Stability of a Fractional-Order Epidemic Model with Nonlinear Incidences and Treatment Rates
- PDF / 777,534 Bytes
- 13 Pages / 595.276 x 790.866 pts Page_size
- 13 Downloads / 157 Views
(0123456789().,-volV)(0123456789().,-volV)
RESEARCH PAPER
Stability of a Fractional-Order Epidemic Model with Nonlinear Incidences and Treatment Rates Abhishek Kumar1 Received: 23 June 2020 / Accepted: 3 August 2020 Ó Shiraz University 2020
Abstract In the present study, Caputo derivative-based a new fractional-order epidemic model is presented along with two explicit saturated incidences and saturated treatment rates. For this, a new fearful population compartment is incorporated into the susceptible-infected-recovered compartmental model, which emphasizes to consider two specific incidence rates: one from susceptible individuals’ compartment to infected individuals’ compartment and another from fearful individuals’ compartment to infected individuals’ compartment. The model is analyzed mathematically for disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability of the model’s equilibria is investigated for local as well as global behaviors. It is investigated that DFE is locally asymptotically stable whenever the basic reproduction number R0 is less than one, and EE exists when R0 crosses one. The EE is proved to be locally stable under certain conditions. Further, global stability behavior is investigated for both equilibria using the basic reproduction number R0 . Finally, numerical results are presented in support of the analytical findings. Keywords Fractional-order epidemic model Saturated incidence rates Holling type II treatment rate Stability Simulation
1 Introduction For the predictions and insights concerning the time-evolution of the epidemics of infectious diseases, several mathematical compartmental models have been proposed by researchers; for example, see (Gumel et al. 2006; Kumar et al. 2020a, b; Kumar and Nilam 2019; Kumar et al. 2019; Huang et al. 2010; Goel et al. 2020a; Alexander et al. 2004; Xu and Ma 2009; Wang 2002; Michael et al. 1999; Goel et al. 2020b). However, in these studies, an integer order compartmental model is considered as a vital tool. Recently, it has been seen that the fractional-order differential equations and their applications have been intensively used in biology, physics, chemistry, biochemistry, engineering, and medicine; see (Matignon 1996; Rostamy and Mottaghi 2016; Ahmed et al. 2006; Ye and Xu 2019; Khan et al. 2020; Atangana 2017; Atangana 2020; Cattani & Abhishek Kumar [email protected]; [email protected] 1
Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida 201310, India
and Pierro 2013; Singh et al. 2018; Jothimani et al. 2019; Valliammal et al. 2019; Gao et al. 2019; Gao et al. 2020a, b, c; Sa´nchez et al. 2020). As opposed to the integer-order or ordinary derivative, which is a local operator, the fractional-order derivative has the main property called memory effect. The state of the integer order systems is a memoryless procedure. But, when a disease emerges and spreads, then the knowledge and experience of individuals about that disease influence their behaviors
Data Loading...
