Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.
It is generally known that the spread of infectious diseases has been a threat to healthy of human beings and other species. In order to prevent and control the transmission of disease (such as hepatitis C, malaria, influenza), pulse vaccination as an effective strategy has been widely studied by many scholars in the study of mathematical epidemiology. In the classical research literature it is usually assumed that the pulse vaccination occurs at fixed moment intervals and total population size remains constant [
On the other hand, the population sizes of all epidemic models with state-dependent pulse control are constant. These types of models have been studied extensively since they are easier to analyze than variable population size models. Obviously, the assum- ption that the total population size which remains constant is reasonable if negligible mortality rate and the disease spread quickly through the population. However, it fails to hold for diseases that are endemic in communities with changing populations, and for diseases which raise the mortality rate substantially. In such situation, we can hardly expect a population remaining constant, and hence more complicated epidemic models with varying population size should be considered. In fact, studies of this type of models have been become a major topic in mathematical epidemiology. For example, an general epidemiological model with vaccination and varying total population was discussed by Yang et al. [
As far as we know, epidemic model with varying total population and state-de- pendent feedback control strategies had never been done in the literatures. Hence, in this paper, the dynamical behavior of an SIRS epidemic model with varying total population and state-dependent pulse control strategy is studied. The main aim is to explore how the state-dependent pulse control strategy affects the transmission of diseases. The remaining part of this paper is organized as follows. In the next section, an SIRS control model is constructed and some preliminaries are introduced, which are useful for the latter discussion. In section 3, we will focus our attention on the existence and orbital stability of disease-free periodic. Finally, some concluding remarks are presented in the last section.
In the study of the dynamic properties of infectious diseases, it was found that when the popularity of disease for a long time total population size change this factor should be considered. In this case, Busenberg et al. [
Here
Since the susceptible individuals are immunity toward certain infectious diseases in the crowd, once infected individuals get into the susceptible groups, this will lead to the outbreak of the diseases. For this reason, we propose a pulse vaccination function as follows
Taking into account pulse vaccination as state-dependent feedback control strategies, model (1) can be extend to the following state-dependent pulse differential equation.
where the critical threshold
The equation for the total population size
It means that total population size
It following from (3) that we can transforms model (2) into the following model for these new variables
Define three threshold parameter as follows
On the dynamics of model (4) without pulse effect has been studied in [
Theorem 1. For model (4) without pulse control, the following result hold true.
1) The disease-free equilibrium
2) When
and
3) The total population
4) When
Based on the above discussions, we just need to discuss cases (a) and (b) in
Considering the similarities of cases (a) and (b), throughout of this paper, we discuss only the case (a). That is, in a increasing population, the number of infected individuals is converges to infinity, while the fraction of infected individuals in population is tending to a nonzero constant
Due to
By the biological background, we only focus on model (5) in the biological meaning region
Let
Firstly, on the positivity of solutions of model (5), we have the following Lemma 1.
Lemma 1. Supposing that
Proof. For any initial value
1) The solution
For this case, due to the endemic equilibrium
case | ||||||
---|---|---|---|---|---|---|
(a) | ||||||
(b) | 0 |
stable, then
2) The solution
For second situation, assume that solution
which contradicts the fact that
The other case is that
which lead to a contradiction with
In order to address the dynamical behaviors of model (5), we could construct two sections to the vector field of model (5) by
and
Choosing section
From the definition of Poincaré map
Obviously, function
Our main purpose in this section is to investigate the existence and orbital stability of periodic solution of model (5). From the geometrical construction of phase space of model (5), we note that the trajectory
Case I: The case of
For this case, it will prove that model (5) possesses a disease-free periodic solution, which is orbitally asymptotically stable.
Suppose
Integrating the first equation of model (7) with the initial condition
where
Assume that
Therefore, model (5) possesses the following disease-free periodic solution, denoted by
where
On the stability of this disease-free periodic solution
Theorem 2. For any
Proof. We assume that section
and
where
From (9), it is concluded that the point sequence
Suppose that
Case II: The case of
For this case, we know that there a point
Theorem 3. For any
then model (5) exists a disease-free periodic solution (8), which is orbitally asympto- tically stable.
For this case, (8) is a disease-free periodic solution of model (5), and the proof of stability is similar to the proof of Theorem 2, we therefore omit here.
In order to explore the effects of the state-dependent pulse control strategies on the transmission of the infectious diseases in a population of varying size, an SIRS epidemic model with varying total population and state-dependent pulse control strategy is proposed and analyzed in this paper. Theoretically analyzing this control model, we find that a disease-free periodic solution always exists and orbitally stable when condition
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research has been partially supported by the Natural Science Foundation of Xinjiang (Grant no. 2016D01C046).
Zhang, F.W. and Nie, L.F. (2016) The Effect of State-Depen- dent Control for an SIRS Epidemic Model with Varying Total Population. Journal of Ap- plied Mathematics and Physics, 4, 1889- 1898. http://dx.doi.org/10.4236/jamp.2016.410191