Abstract
We study the basic dynamical features of a stochastic SIR epidemic model incorporating media coverage. Firstly, we discuss the positivity and boundedness of solutions of the model within deterministic environment and then investigate the asymptotical stability and global stability of equilibria of deterministic model. Secondly, we show that the stochastic model has a unique global positive solution and that this solution oscillates around the equilibria of the deterministic model under certain conditions. Finally, we give some numerical simulations to illustrate our analytical results.
Highlights
Mathematical models plays an important role in the study of epidemiology, which provides understanding of the underlying mechanisms that influence the spread of disease, and in the process, it suggests control strategies
For establishing a mathematical model of disease transmission with the population under study being divided into compartments and with assumptions about the nature and time rate of transfer from one compartment to another, we can formulate our descriptions as compartmental models
In Section, we first study the existence of the global positive solution of the stochastic model ( . ), and we investigate the asymptotic behavior around the equilibria of model ( . )
Summary
Mathematical models plays an important role in the study of epidemiology, which provides understanding of the underlying mechanisms that influence the spread of disease, and in the process, it suggests control strategies. In Section , we first study the existence of the global positive solution of the stochastic model If R < , the disease-free equilibrium E is globally asymptotically stable. Proof Let (S(t), I(t), R(t)) be any positive solution of system Note that if R < , the disease-free equilibrium E is locally asymptotically stable. We conclude that if R < , the disease-free equilibrium E ( /μ, , ) is globally asymptotically stable. For any given initial value (S( ), I( ), R( )) ∈ R +, there is a unique positive solution (S(t), I(t), R(t)) of model From Theorem . we can conclude that if R < and condition ( . ) holds, the solution of Eq ( . ) will fluctuate around the disease-free equilibrium of Eq ( . )
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