An SEIR Epidemic Model with Relapse and General Nonlinear Incidence Rate with Application to Media Impact

Abstract

The aim of this paper is to extend the incidence rate of an SEIR epidemic model with relapse and varying total population size to a general nonlinear form, which does not only include a wide range of monotonic and concave incidence rates but also takes on some neither monotonic nor concave cases, which may be used to reflect media education or psychological effect. By application of the novel geometric approach based on the third additive compound matrix, we focus on establishing the global stability of the SEIR model. Our analytical results reveal that the model proposed can retain its threshold dynamics that the basic reproduction number completely determines the global stability of equilibria. Our conclusions are applied to two special incidence functions reflecting media impact.

Appendix: The Third Additive Compound Matrix

The novel geometric approach based on the third additive compound matrix, in what follows, is briefly presented. Li et al. [22, 38] originally developed the theory, and Cheng et al. [6] applied it to study a generalized cholera epidemiological model. Let us consider the following dynamical system

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{d\mathrm {X}(t)}{dt}=\mathrm {F}(\mathrm {X}), \end{array} \end{aligned}$$

(6.1)

where $\mathrm {F}$: $\mathcal {D}\mapsto \mathbb {R}^n$ is a $\mathbb {C}^1$ function and $\mathcal {D}\subset \mathbb {R}^n$ is a simply connected open set. Here, $\mathrm {X}(t,\ \mathrm {X}_0)$ denotes the solution of (6.1) with the initial condition $\mathrm {X}(0)=\mathrm {X}_0$. Suppose that

$\mathrm{{(A1)}}$ :

there exists a compact absorbing set $\mathcal {K}\subset \mathcal {D}$;

$\mathrm{{(A2)}}$ :

(6.1) has only one positive equilibrium $\mathrm {X}_*\in \mathcal {D}$.

The third additive compound matrix $\mathcal {A}^{[3]}$ for a $4\times 4$ matrix $\mathcal {A}= (a_{ij})$ is given by

$$\begin{aligned} \mathcal {A}^{[3]}=\left( \begin{array}{ccccc} a_{11}+a_{22}+a_{33}\ \ &{} a_{34}\ \ &{} -a_{24}\ \ &{} a_{14}\\ a_{43}\ \ &{} a_{11}+a_{22}+a_{44}\ \ &{} a_{23}\ \ &{} -a_{13}\\ -a_{42}\ \ &{} a_{32}\ \ &{} a_{11}+a_{33}+a_{44}\ \ &{} a_{12}\\ a_{41}\ \ &{} -a_{31}\ \ &{} a_{21}\ \ &{} a_{22}+a_{33}+a_{44} \end{array} \right) . \end{aligned}$$

For a solution $\mathrm {X}(t,\ \mathrm {X}_0)$ of any initial value problem of (6.1), the linearized system is of the following form

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{d\mathrm {Y}(t)}{dt}=\mathrm {J}(\mathrm {X}(t,\ \mathrm {X}_0))\mathrm {Y}(t), \end{array} \end{aligned}$$

(6.2)

and the associated third compound system reads

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{d\mathrm {U}(t)}{dt}=\mathrm {J}^{[3]}(\mathrm {X}(t,\ \mathrm {X}_0))\mathrm {U}(t), \end{array} \end{aligned}$$

(6.3)

where $\mathrm {J}^{[3]}$ stands for the third compound matrix of the Jacobian matrix $\mathrm {J}$ for (6.1).

Lemma 6.1

[6] Let (A1) and (A2) be true and there exists a Lyapunov function $V(\mathrm {X}, \mathrm {U})$, a function $\psi (t)$, and positive constants $h_1$, $h_2$ and l such that

$\mathrm{{(1)}}$ :

$h_1\mid \mathrm {U}\mid \le V(\mathrm {X}, \mathrm {U})\le h_2\mid \mathrm {U}\mid$, $h_1\le \psi (t)\le h_2$,

$\mathrm{{(2)}}$ :

$D_+V\le (\psi (t)-l)V$,

where the total derivative $D_+V$ is taken along the direction of (6.3), then the interior equilibrium $\mathrm {X}_*$ of system (6.1) is GAS.