Elsevier

Mathematical Biosciences

Volume 175, Issue 2, February 2002, Pages 117-131
Mathematical Biosciences

On the stability properties of a stochastic model for phage–bacteria interaction in open marine environment

https://doi.org/10.1016/S0025-5564(01)00089-XGet rights and content

Abstract

In this paper we extend the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment introduced by Beretta and Kuang [Math. Biosci. 149 (1998) 57], allowing random fluctuations around the positive equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically suggesting that the deterministic model is robust with respect to stochastic perturbations.

Section snippets

Introduction and motivation

In recent years, Beretta and Kuang proposed a first simple mathematical model to describe the epidemics induced by bacteriophages (i.e., infected viruses) in marine bacteria populations such as cyanobacteria and heterotrophic bacteria where the environment is the thermoclinic layer of the sea within which bacteriophages and bacteria are assumed to be homogeneously distributed [1]. The experimental evidence of the bacteriophage infection of marine bacteria can be found, for example, in the

Mathematical properties of the deterministic model

For sake of simplicity (1.2) may be written in dimensionless form by rescaling the variables on the carrying capacity Cs=SC,i=IC,p=PCand considering the dimensionless time τ=KCt. The dimensionless equations aredsdτ=as(1−(i+s))−sp,didτ=sp−li,dpdτ=−sp−mp+bli,wherea=αKC,l=λKC,m=μKCare the dimensionless parameters. The initial condition for system (2.2) may be any point inR+03=(s,i,p)∈R3|s⩾0,i>0,p⩾0.For convenience, in the following, time τ is replaced by t as the dimensionless time.

Let E0=(0,0,0)

The stochastic model

Model (2.2) was just a first attempt in the modelling of the phenomenon. A more sophisticated structure that accounts for the latent period of infected bacteria by the introduction of suitable delay terms was studied in [7] and a Campbell-like model on the same topic has been recently considered in [8]. In this last work, stochastic perturbations were introduced in some of the main parameters involved in the model equations. In this paper, instead, we allow stochastic perturbations of the

Stochastic stability of the positive equilibrium

The stochastic differential system (3.1) can be centred at its positive equilibriumE+=s*=mb−1,i*=as*(1−s*)l+as*,p*=al(l−s*)l+as*(which exists provided that s*<1, i.e., b>b*=m+1), by the change of variablesu1=s−s*,u2=i−i*,u3=p−p*.Since E+ is not globally stable for all b>b* but there exists a bc>b* at which a Hopf bifurcation occurs for increasing b (see Proposition 2.3), it looks a very hard problem to derive, by Lyapunov functions methods, asymptotic stability in probability (or in mean square

Numerical experiments and discussion

In order to confirm the stability results of Section 4 we numerically simulated the solution of the SDEs (3.1). It is worth recalling that in problems involving direct simulations of Ito processes it is important trajectories, that is sample paths, of the approximation be close to those of the Ito process, which leads to the concept of strong solution of a stochastic differential equation.

In this paper the approximate strong solution of the Ito system of SDEs (3.1) with initial condition (s0,i0,

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