Modelling and control of a fractional-order epidemic model with fear effect

Energ. Ecol. Environ. (2020) 5(6):421–432 https://doi.org/10.1007/s40974-020-00192-0 ORIGINAL ARTICLE Modelling and control of a fractional-order epidemic model with fear effect Manotosh Mandal1,4 • Soovoojeet Jana2 T. K. Kar4 • Swapan Kumar Nandi3 • 1 Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk, West Bengal 721636, India Department of Mathematics, Ramsaday College, Amta, Howrah, West Bengal 711401, India 3 Department of Mathematics, Nayabasat P.M. Sikshaniketan, Paschim Medinipur, West Bengal 721253, India 4 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India 2 Received: 11 July 2020 / Revised: 6 September 2020 / Accepted: 9 September 2020 / Published online: 26 September 2020 Ó The Joint Center on Global Change and Earth System Science of the University of Maryland and Beijing Normal University 2020 Abstract In this paper, we formulate and study a new fractional-order SIS epidemic model with fear effect of an infectious disease and treatment control. The existence and uniqueness, nonnegativity and finiteness of the system solutions for the proposed model have been analysed. All equilibria of the model system are found, and their local and also global stability analyses are examined. Conditions for fractional backward and fractional Hopf bifurcation are also analysed. We study how the disease control parameter, level of fear and fractional order play a role in the stability of equilibria and Hopf bifurcation. Further, we have established our analytical results through several numerical simulations. Keywords Fractional derivative  Fractional SIS epidemic model  Fractional stability conditions  Fractional Hopf bifurcation  Fear effect  Fractional backward bifurcation 1 Introduction Infectious diseases have become one of the most threatening issues in todays lifestyle. The infectious diseases including chickenpox, measles, cholera, tuberculosis, influenza, SARS, COVID-19, etc., have massive impact & Soovoojeet Jana Manotosh Mandal Swapan Kumar Nandi T. K. Kar than the other types of noninfectious diseases as these types of diseases can be transmitted from one individual to another, some are spread out by bites from insects or animals, and some diseases are acquired by consuming flagitious water or food or being exposed to organisms in the environment. Hence, an infectious disease may spread in a huge region throughout the globe within a very short time period. Due to the improvement of lifestyle of common people and massive enhancement in transportation and globalization, an infectious diseases become pandemic in a less amount of time compared to earlier days (for example Spanish flu in twentieth century had taken a long period of time to become a pandemic compared to the pandemic due to COVID-19 in ongoing time period). Thus, not only to study the dynamics of an infectious disease, but also to control or determining the procedure to control an infectious disease, researchers from various fields have engaged themselves. Several mathematical tools induce a great affect in regulating many infectious diseases. Mathematical modelling is one of the popular and commonly used mathematical tools that can be utilized efficaciously to monitor several infectious diseases. The first effective research work on studying an infectious diseases using mathematical model was probably first done by Kermack and Mckendric (1927). There are several research works where the authors have been studied mathematical models on controlling infectious diseases (see Zhou et al. 2014; Jana et al. 2016, 2017a). In his book, Murray (2002) has analysed theoretical works on some simple SI, SIS, SIR epidemic models. Present days mathematicians formulate more complex epidemic model systems which are almost identical to the real-world problems. In mathematical studies of an epidemic model, proper control strategies are 123 422 significant tools in monitoring and controlling the infectious diseases. In this regard, two most effective and commonly used control tools used in controlling diseases are vaccination and treatment (Kar and Jana 2013a, b). However, isolation (see Jana et al. 2017b), insecticide (to control vector-borne disease)(see Kar and Jana 2013b), etc., are also used in recent days. In the present work, we consider fear factor due to the infectious disease which is an another important parameter for epidemic model. People generally got scared and make a significant distance to prevent the infectious disease. Hence, fear induced by the infectious disease compelled susceptible population to isolate which actually decreases the birth rate of population and also survival of adults is affected consequently. Due to the SARS outbreak in HongKong (SARS started on November 2002; peaked on March 2003; eliminated on June, 2003), the birth rate dramatically had fallen from 8.742 (2002) to 8.436 (2003) and again it increased to 8.558 in 2004 (see Worldbank (2018)). In our work, we formulate a new fractional-order SIS epidemic model with fear effect (Clinchy et al. 2013; Wang et al. 2016) and treatment control to eradicate the disease. Mathematical models based on ordinary differential equations depict the interactions between the population classes (e.g. susceptible-infected), and it is a classical approach in theoretical epidemiology. In recent days, researchers are concerned in developing mathematical model by the fractional-order differential equations because it is an important apparatus for the study of the memory and also some hereditary properties of several biological components and also it has a very close relation to the fractal theory. These are the main advantages of the fractional-order derivatives which are not included in the models based on the ordinary-order derivatives. Therefore, fractional-order derivatives are more naturalistic than the ordinary derivatives. The models constructed by the fractional-order derivatives have been widely applied in different field of research after the some famous books and research works on fractional-order differential equations (Diethelm and Braunschweig 2003; Hilfer 2000; Kilbas et al. 2006; Miller and Ross 1993; Petras 2011; Podlubny 1999; Sabatier et al. 2007; Sengupta et al. 2020; Yadav et al. 2020; Karthikeyan and Arul 2020). However, a fractional-order derivative may be defined in several ways. The most popular and commonly useful definitions of fractional-order derivatives are in the sense of Riemann– Liouville, Grünwald–Letnikov and Caputo definitions (Petras 2011; Podlubny 1999). Since, in the definition due to Caputo, the initial conditions can be expressed in a similar fashion as the integer-order differentiation, it is the most frequently used fractional-order derivative in mathematical modelling. There are very few theories to analyse 123 M. Mandal et al. the dynamical behaviour of the mathematical models with fractional-order de (...truncated)