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Dynamics of Evolutionary Epidemiological SIR Model with Varying Total Population Size

Corresponding Author : M. A. Haque (aminul202@yahoo.com)

Authors : M. A. Haque

Keywords : Epidemic models, resident strain, mutant strain, Lyapunov function, Jacobian Matrix

Abstract :

Infectious diseases have often had a big impact on population sizes and historical events. Therefore, the constant population size models are not often suitable when the natural births and deaths are not balanced or when the disease-related deaths are significant. Moreover mutation of Pathogens is an important phenomenon of the disease dynamics, as pathogens can reproduce quickly. As a consequence, in this paper the evolutionary behavior of SIR model considering variable population size has been studied. The resident strain and mutant strain have been introduced with the SIR model considering varying population size and analyzed their behavior of dynamics. It has shown the disease free equilibrium is locally asymptotically stable if 1 0 < r> r R or 1 0 > m R . The global stability at disease free equilibrium has been shown with the help of Lyapunov function. It has also been investigated when the dynamics of diseases is locally evolutionary stable (LES), and shown that the resident strain is locally evolutionary stable if r m R0 > R0 and the mutant strain is locally evolutionary stable if m r R0 > R0 . In this paper, extensive analyses have been done mathematically and also performed some numerical simulation using Mat lab program and phase portrait analysis with Maple.  

Published on December 31st, 2011 in Volume 14, Issue 2, Applied Sciences and Technology