# 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 and Continuous Dynamical Systems - B, 2016, 21 (2) : 399-415. doi: 10.3934/dcdsb.2016.21.399
##### References:
 [1] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [2] C. Dye and B. G. Williams, Eliminating human tuberculosis in the twenty-first century, J. R. Soc. Interface, 5 (2008), 653-662. doi: 10.1098/rsif.2007.1138. [3] Z. Feng, J. W. Glasser, A. N. Hill, M. A. Franko, R. M. Carlsson, H. Hallander, P. Tull and P. Olin, Modeling rates of infection with transient maternal antibodies and waning active immunity: Applicationto Bordetella pertussis in Sweden, J. Theor. Biol., 356 (2014), 123-132. doi: 10.1016/j.jtbi.2014.04.020. [4] J. Glasser, Z. Feng, A. Moylan, S. Del Valled and C. Castillo-Chavez, Mixing in age-structured population models of infectious diseases, Math. Biosci., 235 (2012), 1-7. doi: 10.1016/j.mbs.2011.10.001. [5] H. W. Hethcote, An age-structured model for pertussis transmission, Math. Biosci., 145 (1997), 89-136. doi: 10.1016/S0025-5564(97)00014-X. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University, Cambridge, 1991. doi: 10.1017/CBO9780511840371. [8] J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4. [9] H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model, J. Math. Biol., 54 (2007), 101-146. doi: 10.1007/s00285-006-0033-y. [10] T. Kuniya and H. Inaba, Endemic threshold results for an agestructured SIS epidemic model with periodic parameters, J. Math. Anal. Appl., 402 (2013), 477-492. doi: 10.1016/j.jmaa.2013.01.044. [11] J. Mossong, N. Hens and M. Jit, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med., 5 (2008), e74. doi: 10.1371/journal.pmed.0050074. [12] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. [13] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

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##### References:
 [1] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [2] C. Dye and B. G. Williams, Eliminating human tuberculosis in the twenty-first century, J. R. Soc. Interface, 5 (2008), 653-662. doi: 10.1098/rsif.2007.1138. [3] Z. Feng, J. W. Glasser, A. N. Hill, M. A. Franko, R. M. Carlsson, H. Hallander, P. Tull and P. Olin, Modeling rates of infection with transient maternal antibodies and waning active immunity: Applicationto Bordetella pertussis in Sweden, J. Theor. Biol., 356 (2014), 123-132. doi: 10.1016/j.jtbi.2014.04.020. [4] J. Glasser, Z. Feng, A. Moylan, S. Del Valled and C. Castillo-Chavez, Mixing in age-structured population models of infectious diseases, Math. Biosci., 235 (2012), 1-7. doi: 10.1016/j.mbs.2011.10.001. [5] H. W. Hethcote, An age-structured model for pertussis transmission, Math. Biosci., 145 (1997), 89-136. doi: 10.1016/S0025-5564(97)00014-X. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University, Cambridge, 1991. doi: 10.1017/CBO9780511840371. [8] J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4. [9] H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model, J. Math. Biol., 54 (2007), 101-146. doi: 10.1007/s00285-006-0033-y. [10] T. Kuniya and H. Inaba, Endemic threshold results for an agestructured SIS epidemic model with periodic parameters, J. Math. Anal. Appl., 402 (2013), 477-492. doi: 10.1016/j.jmaa.2013.01.044. [11] J. Mossong, N. Hens and M. Jit, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med., 5 (2008), e74. doi: 10.1371/journal.pmed.0050074. [12] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. [13] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
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