Elsevier

Chaos, Solitons & Fractals

Volume 40, Issue 2, 30 April 2009, Pages 874-884
Chaos, Solitons & Fractals

Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate

https://doi.org/10.1016/j.chaos.2007.08.035Get rights and content

Abstract

In this paper, the SEIR epidemic model with vertical transmission and the saturating contact rate is studied. It is proved that the global dynamics are completely determined by the basic reproduction number R0(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R0(p, q)  1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out. If R0(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists.

Introduction

Horizontal transmission of infection is the transfer of infection through some direct or indirect contact with (for example, through disease vectors such as mosquitos or other biting insects) infected individuals.

Vertical transmission of disease is the passing of an infection to offspring of infected parents. This mode of transmission plays an important role in the spread of diseases. In recent years, the studies of epidemic models that incorporate vertical transmission and disease caused death have become one of the important areas in the mathematical theory of epidemiology and they have largely been inspired by the works of Busenberg and Cooke (see [1], [2]). In our life, many infectious in nature transmit through both horizontal and vertical modes, such as herpes simplex, Chagas’ disease, rubella, hepatitis B, and, most notorious, AIDS [1], [3].

The incidence of a disease is the number of new cases per unit time and plays an important role in the study of mathematical epidemiology. The general form of a population size dependent incidence should be written as βC(N)SNI, where S and I are, respectively, the numbers of susceptibles and infectives at time t, β is the probability per unit time of transmitting the infection between two individuals taking part in a contact, and C(N) is the ‘unknown’ probability for an individual to take part in a contact. Thus, C(N) is usually called the contact rate, and βC(N), which is the average number of adequate contact which is sufficient for transmission of the infection from an infective to a susceptible. In most literatures, the adequate contact rate frequently takes two forms. One is linearly proportional to the total population size N or βN, so that the corresponding incidence is bilinear form βNSNI=βSI, the other a constant λ, the corresponding incidence λSNI is called standard form. When the total population size N is not too large, since the number of contacts made by an individual per unit time should increase as the total population size N increases, the linear adequate contact rate βN would be suitable. But when the total population size is quite large, since the number of contacts made by an infective per unit time should be limited, or should grow less rapidly as the total population size N increases, the linear adequate contact rate βN is not suitable and the constant adequate contact rate λ may be more realistic. Hence, the two adequate contact rates mentioned above are actually two extreme cases for the total population size N being very small and very large, respectively. More generally, it may be reasonable to assume that the adequate contact rate is a function C(N) of the total population size N. There are some reasonable demands on C(N) that it should be a non-decreasing function of N and that C(N)N should be a non-increasing function of N.

We know that many diseases, such as measles, mumps, tuberculosis, AIDS caused by HIV, SARS, etc., have a latent period, a period during which individuals are exposed to a disease but are not yet infectious. From the view point of mathematical modelling, this leads to SEIR or SEIRS epidemic models. These kinds of models have attracted many authors’ attention and a number of papers have been published. For example, Greenhalgh [4] considered SEIR model that incorporates density dependence in the death rate. Cooke and van den Driessche [5] introduced and studied SEIRS model with two delays. Greenhalgh [6] studied Hopf bifurcations in the models of the SEIRS type with density dependent contact rate and the death rate. Li and Muldowney [7] and Li and Muldowney [8] studied the global dynamics of the SEIR models with a non-linear incidence rate and with a standard incidence rate, respectively. Li et al. [9] analyzed the global dynamics of the SEIR model with vertical transmission and a bilinear incidence. Recently, Zhang and Ma [10] considered the global dynamics of the SEIR model with saturating contact rate.

In this paper, we will consider an SEIR epidemic model with vertical transmission and saturating contact C(N). To incorporate vertical transmission in the SEIR model, it is plausible to assume that a fraction of the offsprings of infected hosts (both E and I) are infected at birth, like adult infected hosts, will stay latent before becoming infectious, and hence the infected birth flux will enter the E class. Moreover, in our paper, we use following saturating contact rate, that is C(N)=αN1+bN, then we see that, for N small, C(N)  αN, whereas for N large, C(N)αb. Furthermore, C(N) is non-decreasing and C(N)N is non-increasing. The demographic structure used in this SEIR model has recruitment and deaths, besides, we also consider the disease-related death rate.

This paper is organized as follows. In Section 2, we formulate the model, and the global stability of the disease-free equilibrium is obtained. In Section 3, the existence and unique of the endemic equilibrium Q is proved. In Section 4, we obtain the local stability of the endemic equilibrium Q. The mathematical framework that is used in the proof of the global stability of endemic equilibrium is outlined in Section 5. Section 6 gives a rigorous treatment of the global stability of the endemic equilibrium. Finally we summarize our findings in Section 7.

Section snippets

The model formulation

We study an SEIR epidemic model in which vertical transmission and saturating contact are incorporated based on the above assumption.

The total population size N(t) is divided into susceptible, exposed (in the latent period), infectious, and recovered, with sizes denoted by S(t), E(t), I(t), and R(t), respectively. Let d0 be a non-negative constant and represents the death rate due to disease. The natural birth rate and death rate are assumed to be identical and denoted by μ. β is the

Existence and uniqueness of endemic equilibria

Global stability of Q0 in T excludes the existence of equilibria other than Q0, thus the study of endemic equilibria is restricted to the case R0(p, q) > 1.

From Section 2, the coordinates of an equilibrium Q  T0 satisfyak(N-E-I-R)=(δω-pδ-qk)h(N),kE=δI,γI=R,1-N=dI,from the last three equations we have,E=δkd(1-N),I=1d(1-N),R=γd(1-N).Substituting (3.2) into the first equation in (3.1) givesa[δωN-(δω-kd)]=d(δω-pδ-qk)h(N).Noting that h(N) = 1 + bN, (3.3) becomesF(N)=(a-bd)δωN+bdpδN+bdqkN-(δω-kd)-d(δω-pδ-qk)

Local stability of the endemic equilibrium Q

Theorem 4.1

If R0(p, q) > 1, then the endemic equilibrium Q of (2.3) is locally asymptotically stable.

Proof

The matrix of the linearization of system (2.3) at the equilibrium Q = (E, I, R, N) is given byJ=-(ω-p)-aIh(N)-aIh(N)+aSh(N)+q-aIh(N)ρk-δ000γ-100-d0-1,whereρ=aIh(N)-abINh2(N)+abI(E+I+R)h2(N),aIh2(N)+abINh2(N),=aIh(N).The characteristic equation of J is(λ+1)(λ3+a1λ2+a2λ+a3)=0,wherea1=(1+δ)+(ω-p)+aIh(N)>0,a2=δ+(ω-p)(1+δ)+a(1+δ)Ih(N)+kaIh(N)-aSh(N)-q=δ+(ω-p)(1+δ)+a(1+δ)Ih(N

A geometric approach to global-stability problems

In this section we briefly outline a general mathematical framework for proving global stability, which will be used in Section 6 to prove Theorem 6.2. The framework is developed in the papers of Smith [13], and Li and Muldowney [14], [15]. The presentation here follows that in [14].

Let x  f(x)  Rn be a C1 function for x in an open set D  Rn. Consider the differential equationx=f(x).Denote by x(t, x0) the solution of (5.1) such that x(0, x0) = x0. We take the following two assumptions:

  • (H1) There

Global stability of the endemic equilibrium

Using the relation R(τ) = 1  S(τ)  E(τ)  I(τ) we can reduce (2.2) to the following equivalent system:dSdτ=1-aSIh(N)-pE-qI-S,dEdτ=aSIh(N)+pE+qI-ωE,dIdτ=kE-δI.It can be verified that the regionΓ={(S,E,I)R+3:0S+E+I1},is positively invariant with respect to (6.1). Here R+3 denotes the non-negative cone of R+3 including its lower-dimensional faces. Thus, system (6.1) is a bounded system. Let ∂Γ and Γ0 denote the boundary and the interior of Γ, respectively.

Let P = (S, E, I) be the unique positive

Conclusions

This paper has considered an SEIR epidemic model that incorporates recruitment and exponential natural death, as well as disease-related death, so that the total population size may vary in time. A distinguishing feature of the SEIR model with the vertical transmission considered here is that there is a saturating contact ratec(N)=αN1+bN.The basic reproduction number R0(p, q) define by (2.5) is a sharp threshold parameter which completely determines the globally dynamics of system (2.2) and the

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    Supported by the NSF of China (No. 10371105; No. 10671166) and the NSF of Henan Province (No. 0312002000).

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