A Discrete Time Analogue for Coupling Within-host and Between-host Dynamics in Environmentally-driven Infectious Disease∗

WANG Jianpeng,TENG Zhidong

(College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

Abstract: In this paper,we study a discrete coupling within-host and between-host model in environmentally-driven infectious disease by the Non-standard finite difference method.We analyze the decoupling models which are divided into fast system and slow system.In the fast system,The basic properties on the positivity and boundedness of solutions and the existence of the infection-free,infected equilibria are established.By using the linearization methods,the local stability of infection-free equilibria and infected equilibria are established.In the slow system,we also prove the existence of endemic equilibrium and the local stability of the equilibria.

Key words:Within-host dynamics;between-host dynamics;NSFD method;basic reproductive number

0 Introduction

As is well known,viruses have caused the abundant types of epidemic and are alive in almost everywhere on Earth,which infect people,animals,plants,and so on.There are a large number of diseases,for example:In fluenza,Hepatitis,HIV,AIDS,SARS,Ebola,MERS,etc.which are caused by viruses.Therefore,it is important to study viral infection.

In recent years,many authors have established and investigated the various kind of viral infection dynamical models.Many important and valuable results are established and successfully applied to the practical viral infection.We refer to[1-3].

In[4],the authors proposed us a coupled within-host and between-host dynamical continuous time model as follow

whereS,I,E,T,T∗andVdenote the numbers of susceptible individuals,infectious individuals,the level of environment contamination,the density of healthy cells,the density of infected cells and parasite load,respectively.The parameter Λ denotes the recruitment rate,β denotes the infection rate of hosts in a contaminated,µ denotes the host natural mortality rate,θ denotes the rate of contamination,γ denotes the clearance rate,kdenotes the infections rate of cells,m and d denotes the natural mortality and infection-induced mortality rates of infected cells,cdenotes the within-host mortality rate of parasites,the functiong(E)denotes the rate at which an average host is inoculated.

In this paper,we discuss the discrete time analogue of above continuous time models(1)and(2),which is established by using Micken’s non-standard finite difference(NSFD)scheme[5−6],the model is proposed as follow

where system(3)denotes slow time system with the slow timet,and system(4)denotes fast time system with the fast times.However,in slow system(3)there is a fast time termV(s+1),and in fast system(4)there is a slow time termg(E(t+1)).Therefore,systems(3)and(4)form a coupling system by termsV(s+1)andg(E(t+1)).whereS(t),I(t),E(t),T(s),T∗(s)andV(s)denote the numbers of susceptible individuals,infectious individuals,the level of environment contamination at timetands,other parameters have the same biological explains as in model(1)-(2).

This paper is organized as follows.In Section 2,we will introduce some assumptions for functiong(E).Next,we will state and prove some basic results on the positivity and ultimate boundedness of solutions with positive initial conditions for system(4).Furthermore,the existence and the stability of the infection-free,infected equilibria are presented.In Section 3,the existence and stability of endemic equilibria for system(3)also is obtained.

1 The analysis of fast system

For coupling system(3)and(4),functiong(E)is assumed to satisfy the following condition

(H)Functiong(E)is de fined for allE≥0 and is continuously differentiable,satis fiesg(0)=0,g0(E)>0 andg00(E)<0 for allE>0.

From the biological background of system(4),it is assumed that any solution(T(s),T∗(s),V(s))of system(4)satis fies the following initial value

Firstly,on the positivity and boundedness of solutions and the existence of nonnegative equilibria for fast system(4)we have the following lemmas.

Lemma 1The solution(T(s),T∗(s),V(s))of system(4)with initial value(5)is positive for alls∈N+and ultimately bounded.

ProofIt is easy to proof this Lemma,hence we omit it.

We de fine the baseline within-host reproduction number as follows

Lemma 2LetE=0 in system(4).Then system(4)always has infection-free equilibrium,0,0),and whenRv0>1,system(4)has a unique infected equilibriumE∗(T∗,T∗∗,V∗).

ProofLetE=0.It is obvious that system(4)has a unique infection-free equilibriumE0(0,0).For infected equilibriumE∗(T∗,T∗∗,V∗),we have

This shows that,whenRv0>1,an infected equilibriumE∗(T∗,T∗∗,V∗)exists.This completes the proof.

LetE>0 in system(4).If(T∗(E),T∗∗(E),V∗(E))is a nonnegative equilibrium of system(4),then we have

Hence,we further have

andT∗(E)satis fies the following equation:

Hence,equation(7)has always two positive real solutions given by the following form:

Owing toa2>0,we have from(9)for anyE≥0,(E)>0 and(E)<0.

In(8),whenE=0 we obtain

Since(E)>0 and(0)≥T0from(10),we obtain)>T0for allE>0.But,from the first equation of(6),we have

This leads to a contradiction.Since)<0 and(0)≤T0from(11),we obtain(E)<0)≤T0for allE≥0.Therefore,whenE>0,system(4)has a unique positive equilibrium

Furthermore,whenE→0+,from(8)and(11),by calculating we can obtain

Summarizing the above discussions,we finally get the following result.

Lemma 3LetE>0 in system(4).Then fast system(4)always has a unique infected equilibriumE1(T∗(E),T∗∗(E),V∗(E)),and

Next,on the local stability of the infection-free equilibrium and infected equilibrium for fast system(4)we have the following theorems.

Theorem 1LetE=0 in system(4).IfRv0<1,then infection-free equilibriumE0is locally asymptotically stable.IfRv0>1,then free-infection equilibriumE0is unstable.

ProofBy using the linearization methods,the coefficient matrix of the linearization system of system(4)at equilibriumE0is

The characteristic equation ofJ(E0)is

where

By computing,WhenRv0<1,we obtainTherefore,free-infection equilibriumE0is locally asymptotically stable.Whenor λ3is more than 1.This implies that free-infection equilibriumE0is unstable.This completes the proof.

Theorem 2LetE=0 in system(4).IfRv0>1,then infected equilibriumE∗(T∗,T∗∗,V∗)is locally asymptotically stable.

ProofThe proof of this theorem is similar to the next theorem,hence we omit it.

Theorem 3LetE>0 in system(4).IfRv0>1,then infected equilibriumE1(T∗(E),T∗∗(E),V∗(E))is locally asymptotically stable.

ProofIn fact,from the proof of Lemma 3,we can know thatT∗0(E)<0.Hence,for allE>0

For conveniently,letT∗(E)=T∗,T∗∗(E)=T∗∗andV∗(E)=V∗.By using the linearization methods,the coefficient matrix of the linearization system of system(4)at equilibriumE1is

The characteristic equation ofJ(E1)is

According to Jury criterion[7],if we can prove the following conditions(1),(2)and(3),then the modules of all roots of equationg(λ)=0 are less than one.

(1)g(1)>0 and(−1)3g(−1)>0; (2)|a3|<1;

(3)|b0|>|b2|⇔−|b0|

Now,we verify conditions(1),(2)and(3).Since

according to(12)andm+kV∗=we have

WhenRv0>1,we obtaing(1)>0.Furthermore,we easily see

Therefore,condition(1)and(2)hold.Finally,we will prove|b0|+b2>0 and|b0|−b2>0.Since

by the simple computing,we have

According to(12),we further have

Therefore,condition(3)holds.This completes the proof.

2 The analysis of slow system

Now,we consider slow system(3).AssumeRv0>1.We further assume that fast and slow systems(3)and(4)are coupled with fast system(4)being near infected equilibriumE1(T∗(E),T∗∗(E),V∗(E)).Since fast system(4)is locally asymptotically stable in infected equilibriumE1,we can chooseV(t+1)=V∗(E(t+1))in slow system(3).Therefore,we know that system(3)changes into the following form

where

SinceN=S+Iremainsconstant,wecaneliminatetheSequationandgetthefollowingtwo-dimensionalequivalent slow system:

where

Here,we give the baseline reproduction numberRh0for the between-host system:

we also give the reproduction numberRhwhen the two subsystem is coupled:

Based on the reproduction numberRh,we have the following lemma.

Lemma4LetRv0>1.IfRh>1.Thentheslowsystem(13)has atleastone endemic equilibrium

ProofAccording to the system(14),we have

where N is a constant(The total population size is not change).Let

Hence,G(E)is an increasing function between

Therefore,ifF(0)>G(0),which is equivalent toRh>1,there is at least one solution∈(0,1).

This completes the proof.

According to the proof of Lemma 4,we have the following relation:

Owing to

Following,we give the stability of the endemic equilibriumE2.

Theorem 4model(13)is locally asymptotically stable.

ProofBy using the linearization methods,the coefficient matrix of the linearization system of system(14)at equilibriumE2is

The characteristic equation ofJ(E2)is

by calculation we obtain

WhenRh0>we havef(1)>0.According to Jury criterion[7],we obtain that two roots λ1and λ2off(λ)=0 satisfy|λ1|<1 and|λ2|<1.Therefore,endemic equilibriumE2is locally asymptotically stable.