(1− I)I + β limn→∞ˆ 10(1− I(pn − a)) I(pn − a)e−µadF(a)− µI170 Yukihiko Nakata and Gergely Röst≥ β(1− I)I + β(1− I)Iˆ 10e−µadF(a)− µI= µI{βµ(1 +ˆ 10e−µadF(a))(1− I)− 1}.From Proposition 3, we have I > 0. Thus, 0 ≥ R0 (1− I)− 1, which impliesthat I ≥ I∗. Similarly we can apply the fluctuation lemma for the sequenceof a solution tending to I. Then we obtain I ≤ I∗. Consequently, I ≥ I∗ ≥ Iholds and we obtain the conclusion. Finally, we obtain the following result, combining the results of globalattractivity of the endemic equilibrium established in Propositions 5 and 7with local stability shown in Proposition 2.Theorem 8. Let us assume that R0 > 1 holds. The endemic equilibrium isglobally asymptotically stable in Y+.5. DiscussionIn the paper [7], Hethcote and van den Driessche propose an SIS typeepidemic model with disease induced death rate. The model developed in[7] allows a general description for the distribution of the infectious periodamong individuals. To proceed the mathematical analysis it is then speci-fied as a step function (2.6), implying that for every infective individual theinfectious period is exactly same. The authors in [7] study stability of thedisease free equilibrium and the endemic equilibrium. In Theorem 5.1 in [7],it is shown that the endemic equilibrium is asymptotically stable if there isno disease induced death rate. We solve the open problem mentioned in thepaper: the endemic equilibrium is globally asymptotically stable if there isno disease induced death rate. Moreover, the global stability holds for themodel with any distribution of the infectious period. The equation (2.4) isalso derived from the model considered in [8] for the case of nonfatal diseases.In the first model considered in [8] the authors assume constant recruitmentfor the total population, differently from the model in [7].In the paper [5], Gripenberg studied a similar integral equation describingdisease transmission dynamics motivated by the study in [3]. Conditions forglobal attractivity of the equilibria are obtained. It involves technical con-ditions concerning kernels denoting variable infectivity and distribution forthe sojourn period. The model considered in [3] is also an SIS type epidemicmodel, where individual’s infectivity varies according to the progression ofthe infection-age (the time elapsed since the infection). The authors in [3]Global stability of an endemic equilibrium 171deduce a characteristic equation for an endemic equilibrium and show thatdestabilization of the endemic equilibrium is possible. In those papers [3, 5],the integral equation is formulated in terms of newly infectives per unit timeat time t.Here, we assume that the maximum infectious period is finite. The as-sumption facilitates our analysis as we have the equation with finite delay[1]. The global stability results would be extended to the case that themaximum infectious period is infinite i.e., h = ∞, using the fading memoryspace [2, 6], see also [4, 11] for the treatment of such cases in epidemiologicalmodels.Acknowledgment. We thank the reviewer for comments which improve theoriginal manuscript. The research was initiated during the second author’svisiting to the University of Tokyo on August, 2015 and completed upon thefirst author’s visiting to Bolyai Institute, University of Szeged on February,2016. The first author’s research visiting was supported by JSPS BilateralJoint Research Project (Open Partnership). The first author was supportedby JSPS Fellows, No.268448 of Japan Society for the Promotion of Scienceand JSPS Grant-in-Aid for Young Scientists (B) 16K20976. The secondauthor was supported by European Research Council StG Nr. 259559 andOTKA K109782.References[1] O. Diekmann, Ph. Getto, and M. Gyllenberg, Stability and bifurcation analysis ofVolterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39(2007/08), 1023–1069.[2] O. Diekmann and M. Gyllenberg, Equations with infinite delay: blending the abstractand the concrete, J. Diff. Eq., 252 (2012), 819–851.[3] O. Diekmann and R. Montijn, Prelude to Hopf bifurcation in an epidemic model: anal-ysis of a characteristic equation associated with a nonlinear Volterra integral equation,J. Math. Biol., 14 (1982), 117–127.[4] S. Gourley, G. Röst, and H. R. Thieme Uniform persistence in a model for bluetonguedynamics, SIAM J. Mathematical Analysis, 46 (2014), 1160–1184[5] G. Gripenberg, On some epidemic models, Quart. Appl. Math., 39 (1981/82), 317–327.[6] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial.Ekvac, 21.1 (1978), 11–41.[7] H. W. Hethcote and P. van den Driessche, An SIS epidemic model with variablepopulation size and a delay, J. Math. Biol., 34 (1995), 177–194[8] H. W. Hethcote and P. van den Driessche, Two SIS epidemiologic models with delays,J. Math. Bio., 40 (2000), 3–26.172 Yukihiko Nakata and Gergely Röst[9] A. Iggidr, K. Niri, and E. Ould Moulay Ely, Fluctuations in a SIS epidemic modelwith variable size population, Appl. Math. Comput., 217 (2010), 55–64[10] B. Liu, Convergence of an SIS epidemic model with a constant delay, Appl. Math.Lett., 49 (2015), 113–118[11] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinitedelay, Math. Biosci. Eng., 5 (2008), 389–402

Global stability of an SIS epidemic model with a finite infectious period

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