Bifurcation and basin stability of an SIR epidemic model with limited medical resources and switching noise

Abstract

Considering the sudden change of environmental disturbance, a stochastic susceptible infectious recovered (SIR) model with Markov jump and the limited medical resources is proposed. Firstly, by a bifurcation analysis of the deterministic SIR model, the maximal medical resource tipping point can be detected to adjust and optimize the medical resource allocation. Then, the impact of environmental disturbance on the basin stability is explored via the first escape probability(FEP). Based on the stochastic averaging of Markov jump process, the SIR epidemic system with switching random excitation is transferred into a probability-weighted Itô stochastic differential equation. Furthermore, the theoretical FEP is solved by the finite difference method and the validity is verified by numerical simulation. It is worth noting that the increase of noise intensity can decrease the basin stability of SIR model, and the existence of switching noise makes a difference in the basin stability compared with the epidemic system without switching intensity.

Introduction

Since the first dynamic model of smallpox introduced by Bernouli in 1760, numerous epidemic models have been proposed and investigated to provide useful insight to enhance our comprehension of complex processes associated with the pathogenesis of diseases [1]. Various versions of the model on the spread of infections diseases have been investigated, for example, the susceptible-infectious-recovered (SIR) epidemic model [2], which has a quite long history. Till now, these epidemic models are still popular in research. The nonlinear epidemic models, where the nonlinearity may comes from social groups with different susceptibilities, nonliear or nonmonotone incidence rate, stage structure or behavioral change of susceptibles, have been modelled and investigated recently [3], [4], [5], [6].

For example, Capasso and Serio [7] proposed a nonlinear saturated incidence function $\beta SI/\left(1+\alpha I\right)$ to represent a crowding effect or protection measure in modeling the cholera epidemics in 1973. Although this incidence rate is more complex, the SIR epidemic models with such nonlinear incidence rates have acquired wide attention because of its great practical significance and rich dynamical behaviors [7]. Considering the maximal capacity for the treatment of patients in public-health systems, Wang [8] introduced a segmented function to describe the treatment, where the treatment rate is proportional to the number of the infective when the maximal capacity of treatment is not reached, or it takes the maximal saturated level. That enhances the effectiveness and accuracy of the SIR model. In addition, the efficiency for treatment will be affected severely if the infective individuals are delayed in treatment. So Zhang [9] intorduced a new continually differentiable treatment function $h\left(I\right)=rI/\left(1+\alpha I\right)$ where the parameter $r$ is the cure rate and the parameter $\alpha$ is used to measure the extent of the effect of the delay for treatment. The results show that a backward bifurcation will take place when the delayed effect of treatment is strong, and a critical value at the turning point is deduced for eradicating the disease. To better understand the effects of limited medical resources and the efficiency of supply of available medical resources on the spread of infectious diseases, Zhou [10] transformed $h\left(I\right)$ into $h^{*}\left(I\right)=\alpha I/\left(\omega +I\right)$, and studied the backward bifurcation and global dynamics of an epidemic model with the saturated treatment function.

The backward bifurcations of epidemic model have received considerable attentions because of the impact on disease control. The backward bifurcation shows that both the disease-free and endemic equilibrium of epidemic system coexist when the reproduction number is less than a unit. That usually indicats a great shift from the stable disease-free state to stable disease state. To delay or avoid the unexpected shifts, most researches are focused on the regime shifts and tipping point of dynamical systems [11], [12]. These results indicate that the tipping phenomena, usually due to bifurcation with smooth changes in the parameter values, are bound up with sudden shifts of dynamic behaviors. In order to delay or restrain the unexpected shifts, it is necessary to study the transient response of the system. The basin stability, as a significant property of the transient response, have been established and applied to many fields such as first escape probability [13] and the mean first passage time [14], [15]. But most researches focus on the unidimensional ecosystem [16], [17], [18], while very few works have been done on the transition behavior of SIR epidemic models.

As a matter of fact, it is well recognized that the epidemic models are often subject to environmental noises. May [19] found the fact that due to environmental fluctuations, the birth rates, death rates, and other parameters involved with the model system exhibit random fluctuations to a greater or lesser extent. To describe the interference of the external irresistible factors, many different types of environmental noises are taken into consideration. For example, the Gaussian white noise and Brownian motion have already been considered [20], [21], [22], and the results indicate that the existence of Gaussian white noise may have a significant impact on the equilibrium [23] and their asymptotic stability [24]. However, some sudden abnormal environment events such as earthquakes and hurricanes can’t be described by Gaussian white noise very well. Therefore, the sochastic SIR model with jump type noises has been introduced to describe the spread of the disease through a popuplation [25], [26], [26], [27], [28]. The telegraph noise, as one of the jump type noises, is usually used to characterize population systems switching from one regime to another [29], [30], [31], which differs by factors such as the nutrition, climatic characteristics or socio-cultural factors [32]. Mostly the switching between the environment regimes is memoryless and the waiting time for the next switching satisfies an exponential distribution [31]. Hence the switching can be modeled by a continuous time Markov jump process within a finite state space. There are already various paper which have looked at the effect of telegraph noise in population systems [31], [33]. The conditions for extinction and persistence, and the stationary distribution of stochastic epidemic models with Markov switching have been investigated [30], [31], [34]. However, only few researches have been done on the transient response of epidemic models, which is important to study the spread of infectious disease.

Motivated by the above works, the SIR model with limited medical resources will be studied from the perspective of the basin stability. This paper is organized as follows. The next section makes a description of the SIR epidemic model with switching noise and treatment function, and discusses the effects of treatment function and incidence rate on bifurcation behavior of deterministic SIR model. Considering the sudden change of environmental disturbance, the effects of switching noise on the basin stability of SIR epidemic system is discussed by the first escape probability and a new stability index in Sec. 3. Our results are discussed and summarized in Section 4.