The impact of nonlinear relapse and reinfection to derive a stochastic threshold for SIRI epidemic model

https://doi.org/10.1016/j.chaos.2020.109897Get rights and content

Highlights

  • we introduce a stochastic SIRI epidemic model with nonlinear relapse.

  • We give sufficient conditions for extinction and persistence of the disease

  • We also study the existence of a stationary distribution and the ergodicity of the solutions.

  • We obtain a stochastic threshold for the disease.

Abstract

In the present work, we introduce a stochastic SIRI epidemic model with nonlinear relapse. We give sufficient conditions for extinction and persistence of the disease. We also study the existence of a stationary distribution and the ergodicity of the solutions. As a special case of our results that under some conditions on noise intensities, we obtain the threshold Rβ for the disease. Finally, we provide some computer simulations to illustrate our theoretical findings.

Introduction

The 21st century has already been marked by major epidemics. Old diseases cholera, plague and yellow fever have returned, and new ones have emerged SARS, pandemic influenza, MERS, Ebola and Zika. Therefore, there is a lot of interest in both understanding diseases and mathematically modelling the spread of epidemics. Mathematical modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak. Epidemics are commonly modeled by using compartmental models where the population is divided into several classes, such as susceptible, infected and removed individuals. The first complete mathematical model was introduced by Kermack and Mckendrick in 1927 [12]. Based on this work, many researches have studied the spread of infectious diseases in a population by compartmental models such as SIS, SIR, SIRS, SEIR or SVIS, see for instance [5], [9], [14], [22]. For some diseases such as tuberculosis and herpes, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class [4]. These types of diseases can be modeled by SIRI models. Tudor (1990) was the first to construct and study a compartmental epidemic model with relapse, which incorporates bilinear incidence rate and constant total population. Sanchez et al. [19] introduced a SIRI epidemic model with nonlinear relapse to describe the dynamics of drinking behaviors generated from contacts between individuals in shared drinking environments. The model is given by the following set of nonlinear differential equations:dS(t)=(μμS(t)βS(t)I(t))dt,dI(t)=((μ+α)I(t)+βS(t)I(t)+δR(t)I(t))dt,dR(t)=(μR(t)+αI(t)δR(t)I(t))dt,1=S(t)+I(t)+R(t), where the population is divided in three classes: S(t), moderate and occasional drinkers, I(t), problem or heavy drinkers, and temporarily recovered, R(t). The parameter β denotes the effective contact rate (infection rate). That is, βS(t)I(t) denotes the rate of transitions from S to I, the result of the frequency dependent interactions between individuals in the classes S and I, μ denotes the recruitment rate of susceptible corresponding to births and immigration, δ denotes the relapse rate. That is, δR(t)I(t) denotes the rate of transitions from R to I, the result of the frequency-dependent interactions between R and I, this nonlinear process assumes that R and I individuals (as well as S individuals) share the same environments, and α denotes the recovery (treatment or education) rate. The basic reproduction number of system (1.1) is given byRα=βμ+α.

Rα is a dimensionless quantity (ratio or number) that can be interpreted as the number of I-individuals “generated” in a population of primarily S-individuals sharing a common environment. Sanchez et al. [19] have shown that the dynamics of system (1.1) depends also on the size of the initial population and the ratiosRp=δβ[1Rα],Rc=δβ[11+μβ2μβμδ].

  • If Rα > 1, then the I-class becomes established. That is, the persistence of a regular drinking class over time is guaranteed.

  • Whenever Rc < Rα < 1 and Rp < 1 or whenever Rα < Rc < 1 the I-class becomes extinct.

  • Whenever Rc < Rα < 1 and Rp > 1 whether or not the I-class becomes established is a function of the initial size of the class of I-individuals.

However, in real life, there is a lot of randomness and hazardous events. This is why stochastic calculus is considered as a solid approach to describe this real random events seen in nature. Exciting results were found thanks to the consideration of stochasticity in epidemic models and population dynamics. The technique of parameter perturbation has been used by a number of researchers [5], [9], [15], [16]. The case of a color noise was investigated by Gray et al. and Settati and Lahrouz [10], [20]. Both of them have accomplished a detailed analysis on asymptotic behavior of an epidemic model under a finite regimes-switching. Motivated by those previous works, in this paper, we consider fluctuations in the environment (see [1], [2], [3], [6], [7], [17]), which are assumed to manifest themselves as fluctuations in parameters β and δ involved in the previous deterministic model (1.1), so thatββ+σ1B˙1(t)andδδ+σ2B˙2(t), where B1(t) and B2(t) are two independent brownian motions with intensities σ1 > 0, σ2 > 0 respectively. Incorporating the above perturbations in the deterministic system (1.1), we get the following stochastic model{dS(t)=(μμS(t)βS(t)I(t))dtσ1S(t)I(t)dB1(t),dI(t)=((μ+α)I(t)+βS(t)I(t)+δR(t)I(t))dt+σ1S(t)I(t)dB1(t)+σ2R(t)I(t)dB2(t),dR(t)=(μR(t)+αI(t)δR(t)I(t)dtσ2R(t)I(t))dB2(t).

From system (1.3), it is obvious thatd(S+I+R)=[μμ(S+I+R)]dt.

With initial condition S(0) > 0, I(0) > 0, R(0) > 0, and S(0)+I(0)+R(0)=1, one can easily show thatS(t)+I(t)+R(t)=1forallt0.

Denote the meaningful domain for system (1.3) byΔ={(x,y,z)R+;x+y+z=1}.

This paper is organized as follows. In Section 2, we introduce some needed results and lemmas which will be used in the following analysis. In Section 3, we investigate the disease extinction of our model (1.3). In Section 4, we explore the persistence in mean. In Section 5, we obtain the existence of a stationary distribution and the ergodicity of solutions for system (1.3). In Section 6, we introduce some examples and numerical simulations to illustrate the analytical results. In the last section, we provide a brief discussion and summary of main results.

Section snippets

Preliminaries

Throughout this paper, let (Ω,F,{Ft}t0,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions (i.e., it is right continuous and increasing while F0 contains all P-null sets). Let X(t) be a regular time-homogeneous Markov process in Rn described by the stochastic differential equationdX(t)=b(X)dt+r=1kσr(X)dBr(t), and the diffusion matrix is defined as followsA(x)=(aij(x)),aij(x)=k=1pσki(x)σkj(x).

We introduce the operator L associated with (1.3) as

Stochastic Disease-Free Dynamics

When studying the behavior of a stochastic epidemic model, the special case when the threshold equal to one still always untreated. Here, we propose an approach based on stopping times to overcome this difficulty. Let us now introduce the two positive numbers.Rδ=δμ+α+12σ22,Rβ=βμ+α+12σ12.

Theorem 3.1

For any initial values (S(0), I(0), R(0)) ∈ Δ, if one of the followings assumptions

(C1)Rδ=1,δβσ22andβσ12

and

(C2)Rβ=1,βδσ12andδσ22

holds, then we getlimtI(t)=0a.s..

Proof

Let 0 < ε < I(0). Define the stopping times

Persistence in mean of the disease

The most interesting topics in studying epidemic modeling concern the extinction and the persistence. The first one has been treated in section 3. In this section, we will show that the disease is persistent in mean.

Theorem 4.1

If Rβ>1andδσ22, then for any given initial value (S(0), I(0), R(0)) ∈ Δ, the solution of system (1.3) has the property that

  • (i)

    limtinf1t0tS(u)duμμ+βa.s.,

  • (ii)

    limtinf1t0tI(u)duμ(β(μ+α)12σ12)β(β12σ12)a.s.,

  • (iii)

    limtinf1t0tR(u)duμα(β(μ+α)12σ12)β(β12σ12)(μ+δ)a.s..

Proof

(i) From the first

Stationary Distribution and Positive Recurrence

In the study of population models, we are also interested in when the disease will persists and prevails in a population. In the deterministic models, the problem can be resolved by showing that the endemic equilibrium is a global attractor or is globally asymptotically stable. But for stochastic models such system (1.3), there is no endemic equilibrium. In this section, we will show the positive recurrence and the existence of stationary distribution for our system (1.3), provided that Rβ>1,

Simulations

In this section, we will introduce some numerical simulations to illustrate our main results. Using the Milstein method developed in [11], we get the following discretization equation of system (1.3):{Sk+1=Sk+[μμSkβSkIk]Δtσ1SkIkΔtτkσ122SkIk(τk21)Δt,Ik+1=Ik+[(μ+α)Ik+βSkIk+δRkIk]Δt+σ1SkIkΔtτk+σ122SkIk(τk21)Δt+σ2RkIkΔtξk+σ222RkIk(ξk21)Δt,Rk+1=Rk+[μRk+αIkδβRkIk]Δtσ2RkIkΔtξkσ222RkIk(ξk21)Δt, where τk, ξk (k=1,2,...) are N(0, 1)-distributed independent random variables.

Example 1

In system (1.3),

Conclusion

This paper is concerned with the dynamics of a stochastic SIRI epidemic model with nonlinear relapse, by perturbing consecutively β and δ. We show sufficient conditions on extinction and persistence of the disease. In addition, we prove the existence of an ergodic stationary distribution to the model (1.3). As consequence of Theorem 3.1, Theorem 3.2, and 4.1, the quantity Rβ is a threshold of system (1.3), if βδσ12 and δσ22, which is not the case for Rδ, for the reason that, we cannot prove

Declaration of Competing Interest

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.

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