# American Institute of Mathematical Sciences

March  2016, 21(2): 399-415. doi: 10.3934/dcdsb.2016.21.399

## Computation of $\mathcal R$ in age-structured epidemiological models with maternal and temporary immunity

 1 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, United States, United States 2 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094 3 National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention, 1600 Clifton Road, NE, Atlanta, GA 30333, United States 4 National Center for Immunization and Respiratory Diseases, 1600 Clifton Road, NE, Atlanta, GA 30333, United States

Received  April 2015 Revised  August 2015 Published  November 2015

For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting re-infection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ${\mathcal R}$ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ${\mathcal R}<1$ and unstable if ${\mathcal R}>1$.
Citation: Zhilan Feng, Qing Han, Zhipeng Qiu, Andrew N. Hill, John W. Glasser. Computation of $\mathcal R$ in age-structured epidemiological models with maternal and temporary immunity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 399-415. doi: 10.3934/dcdsb.2016.21.399
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