Global dynamics of a SEIR model with varying total population size
Introduction
Studies of epidemic models that incorporate disease caused death and varying total population have become one of the important areas in the mathematical theory of epidemiology and they have largely been inspired by the works of Anderson and May (see 1, 2). Most of the research literature on these types of models assume that the disease incubation is negligible so that, once infected, each susceptible individual (in the class S) instantaneously becomes infectious (in the class I) and later recovers (in the class R) with a permanent or temporary acquired immunity. A compartmental model based on these assumptions is customarily called a SIR or SIRS model. Many diseases, however, incubate inside the hosts for a period of time before the hosts become infectious. Models that are more general than the SIR or SIRS types need to be studied to investigate the role of incubation in disease transmission. Using a compartmental approach, one may assume that a susceptible individual first goes through a latent period (and is said to become exposed or in the class E) after infection before becoming infectious. The resulting models are of SEIR or SEIRS type, respectively, depending on whether the acquired immunity is permanent or otherwise.
We assume the population has a homogeneous spatial distribution and the mixing of hosts follow the law of `mass action'. More specifically, we assume that the local density of the total population is a constant though the total population size N(t)=S(t)+E(t)+I(t)+R(t) may vary with time. Here S(t),E(t),I(t) and R(t) denote the sizes of the S,E,I and R classes at any time t, respectively. The per capita contact rate λ, which is the average number of effective contacts with other individual hosts per unit time, is then a constant. A fraction I(t)/N(t) of these contacts is with infectious individuals and thus the average number of relevant contacts of each individual with the infectious class is λI(t)/N(t). The total number of new infections at a time t is given by λI(t)S(t)/N(t). This form of mixing term has been used in the literature under different names. Busenberg and van den Driessche [3] call it proportionate mixing, a term which they attribute to Nold [4]; Mena-Lorca and Hethcote [5] call it standard incidence; de Jong et al. [6] call it true mass-action incidence. This incidence form should not be confused with another form βI(t)S(t) that is often called the simple mass-action incidence (it is called pseudo mass-action incidence in [6]). The recovered hosts are assumed to acquire a permanent immunity so that they will not become susceptible again. This is a technical assumption aimed to reduce the complexity of the mathematical analysis, but is nonetheless a plausible approximation in the case of many viral infections such as in measles, smallpox and rubella. The rate of removal ϵ of individuals from the exposed class is assumed to be a constant so that 1/ϵ can be regarded as the mean latent period. In the limiting case, when ϵ→∞, the latent period is negligible and a SEIR model reduces to a SIR model.
The vital dynamics include exponential natural death with rate constant d and exponential birth with rate constant b. We assume that the infectious individuals suffer a disease-caused mortality with a constant rate α. The equation for the total population size is N′=(b−d)N−αI. If b=d and α=0, N(t) remains a constant and can be normalized to 1. This leads to a SEIR model with constant total population and bilinear incidence rate, which is known to possess a sharp threshold σ=λϵ/(ϵ+b)(γ+b), sometimes called the contact number (see [7]). If σ⩽1 the disease-free equilibrium (1,0,0,0) is globally asymptotically stable, namely, the disease dies out irrespective of the initial configuration; if σ>1, there exists a unique endemic equilibrium which is globally asymptotically stable in the interior of the feasible region and the disease persists, if it initially exists, at an endemic equilibrium state. We refer the reader to 7, 8 for references on SEIRS models with constant total population and to [9] for the proof of the global stability of a unique endemic equilibrium of a SEIR model. In the general case, N(t) may vary with time and the dynamical behavior of the model become more intricate; there is an interplay between the dynamics of the disease and that of the total population. This interplay has been studied in earlier SIR and SIRS models (see 1, 3, 5, 10, 11), and is one of the primary concerns in the present paper.
Research on epidemic models of SEIR or SEIRS type with varying total populations are scarce in the literature. To the authors' knowledge, the present paper is the first that gives a rigorous treatment of the global stability of an unique endemic equilibrium for the fractions of sub-populations and the global dynamics of (S(t),E(t),I(t),R(t)). Several new methods are employed in the present paper to overcome mathematical difficulties that are not present in SIR models. The existence and uniqueness of the endemic equilibrium P* is established without solving explicitly for its coordinates. This makes the verification of the Routh–Hurwitz stability condition a very technical matter. We develop in Lemma 5.1 a new criteria of linear stability using ideas from multilinear algebra. This new criteria is then used to show the local asymptotical stability of P*. The most challenging task is the proof of the global stability of P*. Epidemic models of this type are notorious for the fact that the method of Lyapunov functions has rarely worked for the proof of the global stability of the endemic equilibrium. In the present paper, the global stability is proved by employing the theory of monotone dynamical systems together with a stability criterion for periodic orbits of multidimensional autonomous systems due to Muldowney [12]. This approach is also used in [9] for a SEIR model with constant total population.
Greenhalgh [8] recently studied a class of SEIRS models that incorporate density dependence in the contact rate and natural death rate. Global stability of the disease-free equilibrium and the existence, uniqueness and local asymptotic stability of the endemic equilibrium are proved in [8]. The global stability of the endemic equilibrium, when it is unique, is unresolved in [8]. The model studied in the present paper is a special case of those considered in [8]. We prove the global stability of the unique endemic equilibrium for our model under the restriction α<ϵ. In addition, we present a new method for proving the local stability of the unique endemic equilibrium. Compared with the traditional approach of using Routh–Hurwitz conditions (see, for example, [8]), our method is less technical and more manageable for systems of large number of equations. Our treatment of the disease-free equilibrium and the existence and uniqueness of the endemic equilibrium is standard for models of this type. Similar methods are also used in [8]. Since our model is simpler than those in [8], we are also able to obtain a complete stability analysis for the disease-free equilibrium P0 when σ⩽1, while in [8], this analysis is done only for σ<1.
Cook and van den Driessche [13] introduced and studied SEIRS models with two delays. Greenhalgh [8] studied Hopf bifurcations in models of SEIRS type with density dependent contact and death rates. A recent survey on SEIRS models is given in [8].
Section snippets
Model formulation
A population of size N(t) is partitioned into subclasses of individuals who are susceptible, exposed (infected but not yet infectious), infectious and recovered, with sizes denoted by and R(t), respectively. The sum E(t)+I(t) is the total infected population. Our assumptions on the dynamical transfer of the population are demonstrated in the diagramThe parameter b>0 is the rate for natural birth and d>0 that of natural death. It is assumed that all newborns are susceptible and
The disease-free equilibrium and its global stability
Let σ=λϵ/(ϵ+b)(γ+α+b). Following [5], σ will be called the modified contact number; see Section 7for more discussion of σ. In the following result, the stability of P0 should be understood in the sense of Lyapunov.
Theorem 3.1. The disease-free equilibrium P0=(1,0,0) of (2.4) is globally asymptotically stable in Γ if σ⩽1; it is unstable if σ>1, and the solutions of (2.4) starting sufficiently close to P0 in Γ move away from P0 except that those starting on the invariant s-axis approach P0 along
Existence and uniqueness of an endemic equilibrium
Global stability of P0 in Γ when σ⩽1 precludes the existence of equilibria other than P0; the study of endemic equilibria is restricted to the case σ>1. We remark that σ>1 implies λ>α. This relation will be assumed throughout this and the next two sections.
The coordinates of an equilibrium satisfyand also , and . Adding the above equations leads towhich gives the following range of i*
Local asymptotic stability of the endemic equilibrium
Throughout this section, the relation λ>α is assumed since we are concerned only with the interior equilibrium P*.
To show the asymptotic stability of the equilibrium P*, we use the method of first approximation. The Jacobian matrix of (2.4) at a point P=(s,e,i)∈Γ isWe prove that the matrix is stable, namely, all its eigenvalues have negative real parts. This is routinely done by verifying the Routh–Hurwitz conditions. Since the explicit
Global stability of the endemic equilibrium
In this section, we establish that all solutions of (2.4) in converge to P* when σ>1, which, together with the local stability of P*, implies that P* is globally asymptotically stable in . Note that the relation λ>α holds when σ>1 and thus (2.4) is competitive from Section 2. The following strong Poincaré–Bendixson property follows from Theorem 2.1. Its proof is the same as that of Theorem 4.2 of [9] and thus is omitted.
Theorem 6.1. Suppose that σ>1. Then any non-empty compact omega limit
The dynamics of the population sizes
We now turn to the dynamics of (S(t),E(t),I(t),R(t)) and N(t)=S(t)+E(t)+I(t)+R(t), which are governed by systems (2.1) and (2.2). The fact that R does not appear in the first three equations in (2.1) allows us to study the equivalent systemin its feasible regionIf b<d and α⩾0, or if b⩽d and α>0, (7.1) implies that N(t)→0 monotonically as t→∞ for all solutions with E(0)+I(0)>0, namely, when the disease is
Discussion
This paper has considered a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death, so that the total population size may vary in time. The incidence rate is of the proportionate mixing type frequently used in the literature. The asymptotic behavior of this multidimensional model has been determined as a function of the basic parameters of the system.
The homogeneity of the vector field of the model suggests the way of analyzing the global dynamics; the
Acknowledgements
The research of M.Y.L. is supported in part by NSF grant DMS-9626128 and by a Ralph E. Powe Junior Faculty Enhancement Award from the Oak Ridge Associated Universities. Research of J.R.G. is supported in part by the Mississippi State University Biological and Physical Sciences Research Institute. This work was done when J.K. visited the Department of Mathematics and Statistics at Mississippi State University under the support of a Hungarian Eötvös Fellowship. He also acknowledges the support of
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