Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates
- 331 Downloads
The control of highly contagious diseases is very important today. In this paper, we proposed an SEIR model with Crowley–Martin-type incidence rate and Holling type II and III treatment rates. Dynamics of the spread of infection and its control are performed for both the cases of treatment functions. We have performed the stability and bifurcation analyses of the model system. The sensitivity analysis of all the parameters with respect to the basic reproduction number has been performed. Furthermore, we discussed the optimal control strategy using Pontryagin’s maximum principle and determined the effect of control parameter u on the model dynamics. Moreover, we validate the theoretical results using numerical simulations. Between both the treatment functions, we observe that the implementation of Holling type II treatment is most effective to prevent the spread of diseases. Thus, we conclude that the pervasive effect of treatment not only reduces the basic reproduction number as the control parameter u increases with nonlinear treatment, h(I) but also controls the spread of disease infection among the population.
KeywordsEpidemic model Holling type II and III treatment functions Stability analysis Optimal control
We are thankful to Council of Scientific & Industrial Research (CSIR) India for providing financial support through Project No.- CSIR-25(0277)/17/EMR-II to the first author (RKU).
Compliance with ethical standards
Conflicts of interest
The authors declare that there is no conflict of interests regarding the publication of this article.
The authors state that this research complies with ethical standards. This research does not involve either human participants or animals.
- 2.Anderson, R.M., May, R.M.: Regulation and stability of host-parasite population interactions: I. Regulatory processes. J. Anim. Ecol. 47, 219–247 (1978)Google Scholar
- 3.Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York (1992)Google Scholar
- 6.Bailey, N.T., et al.: The mathematical theory of infectious diseases and its applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE. (1975)Google Scholar
- 7.Binder, S., Levitt, A.M., Sacks, J.J., Hughes, J.M.: Emerging infectious diseases: public health issues for the 21st century. Science 284(5418), 1311–1313 (1999)Google Scholar
- 9.Blower, S.M., McLean, A.R.: Mixing ecology and epidemiology. Proc. R. Soc. Lond. B 245(1314), 187–192 (1991)Google Scholar
- 13.Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8(3), 211–221 (1989)Google Scholar
- 16.Dubey, B., Dubey, P., Dubey, U.S.: Dynamics of an SIR model with nonlinear incidence and treatment rate. Appl. Appl. Math. 10(2), (2015)Google Scholar
- 19.Earn, D.J., Rohani, P., Bolker, B.M., Grenfell, B.T.: A simple model for complex dynamical transitions in epidemics. Science 287(5453), 667–670 (2000)Google Scholar
- 26.Jana, S., Nandi, S.K., Kar, T.: Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment. Acta Biotheor. 64(1), 65–84 (2016)Google Scholar
- 28.Keeling, M.J., Woolhouse, M.E., Shaw, D.J., Matthews, L., Chase-Topping, M., Haydon, D.T., Cornell, S.J., Kappey, J., Wilesmith, J., Grenfell, B.T.: Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science 294(5543), 813–817 (2001)Google Scholar
- 31.Koprivica, V., Stone, D.L., Park, J.K., Callahan, M., Frisch, A., Cohen, I.J., Tayebi, N., Sidransky, E.: Analysis and classification of 304 mutant alleles in patients with type 1 and type 3 Gaucher disease. Am. J. Human Genet. 66(6), 1777–1786 (2000)Google Scholar
- 39.Li, X., Jusup, M., Wang, Z., Li, H., Shi, L., Podobnik, B., Stanley, H.E., Havlin, S., Boccaletti, S.: Punishment diminishes the benefits of network reciprocity in social dilemma experiments. Proc. Natl. Acad. Sci. 115(1), 30–35 (2018)Google Scholar
- 41.May, R.M., Anderson, R.M.: Regulation and stability of host-parasite population interactions: II. Destabilizing processes. J. Anim. Ecol. pp. 249–267 (1978)Google Scholar
- 44.Pontryagin, L.S.: Mathematical theory of optimal processes. Routledge, (1987)Google Scholar
- 54.Wang, Y., Lim, H.: The global childhood obesity epidemic and the association between socio-economic status and childhood obesity (2012)Google Scholar