Existence of periodic solutions of a periodic SEIRS model with general incidence

Abstract

For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when some condition related to ${\mathcal{R}}_0$ holds. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when ${\mathcal{R}}_0<1$. In particular, our main result generalizes the one in Zhang et al. (2012). We also discuss some examples where our results apply and show that, in some particular situations, we have a sharp threshold between existence and non existence of an endemic periodic orbit.

Introduction

In the sequence of the model introduced by Li and Muldowney in  [1], several works were devoted to the study of epidemic models with a latent class. In these models, besides the infected, susceptible and recovered compartments, an exposed compartment is also considered in order to split the infected population into two groups: the individuals that are infected and can infect others (the infective class) and the individuals that are infected but are not yet able to infect others (the exposed or latent class). This division makes the model particularity suitable to include several infectious diseases like measles and, assuming vertical transmission, rubella  [2]. Additionally, if there is no recovery, the model is appropriate to describe diseases such as Chagas’ disease  [3]. This model can also be used to model diseases like hepatitis B and AIDS  [2]. Even influenza can be modeled by a SEIRS model  [4], although, due to the short latency period, it is sometimes more convenient to use the simpler SIRS formulation  [5]. Mathematically, the existence of more than one infected compartment brings some additional challenges to the study of the model.

In this work we focus on the existence and stability of endemic periodic solutions of a large family of periodic SEIRS models contained in the family of models already considered in  [6]. Namely, we will consider models of the form

$$\{\begin{array}{c}{S}^{\prime }=\Lambda \left(t\right)-\beta \left(t\right)\phi (S,N,I)-\mu \left(t\right)S+\eta \left(t\right)R\\ E^{\prime }=\beta \left(t\right)\phi (S,N,I)-(\mu \left(t\right)+ϵ\left(t\right))E\\ I^{\prime }=ϵ\left(t\right)E-(\mu \left(t\right)+\gamma \left(t\right))I\\ R^{\prime }=\gamma \left(t\right)I-(\mu \left(t\right)+\eta \left(t\right))R\\ N=S+E+I+R\end{array}$$

where $S$, $E$, $I$, $R$ denote respectively the susceptible, exposed (infected but not infective), infective and recovered compartments and $N$ is the total population, $\Lambda \left(t\right)$ denotes the birth rate, $\beta \left(t\right)\phi (S,N,I)$ is the incidence into the exposed class of susceptible individuals, $\mu \left(t\right)$ are the natural deaths, $\eta \left(t\right)$ represents the rate of loss of immunity, $ϵ\left(t\right)$ represents the infectivity rate and $\gamma \left(t\right)$ is the rate of recovery. We assume that $\Lambda$, $\beta$, $\mu$, $\eta$, $ϵ$ and $\gamma$ are periodic functions of the same period $\omega$. Naturally, for biological reasons we will take the initial conditions in the set $\{(S,E,I,R)\in {\mathbb{R}}^4:S,E,I,R\ge 0\}$.

Several different incidence functions have been considered to model the transmission in the context of SEIR/SEIRS models. In particular Michaelis–Menten incidence functions, that include the usual simple and standard incidence functions, have the form $\beta \left(t\right)\phi (S,N,I)=\beta \left(t\right)C\left(N\right)SI/N$ and were considered, just to name a few references, in  [7], [8], [9], [10], [11], [12]. The assumption that the incidence function is bilinear is seldom too simple and it is necessary to consider some saturation effect as well as other non-linear behaviors  [13], [14]. The Holling Type II incidence, given by $\beta \left(t\right)\phi (S,N,I)=\beta \left(t\right)SI/(1+\alpha I)$, is an example of an incidence function with saturation effect and was considered for instance in  [15], [16]. Another popular type of incidence, given by $\beta \left(t\right)\phi (S,N,I)=\beta \left(t\right)I^p{S}^q$, was considered in  [17], [13], [18]. Also, a generalization of Holling Type II incidence, $\beta \left(t\right)\phi (S,N,I)=\beta \left(t\right)S{I}^p/(1+\alpha I^q)$, was considered in  [19], [20]. All these incidence functions satisfy our hypothesis (see (P1) to (P6) in Section  2).

The search for periodic solutions and the study of their stability is a very important subject in epidemiology. In fact, in the non-autonomous context, periodic solutions play the same role as equilibriums in the autonomous context. Our main result shows that there exists a positive periodic solution of (1) whenever ${\overline{\mathcal{R}}}_0>1$, where ${\overline{\mathcal{R}}}_0$ is the basic reproductive number of the averaged system, ${inf}_{(0,1]}{\mathcal{R}}_0^{\lambda }>1$, where ${\mathcal{R}}_0^{\lambda }$, $\lambda \in (0,1]$, are the basic reproductive numbers of a family of systems related to system (1) and the determinant of some matrix is not zero, a technical condition required by our method of prove that consists in applying the famous Mawhin continuation theorem. We also prove that, when ${\mathcal{R}}_0<1$, there exists a unique disease-free periodic solution that is globally asymptotically stable. Here, ${\mathcal{R}}_0$ is given by the spectral radius of some operator, obtained by the method developed in  [21] and ${\mathcal{R}}_0^1={\mathcal{R}}_0$. To obtain our result, it is fundamental to have a good result about persistence of the infectives. Fortunately, in  [22] such result is obtained for general epidemiological models and applied to a mass-action SEIRS model. We use this result to obtain persistence in our general incidence case.

For mass-action incidence, in  [23], it is discussed the existence of periodic orbits. It is shown there that, under some condition involving bounds for the periodic parameters, there exists at least a positive periodic orbit. The referred model differs from ours not only because it assumes a particular form for the incidence function, but also because it allows disease induced mortality and it assumes that immunity is permanent. When the disease induced mortality is set to zero (letting $\alpha \equiv 0$), that model becomes a particular case of ours. Thus, when there is no disease induced mortality, Corollary 4 in Section  4 generalizes the main result in  [23].

Although the idea of applying Mawhin’s continuation theorem was borrowed from  [23], we need several nontrivial new arguments to deal with our case. In particular, because we allow temporary immunity, we were forced to use the original four-dimensional system instead of a reduced system.