Abstract
The transmission of infectious diseases has been studied by mathematical methods since 1760s, among which SIR model shows its advantage in its epidemiological description of spread mechanisms. Here we established a modified SIR model with nonlinear incidence and recovery rates, to understand the influence by any government intervention and hospitalization condition variation in the spread of diseases. By analyzing the existence and stability of the equilibria, we found that the basic reproduction number is not a threshold parameter, and our model undergoes backward bifurcation when there is limited number of hospital beds. When the saturated coefficient a is set to zero, it is discovered that the model undergoes the Saddle-Node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2. The bifurcation diagram can further be drawn near the cusp type of the Bogdanov-Takens bifurcation of codimension 3 by numerical simulation. We also found a critical value of the hospital beds bc at and sufficiently small a, which suggests that the disease can be eliminated at the hospitals where the number of beds is larger than bc. The same dynamic behaviors exist even when a ≠ 0. Therefore, it can be concluded that a sufficient number of the beds is critical to control the epidemic.
Highlights
Since the development of the first dynamic model of smallpox by Bernoulli in 1760, various mathematical models have been employed to study infectious diseases [1] in order to reveal the underlying spread mechanisms that influence the transmission and control of these diseases
In the modeling of infectious diseases, the incidence function is one of the important factors to decide the dynamics of epidemic models
The dynamics of models are relatively simple and almost determined by the basic reproduction number R0: the disease will be eliminated if R0 < 1, otherwise, the disease will persist
Summary
Since the development of the first dynamic model of smallpox by Bernoulli in 1760, various mathematical models have been employed to study infectious diseases [1] in order to reveal the underlying spread mechanisms that influence the transmission and control of these diseases. The dynamics of models are relatively simple and almost determined by the basic reproduction number R0: the disease will be eliminated if R0 < 1, otherwise, the disease will persist Intervention strategies, such as isolation, quarantine, mask-wearing and medical report about emerging infectious diseases, play an vital role in controlling the spread, sometimes contributing to the eradication of diseases. A number of compartmental models have been formulated to explore the impact of intervention strategies on the transmission dynamics of infectious diseases. Medical treatments, determining how well the diseases are controlled, are normally expressed as constant recovery rates in the current models.
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