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

Zhilan  Feng; Qing  Han; Zhipeng Qiu; Andrew N.  Hill; John W.  Glasser

doi:10.3934/dcdsb.2016.21.399

Article Contents

2016, Volume 21, Issue 2: 399-415. Doi: 10.3934/dcdsb.2016.21.399

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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
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
4.
National Center for Immunization and Respiratory Diseases, 1600 Clifton Road, NE, Atlanta, GA 30333

Received: April 2015

Revised: August 2015

Published: March 2016

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Abstract

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 ${\mathcal R}>1$.

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Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 92D25.

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