# American Institute of Mathematical Sciences

2009, 6(2): 321-332. doi: 10.3934/mbe.2009.6.321

## Epidemic models with differential susceptibility and staged progression and their dynamics

 1 Theoretical Division, MS-B284, Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, United States 2 Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received  December 2007 Revised  July 2008 Published  March 2009

We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each stage. We consider the contact rates to be proportional to the total population or constant such that the infection rates have a bilinear or standard form, respectively. We derive explicit formulas for the reproductive number $R_0$, and show that the infection-free equilibrium is globally asymptotically stable if $R_0<1$ when the infection rate has a bilinear form. We investigate existence of the endemic equilibrium for the two cases and show that there exists a unique endemic equilibrium for the bilinear incidence, and at least one endemic equilibrium for the standard incidence when $R_0>1$.
Citation: James M. Hyman, Jia Li. Epidemic models with differential susceptibility and staged progression and their dynamics. Mathematical Biosciences & Engineering, 2009, 6 (2) : 321-332. doi: 10.3934/mbe.2009.6.321
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