Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds
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Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds Milliward Maliyoni1 Received: 22 July 2019 / Accepted: 17 August 2020 © Springer Nature B.V. 2020
Abstract Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuoustime Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes. Keywords Markov chain · West Nile virus · Multitype branching process · Finitetime extinction Mathematics Subject Classification 91B70 · 92B05 · 92D25 · 92D30
* Milliward Maliyoni [email protected] 1
Mathematical Sciences Department, University of Malawi, Chancellor College, P. O. Box 280 Zomba, Malawi
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M. Maliyoni
1 Introduction West Nile virus (WNV) is an emerging mosquito-borne arbovirus that belongs to the Flavivirus genus (Bergsman et al. 2016; Mackenzie et al. 2004). WVN can cause fatal neurological diseases by infecting the central nervous system of various host species (Chen et al. 2016; Hayes and Gubler 2006). The transmission cycle of WNV involves wild and domestic birds as primary hosts and mosquitoes, mainly of the Culex genus, as primary vectors of the virus even though the different Culex species responsible for transmission vary greatly by region (Bergsman et al. 2016; Chen et al. 2016; Chevalier et al. 2014). The virus can infect multiple bird species which is a key factor in how it has spread so rapidly worldwide (Bergsman et al. 2016). For instance, WVN was introduced to the Middle East by migrating white storks (Bergsman et al. 2016; Chatterjee et al. 2008). In North America, the first recorded epidemic of WNV was detected in New York State in 1999 and spread rapidly across North America and into Latin America and the Caribbean (Chen et al. 2016; Marfin and Gubler 2001; Wonham and Lewis 2008) presumably after being introduced by migratory birds (Chen et al. 201
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