American Institute of Mathematical Sciences

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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

Zhilan Feng 1, , Qing Han 1, , Zhipeng Qiu 2, , Andrew N. Hill 3, and  John W. Glasser 4,

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 ${\mathcal R}>1$.
Keywords: multiple infections., Age-structured epidemiological model, partial immunity, reproduction numbers.
Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 92D2.
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
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, (2000).   Google Scholar

[2]

C. Dye and B. G. Williams, Eliminating human tuberculosis in the twenty-first century,, J. R. Soc. Interface, 5 (2008), 653.  doi: 10.1098/rsif.2007.1138.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2014.04.020.  Google Scholar

[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.  doi: 10.1016/j.mbs.2011.10.001.  Google Scholar

[5]

H. W. Hethcote, An age-structured model for pertussis transmission,, Math. Biosci., 145 (1997), 89.  doi: 10.1016/S0025-5564(97)00014-X.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[7]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,, Cambridge University, (1991).  doi: 10.1017/CBO9780511840371.  Google Scholar

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[9]

H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model,, J. Math. Biol., 54 (2007), 101.  doi: 10.1007/s00285-006-0033-y.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2013.01.044.  Google Scholar

[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).  doi: 10.1371/journal.pmed.0050074.  Google Scholar

[12]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time-heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

show all references

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, (2000).   Google Scholar

[2]

C. Dye and B. G. Williams, Eliminating human tuberculosis in the twenty-first century,, J. R. Soc. Interface, 5 (2008), 653.  doi: 10.1098/rsif.2007.1138.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2014.04.020.  Google Scholar

[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.  doi: 10.1016/j.mbs.2011.10.001.  Google Scholar

[5]

H. W. Hethcote, An age-structured model for pertussis transmission,, Math. Biosci., 145 (1997), 89.  doi: 10.1016/S0025-5564(97)00014-X.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[7]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,, Cambridge University, (1991).  doi: 10.1017/CBO9780511840371.  Google Scholar

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[9]

H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model,, J. Math. Biol., 54 (2007), 101.  doi: 10.1007/s00285-006-0033-y.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2013.01.044.  Google Scholar

[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).  doi: 10.1371/journal.pmed.0050074.  Google Scholar

[12]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time-heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

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