American Institute of Mathematical Sciences

Advanced
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

[1]

Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044
[2]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[3]

Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641
[4]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019226
[5]

Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 577-599. doi: 10.3934/mbe.2012.9.577
[6]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[7]

Georgi Kapitanov, Christina Alvey, Katia Vogt-Geisse, Zhilan Feng. An age-structured model for the coupled dynamics of HIV and HSV-2. Mathematical Biosciences & Engineering, 2015, 12 (4) : 803-840. doi: 10.3934/mbe.2015.12.803
[8]

Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
[9]

Xichao Duan, Sanling Yuan, Kaifa Wang. Dynamics of a diffusive age-structured HBV model with saturating incidence. Mathematical Biosciences & Engineering, 2016, 13 (5) : 935-968. doi: 10.3934/mbe.2016024
[10]

Diène Ngom, A. Iggidir, Aboudramane Guiro, Abderrahim Ouahbi. An observer for a nonlinear age-structured model of a harvested fish population. Mathematical Biosciences & Engineering, 2008, 5 (2) : 337-354. doi: 10.3934/mbe.2008.5.337
[11]

Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23
[12]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69
[13]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[14]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[15]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[16]

Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789
[17]

Yicang Zhou, Paolo Fergola. Dynamics of a discrete age-structured SIS models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 841-850. doi: 10.3934/dcdsb.2004.4.841
[18]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501
[19]

Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva. An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences & Engineering, 2010, 7 (1) : 123-147. doi: 10.3934/mbe.2010.7.123
[20]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences