# Competitive Coexistence in a Two-Strain Epidemic Model with a Periodic Infection Rate

Authors: Chentong Li; Yingying Zhang; Yicang Zhou
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this article, we study the global dynamical behavior of a two-strain SIS model with a periodic infection rate. The positivity and boundedness of solutions are established, and the competitive exclusion conditions are given for the model. The conditions for the global stability of the disease-free equilibrium and persistence of the model are obtained. The conditions of coexistence in this model are also found. Finally, the conditions of uniqueness of the solution are proved.


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## Body

## 1. Introduction

Since the first work [1] about the mathematical epidemic model published, there were plenty of results of infectious diseases in modeling and dynamics. These epidemic models usually include two important parameters, the infection rate and recovery rate. Many of the epidemic models focus on the disease eradication and persistence. Among these models, some are applied successfully on infectious disease forecast and control [2–6]. In the analysis of the epidemic model, it is found that infectious diseases often fluctuate over time and exhibit periodic behavior [7–9]. This fluctuation is also observed in the real data [10, 11]. The influenza data from CDC [10] show that the number of influenza cases per week oscillates with a period between two peaks of one year and measles data [11] illustrate the period of two years. The periodic behavior of the incidence of many infectious diseases is caused by the influence of temperature and humidity on virus [12, 13]. In [14, 15], the authors introduced the epidemic models that had periodic coefficients to illustrate the periodical phenomenon. An epidemic SIS model with a periodic infection rate was first considered by Hethcote [15]. Furthermore, Dietz [14] considered SIR and SEIR models with a periodic infection rate.The causative agents of many diseases are represented by multiple genetically distinct variants. Early autonomous multistrain models suggested that competitive exclusion is the only possible result of the competition of many strains [16]. However, these models disregard many mechanisms such as coinfection [17], mutation [18], cross immunity [19], and periodicity [20], which were proposed as possible mechanisms that could support diversity of causative agents. Castillo-Chavez et al. [21] established a sexually transmitted disease model with two competing strains and found the conditions of coexistence equilibrium and its global stability. The interesting work finished by Nuño et al. [19] considered a two-strain influenza model with cross immunity, and their simulation results showed that there might be up to four coexistence equilibria for their model. Another result about multiple-strain model was finished by Martcheva [20]. In this work, the author introduced a nonautonomous multistrain epidemic model without susceptible individuals and found the coexistence and persistence conditions of this model. All of these results also show that the analysis of multiple-strain model can extend our knowledge about the mechanism of the multiple-strain coexistence and competitive exclusion.In this paper, we investigate a two-strain SIS model with a periodic infection rate(1)dSdt=λ−β1tSI11+S−β2tSI21+S+γ1I1+γ2I2−dS,dI1dt=β1tS1+S−k1I1,dI2dt=β2tS1+S−k2I2,where S is the number of susceptible individuals and Iii=1,2 is the number of individuals that are infected by the disease strain i. 0<βit<Mi=1,2 is the infection rate of strain i, and it is a periodic function with the period T. “M” here is the upper bound of βiti=1,2. d>0 is the death rate of susceptible individuals. ki=γi+d+αi>0, where γi>0 is the recovery rate of the infected individuals of strain i, i=1,2. di>d>0 is the death rate of the infected individuals of strain i, and αi>0 is the disease introduced death rate.One of the most important concepts about the dynamics of epidemic models is the basic reproductive numberR0, which represents the expected number of secondary cases produced by a typical infected individual in a fully susceptible population [22, 23]. According to the definition, it is easy to conclude that an epidemic will never occur when R0<1, and instead it will occur if R0>1. In this manuscript, we introduce two basic reproductive numbers R01 and R02 for our two-strain model and use these two values as the thresholds to analyze the dynamic behaviors of model (1).This paper is organized as follows: InSection 2, we analyze the positivity and boundedness of model (1), as well as the global stability of the disease-free equilibrium. The competitive exclusion conditions are also gained in this section. In Section 3, we study the coexistence and stability of T-periodic positive solution of this model. In Section 4, the conditions of uniqueness of the solution are found. Finally, the numerical simulations are done to illustrate our results.


## 2. The Global Stability of the Disease-Free Equilibrium and the Competitive Exclusion

We first give the positivity and boundedness of model (1).Theorem 1.
Solutions of model (1) with positive initial conditions are positive and bounded.Proof.
Assume that the initial condition isS0,I10,I20 with S0≥0,I10≥0,I20≥0, then we have(2)I1t=exp∫0tβ1τSτ1+Sτ−k1dτI10≥0,I2t=exp∫0tβ2τSτ1+Sτ−k2dτI20≥0.
The solutionsI1t and I2t are nonnegative, and they are positive when I10≠0 and I20≠0.
Next, we prove the positivity ofSt with positive initial values. Assuming the contrary and letting t0>0 be the first time such that St0=0, by the first equation of model (1), we have(3)dStdtt=t0=λ+γ1I1t0+γ2I2t0>0,which implies St<0 for t∈t0−ε,t0 and sufficient small ε>0. This contradicts that t0 is the first time such that St0=0. It follows that St>0 for t>0.
Next, we prove the boundedness. LetNt=St+I1t+I2t and α, we have(4)dNtdt=λ−dNt−α1I1t−α2I2t≤λ−dNt.
If the initial condition0≤N0≤λ/d holds, then by the comparison theorem(5)Nt≤λd−λd−N0e−dt≤λd.
This inequality shows that the solution of model (1) is bounded.
This completes the proof.Remark 1.
In the rest of this paper, we assume thatS0,I10,I20∈D, where D=S,I1,I2S≥0,I1≥0,I2≥0,0≤S+I1+I2≤λ/d.
Model (1) has a disease-free equilibrium E0=λ/d,0,0=SE,0,0. So, we define the basic reproductive number of strain i as R0i=μiSE/ki, where μiSE=1/T∫0TβitSE/1+SEdt,i=1,2. Then, we have the following theorem.Theorem 2.
For model (1), if R01<1 and R02<1, then the disease-free equilibrium E0 is globally asymptotically stable.Proof.
The solution of model (1) satisfies(6)Iit=Ii0exp∫0tβiτSτ1+Sτ−kidτ,i=1,2.
For any givent≥0, there exists an integer number n and a real number s such that t=nT+s, where 0≤s<T and n≥0. The inequality St≤Nt≤λ/d implies that(7)Iit≤Ii0exp∫0nTβiτSE1+SE−kidτexp∫nTnT+sβiτSE1+SE−kidτ=Ii0expkiR0i−1nexpSE1+SE∫0sβiτdτ−ki,which implies that limt⟶∞Iit=0; that is, for any ε>0, there exists a Ti>0, such that Iit<ε. Let T0=maxT1,T2 and β0=max0≤t≤Tβ1t+β2t. If t>T0, the first equation of (1) satisfies(8)dSdt≥λ−β0ε−dS.
The comparison theorem implies that(9)St≥λ−β0εd−λ−β0εd−S0e−dt.
SinceSt≤Nt≤λ/d, we have limt⟶∞St=λ/d.
The limitslimt⟶∞St=λ/d and limt⟶∞Iit=0i=1,2 show that E0 is globally attractive.
The Jacobin matrix at the disease-free equilibriumE0=SE,0,0 is(10)J=−dγ1−β1tSE1+SEγ2−β2tSE1+SE0β1tSE1+SE−k1000β2tSE1+SE−k2.
The three characteristic multipliers are(11)e−Td,e∫0Tβ1tSE/1+SE−k1dt,e∫0Tβ2tSE/1+SE−k2dt.
The conditionR0i<1i=1,2 implies that if e∫0TβitSE/1+SE−kidt<1i=1,2, then we claim that the disease-free equilibrium is globally asymptotically stable.We give the competitive exclusion conditions of model (1).Theorem 3.
(i) Ifβ1t/β2t<k1/k2 hold, then all solutions of model (1) with S0,I10,I20∈D satisfies limt⟶∞I1t=0. (ii) If β1t/β2t>k1/k2 hold, then all solutions of model (1) with S0,I10,I20∈D satisfies limt⟶∞I2t=0.Proof.
To prove statement (i), we letm=k1/k2 and consider the function(12)VS,I1,I2=I1I2−m.
We have(13)dVS,I1,I2dt=β1t−mβ2tS1+SVS,I1,I2.
The conditionβ1t/β2t<m implies that(14)β1t−mβ2tSt1+St<0,which follows(15)I1tI2t−m=I1t0I2t0−mexp∫0tβ1τ−mβ2τSτ1+Sτdτ.
Let(16)D0=S,I1,I2dVS,I1,I2dt=0,S,I1,I2∈D.Then D0=S,I1,I2I1=0 is an invariant set of model (1). By Lasalle’s principle, we know that all solutions of model (1) in D satisfy limt⟶∞I1t=0.
Using similar argument, we can obtain statement (ii).These two conditions are usually called the principle of competitive exclusion [24, 25].


## 3. Coexistence

This section discusses the persistence of model (1).Theorem 4.
For model (1), if R01>1 and R02>1, then there exists a positive constant δ∗>0, such that for at least one strain It satisfies limsupt⟶∞It≥δ∗.Proof.
AsSE=λ/d is a global attractor for dS/dt=λ−dS, we can choose small enough δ>0 that for the system,(17)dSdt=λ+γ1+γ2−β1tS1+S−β2tS1+Sη−dS.
If0<η<δ, then ∀ε>0. There exists a time T that when t>T, St−SE<ε. Without loss of generality, from the conditions R01=1/k1T∫0Tβ1tSE/1+SE>1 and R02=1/k2T∫0Tβ2tSE/1+SE>1, we can choose δ∗>0 and small ε that, for any t>T, St>SE−ε, 1/k1T∫0Tβ1tSE−ε/1+SE−ε>1, and 1/k2T∫0Tβ2tSE−ε/1+SE−ε>1.
We prove by contradiction. Assume both of these two strainsI1 and I2 satisfy limsupt⟶∞I1t<δ∗ and limsupt⟶∞I2t<δ∗. Without loss of generality, we assume I1t<δ∗ and I2t<δ∗.
As a consequence, for allt≥T, there hold(18)I1′t≥β1tSE−ε1+SE−ε−k1I1,I2′t≥β2tSE−ε1+SE−ε−k2I2,and it is easy to see that limt⟶∞I1t=+∞ and limt⟶∞I2t=+∞, which lead to a contradiction. Hence, the conclusion holds.Moreover, ifβ2t/β1t=k2/k1=m hold for all t≥0, then it is easy to get that(19)dI2dI1=mI2I1.Therefore,(20)I2=QI1m,where Q is a constant which is determined by the initial values I10,I20. Then, the dynamical behavior of model (1) is equivalent to that of the following model:(21)dSdt=λ+γ1−β1tS1+SI1+Qγ2−mβ1tS1+SI1m−dS,dI1dt=β1tS1+S−k1I1.We can get the following theorem about the above model.Theorem 5.
IfR01>1, then there exists a positive constant δ>0; then (1) all solutions of model (21) with positive initial conditions will satisfy liminft⟶∞I1t≥δ, and model (21) will admit at least one positive periodic solution.Proof.
SinceR01=μ1SE/k1>1, there exists a small enough η>0 such that μ1SE−η>k1. Let us consider the following equation:(22)dxdt=λ−β1tδ−mQβ1tδm−dx,and the solution of this equation (22) is(23)xt=SE+C0e−dt−δ+mQδm∫0tβ1τedτ−tdτ,where C0 is a constant. As 0<β1t<M, the following inequalities hold:(24)SE−δ+mQδmMd≤limt⟶∞xt≤SE.
Thus, we can fix a small enough numberδ=δ∗>0 such that limt⟶∞xt>SE−η.
Suppose, by contradiction, thatlimsupt⟶∞I1t<δ∗. Without loss of generality, we can assume that I1t<δ∗, it follows 0≤I1t<δ∗. By the first equation of model (21), we have the following inequality:(25)S′t≥λ−β1tδ∗−mQβ1tδ∗m−dS.
Thus, there exists a timeT∗; for all t>T∗, we have St>SE−η. As a consequence, for all t≥T, there holds(26)I1′t≥βtSE−η1+SE−η−kI1.
SinceμSE−η>k, it is easy to see that limt⟶∞I1t=+∞, which leads to a contradiction. Hence, limsupt⟶∞I1t≥δ∗.
Define(27)X=R+2,X0=R+×intR+,∂X0=X/X0,where R+=xx≥0 and intR+=xx>0. Then, assume ut,S0,I10 is the solution of model (21) and pS0,I10=uT,S0,I10 is the Poincare map.
E0 is globally attractive in ∂X0 for p, and E0 is isolated invariant sets in X with WsE0∩X0=∅. Clearly, every orbit in ∂X0 converges to E0. By Theorem 1.3.1 in [26], for a stronger repelling property of ∂X0, we conclude that p is uniformly persistent with respect to X0,∂X0. Thus, Theorem 3.1.1 in [26] implies the uniform persistence of the solutions of model (21) with respect to X0,∂X0. By Theorem 1.3.6 in [26], p has a fixed point S^0,I1^0∈X0. Then S^0∈R+ and I1^0∈intR+. We further claim that S^0∈R+/0. Suppose S^0=0, then we have λ+γ1I1^0+γ2QI1m^0=0, which is a contradiction. So from model (21), we can get ut,S^0,I1^0∈intR+2,∀t>0. Then S^0,I1^0 is a component-wise positive fixed point of p. Thus, S^t,I1^t is a positive T-periodic solution of model (21).Remark 2.
It is easy to conclude that when the condition∀t,β2t/β1t=k2/k1=m and R01>1 hold, there exist at least one persistent solution St,I1t,andI2t with respect to I2t=QI1tm and Q≥0. Moreover, when Q=0, model (21) becomes the plane system that I2t=0.
Then, we consider a more special case that the infection rate of model (21) satisfies β1t=b0+εcost, where ε>0 is a small number. When R01>1 and ε=0, model (21) becomes an autonomous model(28)dSdt=λ+γ1−b0S1+SI1+Qγ2−mb0S1+SI1m−dS,dI1dt=b0S1+S−k1I1.
The positive equilibriumE1=S∗,I∗ for model (28) satisfies(29)b0S∗1+S∗=k1,γ−b0S∗1+S∗I∗+Qγ2−mb0S∗1+S∗I∗m=dS∗−λ.
Asb0S∗/1+S∗=k1>γ1, mb0S∗/1+S∗=k2>γ2 and R01>1 implies dS∗−λ>0, so there exists unique positive equilibrium E1=S∗,I∗ for model (28). It is easy to check the Jacobin matrix at equilibrium E1 is(30)−b0I∗1+S∗2−mQb0I∗m1+S∗2−dγ1−b0S∗1+S∗+mQγ2−mb0S∗1+S∗I1∗m−1b0I∗1+S∗20,and its two eigenvalues λ1,λ2 satisfy(31)λ1+λ2=−b0I∗1+S∗2−mQb0I∗m1+S∗2−d<0,λ1λ2=b0I∗1+S∗2γ1−b0S∗1+S∗+mQγ2−mb0S∗1+S∗I1∗m−1>0,when R01>1, which implies that the real parts of λ1 and λ2 are negative. So the unique equilibrium E1 is asymptotically stable.
The right-hand side of model (21) fS,I,μ can be illustrated as(32)fS,I,b1=λ+γ1−b0S1+SI1+Qγ2−mb0S1+SI1m−dSb0S1+S−k1I1+ε−costSI11+S−QmcostSI1m1+ScostSI11+S.
It is easy to seefS,I,ε is analytic with S,I∈R+ and ε∈R, and when ϵ=0, model (21) becomes (28). Then by Theorem 1.1 in Chapter 14 in [27], there exists a T-periodic solution St,It near equilibrium E1 for small ε, and by Theorem 1.2 in Chapter 14 in [27] , this solution St,It is asymptotically stable.
Thus, we have the following theorem.Theorem 6.
For model (21), if R01>1 and β1t=b0+εcost, where ε>0 and small, then there exists a asymptotically stable positive 2π-periodic solution St,It for model (21).


## 4. Uniqueness of the Solution on the Surface

In this section, we mainly talk about the uniqueness of the solution of model (21), and the solution is on the surfaceI2=QI1m. We consider the model(33)dSdt=λ+γ1−βtS1+SI+Qγ2−mβtS1+SIm−dS,dIdt=βtS1+S−kI,where βt=b0+εcost and b0=8,λ=1,γ1=γ2=1,d=1,k=2,Q=2,andm=2.Theorem 7.
Whenε is small enough, model (5) exhibits unique 2π-periodic solution.Proof.
Assume system (33) has a solution(34)Spt=S0t+εS1t+ε2S2t+ε3S3t+…,Ipt=I0t+εI1t+ε2I2t+ε3I3t+….
Using (34) in model (35), and equating the powers of ε, we get the following equations:(35)dS0tdt=λ+γ1−b0S0t1+S0tI0t+Qγ2−mb0S0t1+S0tI0tm−dS0t,dI0tdt=b0S0t1+S0t−kI0t,dSjtdtdIjtdt=JSjtIjt+FjtGjt,where Fj and Gj are periodic functions of the period 2π, and they are only relevant to S0t,I0t,…,Sj−1t,Ij−1t, and(36)J=−2b0QI0t21+S0t2−b0I0t1+S0t2−dγ1−b0S0t1+S0t+2Qγ2−2b0S0t1+S0tI0tb0I0t1+S0t2b0S0t1+S0t−k.
We setb0=8,λ=1,γ1=γ2=1,d=1,k=2,andQ=2 and obtain(37)S0t=13,I0t=13,
The fundamental matrixΦt of the linear homogeneous differential equation system of system (35) is(38)Φt=ϕ11t,ϕ12tϕ21t,ϕ22t,where(39)ϕ11t=e−7/4tcos1/4t23−723e−7/4tsin1/4t2323,ϕ12t=−1223e−7/4tsin1/4t2323,ϕ21t=623e−7/4tsin1/4t2323,ϕ22t=e−7/4tcos1/4t23+723e−7/4tsin1/4t2323.
The inverse matrix ofΦt is Ψt:(40)Ψt=ψ11t,ψ12tψ21t,ψ22t,where(41)ψ11t=123723sin1/4t23+23cos1/4t23e−7/4tcos1/4t232+sin1/4t232,ψ12t=1223sin1/4t2323e−7/4tcos1/4t232+sin1/4t232,ψ21t=−623sin1/4t2323e−7/4tcos1/4t232+sin1/4t232,ψ22t=123−723sin1/4t23+23cos1/4t23e−7/4tcos1/4t232+sin1/4t232.
We obtain the solution of system (35) (42)SjtIjt=Φtc1jc2jΨt∫0tΨτFjτGjτdτ.
By periodicity, we obtain(43)c1jc2j=Φ0−Φ2π−1Φ2π∫02πΨτFjτGjτdτ.
After simple calculation, we getdetΦ0−Φ2π−1≈1≠0. Therefore, we obtain the unique existence of the periodic solution of period 2π of model (33).


## 5. Numerical Simulation

Numerical simulations are performed to illustrate the dynamic behaviors of model (1) and (21).Figures1(a) and 1(b) show the solution of (21) under the conditions R01=1.5 and R01=2. From these two figures, we can see that there exists one stable periodic solution of model (21) under different values of R0, and when the value of R0 becomes larger, the mean value of It becomes larger.Figure 1
The simulated solution of model (21) under the conditions (a) R01=1.5 and (b) R01=2.
(a)(b)Figures2(a) and 2(b) show the results of Theorem 3. When the competitive exclusion condition holds, one strain will drive another into extinction.Figure 2
The simulated solution of model (1) under the conditions (a) β1t/k1=3+sint and β2t/k2=2.85+0.95sint and (b) β1t/k1=2.72+0.91sint and β2t/k2=3+sint.
(a)(b)Figures3(a) and 3(b) show the coexistence of model (1) under the condition β1t/k1=β2t/k2. Figure 3(a) is under the initial condition S0,I10,I20=2,1,2 and Figure 3(b) is S0,I10,I20=2,2,1.Figure 3
The simulated solution of model (1) with β1t/k1=β2t/k2=3+sint and the initial conditions (a)S0,I10,I20=2,1,2 and (b)S0,I10,I20=2,2,1.
(a)(b)Figure4(a) is the solution of model (1) with β1=3+sint,β2=3+cost,λ=1,d=1,k1=1,γ1=0.1,k2=1,γ2=0.1, and R01=R02. Figure 4(b) is the solution of model (1) with β1=3+sint,β2=3.5+sint,λ=1,d=1,k1=1.2,γ1=0.1,k2=1.401,γ2=0.1, R01=1.25, R02=1.2491, and R01≠R02. These two figures illustrate that the basic reproductive numbers of different strains cannot determine the coexistence of model (1). One strain can drive another into extinction even with the same reproductive number, and the two strains can coexist with different basic reproductive numbers.Figure 4
(a) The simulated solution of model (1) with β1=3+sint,β2=3+cost,λ=1,d=1,k1=1,γ1=0.1,k2=1,andγ2=0.1. (b) The stable simulated solution of model (1) with β1=3+sint,β2=3.5+sint,λ=1,d=1,k1=1.2,γ1=0.1,k2=1.401,andγ2=0.1.
(a)(b)


## 6. Discussion

In this paper, we study the dynamical behaviors of a two-strain SIS model with a periodic infection rate and get its persistence, competitive exclusion, and coexistence conditions. We gain the global stability conditions of the disease-free equilibriumE0 and establish the competitive exclusion condition of the two strains. The coexistence and uniqueness are also discussed and proved. Our results lead to a new insight into the mechanism of two strains interaction and provide a new approach to investigate the inference of the periodic infection rate on the coexistence of two strains. It is worth to mention that the analysis in this article can be applied to the n-strain epidemic model with a periodic infection rate and obtain similar results.



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