Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate

In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population.


Introduction
COVID-19 affects the human population worldwide and is caused by the SARS-COV-2 virus. The extent of COVID-19 is reduced in a few countries due to the development of vaccination, wearing of proper face masks, frequently hand washing, maintaining social distancing, avoiding public and crowded places, advertisements, adequate medical facilities, and following the guideline given by the World Health Organization (WHO) and national governments. But still, significant new cases of COVID-19 are registered in many regions across the world.
So far, there have been many attempts to study the effects of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of disease, and limitation in the treatment of COVID-19 by using elaborate models with five to eight compartments classes. We proposed an SIR model with nonlinear infection and treatment rates to overcome this increased complexity.
Infection and treatment rates play a vital role in reducing the dimension of the COVID-19 model, so it turns to enable better analysis. The WHO declared COVID-19 as a global pandemic in march 2020 WHO (2020). In this situation, it is unreasonable to choose a bilinear infection rate to formulate the COVID-19 model. Hence to overcome this disadvantage we chose the nonlinear infection rate introduced by Beddington and DeAngelis Beddington (1975), DeAngelis et al (1975). The Beddington-DeAngelis infection rate is defined as where β is the transmission rate of disease, ξ 1 is a measure of inhibition for the susceptible population, and ξ 2 is a measure of inhibition for the infected population. The common population-wide effort to reduce transmission of COVID-19 is to impose a lockdown. One prevention measure used by susceptible populations includes proper use of face masks and social distancing to reduce disease transmission. Prevention measures taken by the infected population include quarantine and selfisolation so as to reduce transmission or aid recovery from the disease. In our model, β is used to capture the impact of lockdown, ξ 1 is used to capture the impact of face masks and social distancing, and ξ 2 is used to represent the impact of quarantining.
The selection of treatment rate is also very crucial in formulating a COVID-19 model. During the initial phase of COVID-19, there were multiple crises in accessing hospitals, obtaining vaccines, securing oxygen cylinders, and other medical resources. Over time, the situation improved slowly in terms of the availability of hospitals, vaccines, and other medical facilities. To describe this situation, we chose the Holling type-II treatment due to its ability to capture this scenario. Zhou and Fan introduced the Holling type-II treatment rate and defined it as Zhou and Fan (2012) where j is the treatment rate of the disease, and ρ is the limitation in treatment availability.
To the best of our knowledge, SIR models with a Beddington-DeAngelis infection rate, a Holling type-II treatment rate with Mittag-Leffler law have not been used to examine the effects of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of disease, and limitation in treatment resources on COVID-19.
The manuscript is organized as follows: in section 2, we provide a necessary definition of the non-local and non-singular Atangana-Baleanu operator. In section 3, the SIR model with the Mittag-Leffler law is formulated. The qualitative properties and local and global asymptotic stability of the model, under some conditions, are discussed in section 4. In section 6, the numerical results are discussed. Finally, some conclusions are provided in section 7.

Preliminaries
In this section, we provide a necessary definition of fractional calculus that is used in the development and numerical simulation of the model Atangana and Baleanu (2016).
where a a = -+ is the Riemann-Liouville fractional integral.
Definition 4. The Laplace transform of the ABC derivative of a function f (t) of order a > 0 is defined as

Formulation of a mathematical model
Here, we consider an SIR model with Beddington-DeAngelis type infection rate and Holling type II treatment rate. The basic SIR model is described by the following system of differential equations.
with the given initial conditions S(0) = S 0 0, I(0) = I 0 0, and R(0) = R 0 0. The variables, parameters, and biological interpretation are given in table 1. The susceptible populations in compartment models use their memory to prevent infection. Due to this reason, the fractional-order model achieved high attention in modeling a biological process. Hence we convert the classical model (7) into the fractional-order one to enhance the accuracy of the model Podlubny (1998), Magin (2004), Hanert et al (2011), Baleanu et al (2012, Joshi and Jha (2018, 2021a, 2021b, 2022. However, there are many other definitions of the fractional derivative available to deal with the memory effect. The Caputo derivative is well known and widely used in the literature for the development of a fractional order model. The authors in Swati (2022) have studied the Caputo fractional order SIR model with BeddingtonDeAngelis incidence and Holling type II treatment rate. But the Caputo derivative consists of a singular power kernel that leads to many complexities in the problem. To overcome this disadvantage, we consider the ABC fractional derivative that consists of a non-singular Mittag-Leffler kernel Atangana and Baleanu (2016). Thus the classical model (7) is transformed to the fractional-order in the ABC sense as where S(0) = S 0 , I(0) = I 0 , and R(0) = R 0 .

Model analysis
In the epidemiological model, we must ensure that the solution is non-negative and bounded for all time. Let Then the generalized mean value theorem states that Proof. First, we show that the solutions of model (8) are positively invariant in the region Ω. We have g j r Hence for all  t 0, and from lemma 1, we concluded that all the solutions of the model (8) are positively invariant. Next, we prove the boundedness of the model (8) with the framework of equation (3).
By summing up all the equations of the model (8), we have

Consider the following initial value problem
Using equation (6), we obtain a a t ma t ma = L - Further, equation (14) can be simplified as Applying the inverse Laplace transform to (15) and using lemma 2 we obtain m a m m t mt It is observed that as t → + ∞ , and using the fact that  Proof. In model (8), the susceptible and infected populations are independent of the recovered populations. Due to this fact, it is convenient to drop the third equation of the model (8). Hence model (8) can be recast as follows Thus, to find the E 0 and * E we solve the following equations Now combining equations (17) and (18) we have the following algebraic system ) Further information about * E is provided in theorem 9.

Basic reproduction number
Here, we obtain the basic reproduction number (R 0 ) for the proposed model (8). In disease dynamics, R 0 captures the impact of model parameters on the spread of disease and so it plays a key role in developing any strategies for disease control. To find R 0 using a next-generation matrix method Diekmann et al (2010), we consider the following system = - The Jacobian matrices of new infected terms, F(x), and other transfer terms, Therefore, R 0 is the spectral radius of the matrix FV −1 and is defined by the following expression b m x j m h g Next, we derive the local stability analysis of the model (8) to determine whether the disease persists or not.  (8) is locally asymptotically where the components of J are given by According to the fundamental theorem of algebra, the cubic equation (32) possesses at most three real nonnegative roots. Here we focus on finding the endemic equilibrium. Using Descartes rule of signs Wang (2004), the equation (32) possesses a unique non-negative I * under the following conditions: Then the Routh-Hurwitz stability criteria Matignon (1996) confirm that the roots of equation (40) have negative real parts provided that a i > 0, i = 1, 2. Thus, the endemic equilibrium point E * = (S * , I * ) of the model (8) (8) Applying the ABC derivative on both the side of equation (43), we have Then by using the linearity property and lemma 8 on equation (44), we have Next, we discuss two cases based on j.
This means that j m h g m h g Further modification gives = j m h g m h g Otherwise, it is unstable.
Proof. Consider a candidate positive definite Lyapunov function W  + L S I R , : Applying the ABC derivative to both the side of equation (53), we have Then by using the linearity property and lemma 8 on equation (54), we have

Numerical results
In this section, we perform a numerical simulation of the model (8) to validate the analytical work. The numerical values of the biological parameters used in the simulation are defined in table 2.
For the given set of values, we calculated the endemic equilibrium point of the model (8) as E * = (S * , I * ) = (52.6, 17.97). This value satisfies the result of theorem 10. Next, to show that E * is globally asymptotically stable, we consider a random set of initial conditions as defined in table 3. Figure 1 is simulated for seven random sets of initial conditions for the susceptible and infected population. All solution trajectories corresponding to various different initial conditions approach the equilibrium E * = (52.6, 17.97) asymptotically, irrespective of the starting point. Thus from this numerical result, it is observed that E * is globally asymptotically stable.
The initial conditions used in the numerical simulation are chosen as S 0 = 94, I 0 = 6, and R 0 = 0 for susceptible, infected, and recovered populations, respectively. Using the numerical scheme developed by M. Toufik and A. Atangana Toufik and Atangana (2017), model (8) is simulated to observe the effects of face mask, social distancing, quarantine, lockdown, immigration, treatment rate of disease, limitation in treatment resources, and also different orders of ABC derivative. The order of the ABC derivative is chosen as α = 0.8, 0.9, 1 in figures 2(a) to (c), respectively. Figure 2 shows the transmission dynamics of the state variables with an arbitrarily chosen time interval of 300 days. It is observed from figure 2(a), that as time increases the susceptible population is decreased and as a consequence, the infected population is increasing. Finally, the susceptible and infected populations move to achieve a steady state equilibrium E * . A sharp fall and pick up is observed in the susceptible, infected, and recovered population in the case of the ABC derivative with α = 1. As the fractional order decreases, the sharp fall and pick up gradually vanish due to the non-local and non-singular effects of the ABC derivative and faster attainment of the equilibrium steady state. Figure 3 illustrates the effect of preventive measures taken by susceptible populations, such as the proper use of face masks and social distancing, on the transmission dynamics of the infected population. It is observed from figure 3(a)-(c), that if the use of face masks and social distancing is properly followed, so that the value of ξ 1 is  Table 3. Random set of initial conditions. Initial condition (IC)  IC1  IC2  IC3  IC4  IC5  IC6  IC7   S 0  20  30  11  15  60  70  80  I 0  3  8  27  35  40  30  20 increased, then the infected population dramatically decreases, and moves towards an equilibrium state. The profile suddenly attains a pick-up for the first 50 days and then decreases but the pick-up gradually vanishes due to the ABC derivative as it includes the memory of infected populations. Figure 4 illustrates the effect of preventive measures taken by the infected population, such as quarantine and self-isolation, on the transmission dynamics of the infected population. It is observed from figures 4(a)-(c) that if the quarantine and self-isolation for infective populations are properly followed, so that the value of ξ 2 is increased, then the infected population dramatically decreases. The decrease in the infected population means a corresponding increase in the susceptible or recovered populations.
The common population-wide effort to reduce transmission of COVID-19 is to impose a lockdown. Figures 5(a)-(c) show the effect of lockdown on the infected population for ABC derivatives α = 0.8, 0.9, 1 respectively. It is observed from figures 5(a)-(c), that as the lockdown is imposed, then the disease transmission rate is decreased, and ultimately the infected population is decreased. Also, from figure 5 it is observed that the lockdown and memory effect plays a vital role in reducing the infected populations.
Sealing borders, closing entry into a country, or restricting air, sea, and other forms of travel, are common measures used to reduce infection rates. Figure 6 shows the effect of travel (i.e. immigration) on the infected population over time t. It is observed from figures 6(a)-(c) that as immigration is restricted, there is a dramatic reduction in the infected populations. As the value of Λ is increased, so there is a sudden rise in the infected population due to the new arrival of an infected immigrant population. Figure 7 shows the effect of treatment rates on the transmission dynamics of the infected population. It is observed from figures 7(a)-(c), that with an increase in treatment rates such as better access to hospital facilities, vaccination, awareness, and cure, that is with an increasing value of j, then there are dramatic falls in the infected population number. Also, it is observed that in figure 7(c) the infected population rises suddenly and attains a sharp pick as compared to figure 7(a). It can be interpreted that this situation arises from the ABC derivative. Due to the ABC derivative or memory of the infected population, a gradual reduction in the initial days and a sharp pick is observed. Figure 8 shows the effect of a limitation in treatment availability on the infected population. It is observed from figures 8(a)-(c), that with a limitation in a hospital facility, lack of a vaccine or lack of oxygen cylinder, and so an increase in ρ, then the infected population increases.

Conclusion
In this paper, we developed an SIR model with the ABC derivative to investigate the effects of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of disease, and limitation in treatment resources. A Beddington-DeAngelis infection rate and Holling type-II treatment rate were used to capture the impact of model parameters on the infected population. The SIR model (8) with the ABC derivative has possessed two equilibrium points, namely disease-free and endemic equilibrium points. The disease-free equilibrium point of the model (8) is locally asymptotically stable if R 0 < 1. The endemic equilibrium point of the model (8) is locally asymptotically stable if R 0 > 1 and under the conditions given in inequality (36). The disease-free equilibrium  (8) is globally asymptotically stable if R 0 1 and j = 0. When j ≠ 0, the model is globally asymptotically stable if R 1 1. The endemic equilibrium point of the model (8) is globally asymptotically stable if R 0 > 1 and under the conditions of inequality (51). The global stability of the endemic equilibrium point has been also shown by the numerical results. From the numerical results, it has been observed that face masks and social distancing reduce the infected population. Also, quarantine, lockdown, restricted immigration, and effective treatment rates significantly reduce the infected population over time, whereas limitation in the availability of treatment raises the infected population. Furthermore, the numerical results suggest that COVID-19 can be reduced or eliminated from the community by adhering to the guideline of WHO such as the proper wearing of face masks, social distancing, quarantine, lockdown, and restricted immigration. The non-local and non-singular properties of the ABC derivative have advantages compared with integer-order derivative models in which we require more compartments to capture the disease complexity. The present model can be further used to develop a control strategy or predict the number of the infected population in future pandemics.