The Effects of a Backward Bifurcation on a Continuous Time Markov Chain Model for the Transmission Dynamics of Single Strain Dengue Virus

Adnan Khan, Muhammad Hassan, Mudassar Imran

Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore, Pakistan.

DOI: 10.4236/am.2013.44091                

Abstract

Global incidence of dengue, a vector-borne tropical disease, has seen a dramatic increase with several major outbreaks in the past few decades. We formulate and analyze a stochastic epidemic model for the transmission dynamics of a single strain of dengue virus. The stochastic model is constructed using a continuous time Markov chain (CTMC) and is based on an existing deterministic model that suggests the existence of a backward bifurcation for some values of the model parameters. The dynamics of the stochastic model are explored through numerical simulations in this region of bistability. The mean of each random variable is numerically estimated and these are compared to the dynamics of the deterministic model. It is observed that the stochastic model also predicts the co-existence of a locally asymptotically stable disease-free equilibrium along with a locally stable endemic equilibrium. This co-existence of equilibria is important from a public health perspective because it implies that dengue can persist in populations even if the value of the basic reproduction number is less than unity.

Keywords

Epidemiology; Dengue Fever; Backward Bifurcation; Stochastic Model

Conflicts of Interest

The authors declare no conflicts of interest.

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