Complex Dynamics of an SIR Epidemic Model with Saturated Incidence Rate and Treatment
- PDF / 690,567 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 66 Downloads / 204 Views
Complex Dynamics of an SIR Epidemic Model with Saturated Incidence Rate and Treatment Soovoojeet Jana1 • Swapan Kumar Nandi2 T. K. Kar3
•
Received: 26 March 2015 / Accepted: 3 November 2015 Springer Science+Business Media Dordrecht 2015
Abstract This paper describes a traditional SIR type epidemic model with saturated infection rate and treatment function. The dynamics of the model is studied from the point of view of stability and bifurcation. Basic reproduction number is obtained and it is shown that the model system may possess a backward bifurcation. The global asymptotic stability of the endemic equilibrium is studied with the help of a geometric approach. Optimal control problem is formulated and solved. Some numerical simulation works are carried out to validate our analytical results. Keywords Epidemiology Basic reproduction number Global stability Backward bifurcation Optimal treatment control
1 Introduction Rapid enhancement of medical sciences enables us to detect and control different deadly disease in a better way than it was in earlier days. But infectious diseases on human populations, mainly spread due to the direct contact or through another & Soovoojeet Jana [email protected] Swapan Kumar Nandi [email protected] T. K. Kar [email protected] 1
Department of Mathematics, Abhedananda Mahavidyalaya, Sainthia, Birbhum, West Bengal 731234, India
2
Department of Mathematics, Nayabasat P. M. Sikshaniketan, Paschim Medinipur, West Bengal 721253, India
3
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India
123
S. Jana et al.
medium, still threaten to become epidemic in different parts of the globe. Some different control tools are developed and some are developing to control different infectious diseases but as time progresses, some new and deadly infectious diseases, with complex nature and structure, are entering in the human society. Therefore, researchers and experts from different fields other than medicine have engaged themselves to find out the optimal way to control a disease, if complete eradication is not achieved. Mathematical modeling is an important and effective tool to study both the nature and control methodology of an infectious disease. Almost 250 years ago, Bernoulli (1760) presented some works on human epidemiology with the help of mathematical models. Later on, towards the beginning of 2nd quarter of the twentieth century, Kermack and McKendric (1927) presented the classical SIR model on epidemiology. Recently, some good works have been done and some are going on to control the effect of infectious diseases with the help of mathematical modeling (see Eckalbar and Eckalbar 2011; Hosono and Ilyas 1995; Hu et al. 2012; Kar and Jana 2013a; Wang 2006; Wang et al. 2012; Zhou and Fan 2012). The form of disease transmission from an infected person to a healthy person is really an important task to sort out. The most common and usual form of this transmission is the mass action form where it is assumed t
Data Loading...
