### Global analysis for a two-strain avian influenza model with distributed delay and environmental transmission

#### Abstract

A two-strain avian influenza model with distributed delay and environmental spread in humans is investigated. The model describes well the transmission of avian influenza between poultry and humans. In this study, we introduce the behavior of both high pathogenic avian influenza (HPAI) as strain two and low pathogenic avian influenza (LPAI) as strain one in a domestic poultry population. We also include the distribution of the strain two through the contaminated environment. We compute the strain reproduction numbers $\mathcal R_1$, $\mathcal R_2$ and the invasion $\hat{\mathcal R}_1$, $\hat{\mathcal R}_2$. We find that besides the disease-free equilibrium, there exist a dominance equilibrium for each strain and many coexistence equilibrium of both strain one and strain two if $\mathcal R_1=\mathcal R_2$. Using a Lyapunov functional, we are able to establish global stability of the disease-free equilibrium if $\max \{\mathcal R_1, \mathcal R_2\}<1$. If $\mathcal R_i$, the reproduction number of strain $i$ is larger than one, then a single-strain equilibrium, corresponding to strain $i$ exists. This single-strain equilibrium is locally stable whenever $\hat{\mathcal R}_i>1$. Using a Lyapunov functional, we establish that the corresponding single-strain equilibrium $\varepsilon_i $ is globally stable. When $\mathcal R_1=\mathcal R_2>1$ and $\hat{\mathcal R}_1=\hat{\mathcal R}_2=1$, there are perhaps many coexistence equilibria of both strain one and strain two. Environmental transmission to humans may explain why avian influenza A (H7N9) virus has appear in humans in different places in China in 2013 and 2014.

**Published:**2017-02-13

**How to Cite this Article:**Yan-Xia Dang, Juan Wang, Maia Martcheva, Xue-Zhi Li, Global analysis for a two-strain avian influenza model with distributed delay and environmental transmission, Commun. Math. Biol. Neurosci., 2017 (2017), Article ID 4 Copyright © 2017 Yan-Xia Dang, Juan Wang, Maia Martcheva, Xue-Zhi Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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