Epidemic models  with age  of  infection, indirecttransmission and incomplete treatment

Liming Cai; Maia Martcheva; Xue-Zhi Li

doi:10.3934/dcdsb.2013.18.2239

Article Contents

2013, Volume 18, Issue 9: 2239-2265. Doi: 10.3934/dcdsb.2013.18.2239

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Epidemic models with age of infection, indirect transmission and incomplete treatment

1.
Department of Mathematics, Xinyang Normal University, Xinyang 464000
2.
Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville

Received: June 30, 2012

Revised: June 30, 2013

Published: October 2013

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Abstract

An infection-age-structured epidemic model with environmental bacterial infection is investigated in this paper. It is assumed that the infective population is structured according to age of infection, and the infectivity of the treated individuals is reduced but varies with the infection-age. An explicit formula for the reproductive number $ \Re_0$ of the model is obtained. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$. It is also shown that if the reproduction number $\Re_0>1$, then the system has a unique endemic equilibrium which is locally asymptotically stable. Furthermore, if the reproduction number $\Re_0>1$, the system is permanent. When the treatment rate and the transmission rate are both independent of infection age, the system of partial differential equations (PDEs) reduces to a system of ordinary differential equations (ODEs). In this special case, it is shown that the global dynamics of the system can be determined by the basic reproductive number.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25.

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