Abstract
Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.
Article PDF
Similar content being viewed by others
References
Oaks, J. S. C., Shope, R. E. and Lederberg, J., Emerging infections: Microbial Threats to Health in the United States, National Academies Press, New York, 1992.
Brauer, F. and Castillo-Chavez, C., Mathematical Models in Population Biology and Epidemiology, Text in Applied Mathematics, 40, Springer-Verlag, New York, 2001.
Gubler, D. J., Resurgent vector-borne diseases as a global health problem, Emerg. Infect. Dis., 4, 1998, 442–450.
Mcnicoll, R. B. C., The World Health Report 1996: Fighting Disease, Fostering Development; Report of the Director-General. by World Health Organization, Population and Development Review, 23, 1997, 203–204.
Anderson, R. M., Anderson, B. and May, R. M., Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, 1992.
Zhang, Z., Kundu, S., Tripathi, J. P. and Bugalia, S., Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays, Chaos Solitons Fractals, 131, 2020, 109483.
Hethcote, H. W. and Driessche, P. V. D., Some epidemiological models with nonlinear incidence, J. Math. Biol., 29, 1991, 271–287.
Kermack, W. O. and McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proceedings of the royal society of london., Series A, 115, 1927, 700–721.
Hethcote, H. W., Stech, H. W. and Driessche, P. V. D., Periodicity and stability in epidemic models: A survey, Differential Equations and Applications in Ecology, Epidemics, and Population Problems, Elsevier Academies Press, New York-London, 1981, 65–82.
Sun, G. Q., Jusup, M., Jin, Z., et al., Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Phys. Life Rev., 19, 2016, 43–73.
Li, L., Patch invasion in a spatial epidemic model, Appl. Math. Comput., 258, 2015, 342–349.
Guo, Z. G., Song, L. P., Sun, G. Q., et al., Pattern dynamics of an SIS epidemic model with nonlocal delay, Int. J. Bifurcation Chaos, 29, 2019, 1950027.
Sun, G. Q., Wang, C. H., Chang, L. L., et. al., Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61, 2018, 200–215.
Andersson, H. and Britton, T., Heterogeneity in epidemic models and its effect on the spread of infection, J. Appl. Probab., 35, 1998, 651–661.
Bowden, S. and Drake, J., Ecology of multi-host pathogens of animals, Nat Education Knowledge, 4, 2013, 5–5.
Anderson, R. M. and May, R. M., Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes, Math. Med. Biol., 1, 1984, 233–266.
Luo, X., Yang, J., Jin, Z. and Li, J., An edge-based model for nonMarkovian sexually transmitted infections in coupled network, Int. J. Biomath., 13, 2020, 2050014.
Lajmanovich, A. and Yorke, J. A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28, 1976, 221–236.
Li, M. T., Sun, G. Q., Zhang, J. and Jin, Z., Global dynamic behavior of a multigroup cholera model with indirect transmission, Discrete Dyn. Nat. Soc., 2013, 2013, 703826.
Hyman, J. M., Li, J. and Stanley, E. A., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 1999, 77–109.
Beretta, E. and Capasso, V., Global stability results for a multigroup SIR epidemic model, T. G. Hallam, L. J. Gross, S. A. Levin (Eds.), Mathematical Ecology, World Scientific, Singapore, 1986, 317–342.
Hethcote, H. W., An immunization model for a heterogeneous population, Theor. Popul. Biol., 14, 1978, 338–349.
Huang, W., Cooke, K. L. and Castillo-Chavez, C., Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52, 1992, 835–854.
Lin, X. and So, W. H., Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations, J. Aust. Math. Soc., 34, 1993, 282–295.
Thieme, H. R., Local stability in epidemic models for heterogeneous populations, Math. Med. Biol., 97, 1985, 185–189.
Thieme, H. R., Mathematics in Population Biology, Princeton University Press, Princeton, 2018.
Magal, P., McCluskey, C. and Webb, G., Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89, 2010, 1109–1140.
Feng, Z., Huang, W. and Castillo-Chavez, C., On the role of variable latent periods in mathematical models for tuberculosis, J. Dyn. Differ. Equ., 13, 2001, 425–452.
Thieme, H. R. and Castillo-Chavez, C., How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53, 1993, 1447–1479.
Inaba, H. and Sekine, H., A mathematical model for Chagas disease with infection-age-dependent infectivity, Math. Biosci., 190, 2004, 39–69.
Alexander, M. E., Moghadas, S. M., Röst, G. and Wu, J., A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70, 2008, 382–397.
Röst, G., SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5, 2008, 389–402.
McCluskey, C. C., Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6, 2009, 603–610.
Li, M. Y., Shuai, Z. and Wang, C., Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361, 2010, 38–47.
Chen, F., Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162, 2005, 1279–1302.
Chen, F., Yang, J., Chen, L. and Xie, X., On a mutualism model with feedback controls, Appl. Math. Comput., 214, 2009, 581–587.
Fan, M., Wang, K., Wong, P. J. and Agarwal, R. P., Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Mathematica Sinica, 19, 2003, 801–822.
Gopalsamy, K. and Weng, P. X., Feedback regulation of logistic growth, International Journal of Mathematics and Mathematical Sciences, 16, 1993, 177–192.
Li, Z., Han, M. and Chen, F., Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays, Nonlinear Anal.-Real World Appl., 14, 2013, 402–413.
Niyaz, T. and Muhammadhaji, A., Positive periodic solutions of cooperative systems with delays and feedback controls, International Journal of Differential Equations, 2013, 2013, 502963.
Xiao, Y. N., Tang, S. Y. and Chen, J. F., Permanence and periodic solution in competitive system with feedback controls, Math. Comput. Model., 27, 1998, 33–37.
Fan, Y. H. and Wang, L. L., Global asymptotical stability of a Logistic model with feedback control, Nonlinear Anal-Real World Appl., 11, 2010, 2686–2697.
Yang, K., Miao, Z., Chen, F. and Xie, X., Influence of single feedback control variable on an autonomous Holling-II type cooperative system, J. Math. Anal. Appl., 435, 2016, 874–888.
Li, H. L., Zhang, L., Teng, Z., et al., Global stability of an SI epidemic model with feedback controls in a patchy environment, Appl. Math. Comput., 321, 2018, 372–384.
Sun, R., Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60, 2010, 2286–2291.
Shang, Y., Global stability of disease-free equilibria in a two-group SI model with feedback control, Nonlinear Anal. Model. Control, 20, 2015, 501–508.
Tripathi, J. P. and Abbas, S., Global dynamics of autonomous and nonautonomous SI epidemic models with nonlinear incidence rate and feedback controls, Nonlinear Dyn., 86, 2016, 337–351.
Guo, H., Li, M. Y. and Shuai, Z., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14, 2006, 259–284.
Guo, H., Li, M. and Shuai, Z., A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136, 2008, 2793–2802.
Kuniya, T., Global stability of a multi-group SVIR epidemic model, Nonlinear Anal.-Real World Appl., 14, 2013, 1135–1143.
Li, M. Y. and Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248, 2010, 1–20.
Shuai, Z. and Driessche, P. V. D., Global stability of infectious disease models using lyapunov functions, SIAM J. Appl. Math., 73, 2013, 1513–1532.
Li, M. T., Jin, Z., Sun, G. Q. and Zhang, J., Modeling direct and indirect disease transmission using multi-group model, J. Math. Anal. Appl., 446, 2017, 1292–1309.
Shuai, Z. S. and Driessche, P. V. D., Global dynamics of cholera models with differential infectivity, Math. Biosci., 234, 2011, 118–126.
Horn, R. A. and Johnson, C. R., Matrix Analysis, Cambridge university press, Cambridge, 2012.
Moon, J. W., Counting Labelled Trees, Canadian Mathematical Congress, Montreal, 1970.
Berman, A. and Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1994.
FV, A. and Haddock, J., On determining phase spaces for functional differential equations, Funkcialaj Ekvacioj, 31, 1988, 331–347.
Kolmanovskii, V. and Myshkis, A., Applied theory of functional differential equations, Mathematics and its Applications (Soviet Series), 85. Kluwer Academic Publishers Group, Dordrecht, 1992.
Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, John Wiley & Sons, New York, 1989.
Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A., On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 1990, 365–382.
LaSalle, J. P., The stability of dynamical systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, 1976.
Freedman, H. I., Ruan, S. and Tang, M., Uniform persistence and flows near a closed positively invariant set, J. Differ. Equ., 6, 1994, 583–600.
Li, M. Y., Graef, J. R., Wang, L. and Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160, 1999, 191–213.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (No. 12022113), Henry Fok Foundation for Young Teachers, China (No. 171002), Outstanding Young Talents Support Plan of Shanxi Province, Science and Engineering Research Board (SERB for short), India (No. ECR/2017/002786), UGC-BSR Research Start-Up-Grant, India (No. F.30-356/2017(BSR)), and Senior Research Fellowship from the Council of Scientific and Industrial Research (CSIR for short), India (No. 09/1131(0006)/2017-EMR-I).
Rights and permissions
About this article
Cite this article
Bajiya, V.P., Tripathi, J.P., Kakkar, V. et al. Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age. Chin. Ann. Math. Ser. B 42, 833–860 (2021). https://doi.org/10.1007/s11401-021-0294-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-021-0294-1