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. 2017:2017:4820183.
doi: 10.1155/2017/4820183. Epub 2017 Jul 6.

Threshold Dynamics of a Stochastic SIR Model with Vertical Transmission and Vaccination

Anqi Miao  1 , Jian Zhang  1 , Tongqian Zhang  1   2 , B G Sampath Aruna Pradeep  3
Affiliations

Threshold Dynamics of a Stochastic SIR Model with Vertical Transmission and Vaccination

Anqi Miao et al. Comput Math Methods Med. 2017.

Abstract

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control.

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Figures

Figure 1
Figure 1
The compartmental diagram for the SIR model with vertical transmission and vaccination.
Figure 2
Figure 2
Illustration for the deterministic SIR system where β = 0.8, p = 0.6, b = 0.2, b′ = 0.4, and γ = 0.2.
Figure 3
Figure 3
Comparison of the deterministic system and stochastic system, where m = 0.2, β = 0.8, p = 0.6, b = 0.2, b′ = 0.4, γ = 0.2, σ = 0.9, and R0 = 1.4545 > 1.
Figure 4
Figure 4
Comparison of the deterministic system and stochastic system, where m = 0.2, β = 0.8, p = 0.6, b = 0.2, b′ = 0.4, γ = 0.2, σ = 0.85, R∗ = 0.9972, and R0 = 1.4545 > 1.
Figure 5
Figure 5
Comparison of the deterministic system and stochastic system, where m = 0.2, β = 0.8, p = 0.6, b = 0.2, b′ = 0.4, γ = 0.2, σ = 0.2, R∗∗ = 1.4091, and R0 = 1.4545 > 1.
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