Abstract
In this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number R_{0} which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.
Highlights
To the best of our knowledge, vaccination is one of the most effective ways to treat and prevent diseases
With the development of information technology, media reports play an important role in the prevention and control of diseases, for example, during the outbreak of SARS and H1N1, media reports effectively stopped the spread of the disease and provided scientific and reasonable preventive measures for people [6,7,8,9]
We study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage
Summary
To the best of our knowledge, vaccination is one of the most effective ways to treat and prevent diseases. During the outbreak of SARS in 2003 [2], H1N1 influenza pandemic in 2009 [3], and H7N9 influenza in 2013 [4], unprecedented mass influenza vaccination programs were launched by a large number of countries to timely immunize as many people as possible Those strategies greatly controlled the spread of infection and decreased the incidence rate [5]. Cui et al [13] presented an SEI epidemic model with incidence rate βe–mISI and found the disease can be controlled when the media impact is stronger. In [29], Yang et al studied the global threshold dynamics for a stochastic SIS epidemic model incorporating media coverage and gave the basic reproduction number which determines the persistence or extinction of the disease. 3.2 Existence and uniqueness of positive solution Theorem 3.1 There is a unique solution (S(t), I(t)) of model (1.3) on t ≥ 0 for any initial value (S(0), I(0)) ∈ R2+, and the solution will remain in R2+ with probability one, namely, (S(t), I(t)) ∈ R2+ for all t ≥ 0 almost surely
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