The Stability Analysis and Transmission Dynamics of the SIR Model with Nonlinear Recovery and Incidence Rates

Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No. 1888, Adama, Ethiopia Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’Sik, Hassan II University, P. O. Box 7955, Sidi Othman, Casablanca, Morocco Department of Computing and Technology, Abasyn University, Peshawar Ring Road, Peshawar 25000, Khyber Pakhtunkhwa, Pakistan


Introduction
Infectious diseases are key troubles for human beings [1,2]. ey can cause disability, mortality as well as produce economic and social problems for society. Infectious diseases are disorders which are happened by organisms such as fungi, parasites, viruses, or bacteria. Several organisms exist inside and on our bodies. Some infectious diseases can be transmitted from person to person, and some are passed by insects or other animals. Symptoms and signs di er depending on the organism causing the infection. Mild infections may respond to rest and home medication, while some dangerous diseases might require hospitalization. Malaria, measles, pneumonia, diarrheal diseases (cholera), HIV, tuberculosis (TB), and more recently COVID-19 are the major deadly infectious diseases [3,4].
Infectious diseases create a constant threat to the health of individual and public. TB is one of the dangerous infectious diseases that cause many deaths globally. In undeveloped countries, conventional methods are utilized in diagnosing TB [5][6][7]. COVID-19 is an infection that can cause lung problems such as pneumonia and, in the most rigorous cases, acute respiratory syndrome. e COVID-19 is an infectious sickness that can be spread through aerosols of droplets and by direct contact with the people [8][9][10][11][12][13][14].
is disease has extended rapidly over the world after it was displayed to have a more signi cant level of irresistible and pandemic threat than SARS. Because of the sharp increment level of spread, the WHO articulated the COVID-19 outbreak as a pandemic on 11 March 2019. e indications of COVID-19 are cough, fever, exhaustion, sputum, cerebral pain, lymphopenia, and run dyspnoea [15]. In serious cases, COVID-19 can cause pneumonia and even death [16]. To stop the spreading of COVID-19 , different governments estimated different preventive measures like wearing a mask, remaining six feet social distance, washing hands regularly, and staying away from sick individuals. Vaccines also serve a serious role to prevent people from infectious diseases and contribute toward controlling the spread of the disease. e vaccinated people also need to be attentive of the additional defensive behaviors required to manage the disease. e mathematical modeling is a helpful device to concentrate on the procedure that how an irresistible infection can reach out into a population. e researchers are using fractional order [17,18] as well as integer order [19][20][21][22][23][24] mathematical models to discuss epidemic infectious diseases; however, mathematical models of integer order are among the most studied problems in the world. e disparate kinds of recovery and incidence rates play a significant role to examine the dynamical behavior of epidemic disease models. Many researchers have used these saturated rates in their works [19][20][21][22][23][24][25][26][27][28][29][30]. For instance, the standard bilinear incidence rates βSI have been regularly utilized in epidemiological models [19][20][21][22][23]. In Ref. [24], the SEIR epidemic model with nonlinear incidence rate, vaccination, and quarantine strategies is provided. Yusuf and Benyah [25] have presented and broke down a discrete SIR epidemic model with nonlinear recovery rates. Anderson and May [26] offered the nonlinear incidence rate aSI/1 + aI which got drenched due to a cluster of infective persons at a high level of disease. One more nonlinear incidence rate aSI/1 + a 1 S + a 2 I is extensively presented in Refs. [27][28][29]. In Ref. [30], the author modified the above incidence rate to nonlinear recovery rate aSI/1 + a 1 S + a 2 I 2 e nonlinear recovery rate can display rich dynamics such as saddle-node, backward, Hopf, and Bogdanov-Takens bifurcations. e bilinear incidence rates do not take into account the impact of the preventive measures such as mask-wearing, quarantine, and isolation, which play an important role to manage the spread of an infectious disease. Continuous compression and impact techniques also make the feedback of the incidence rate more leisurely compared to the standard bilinear structure aSI erefore, it is crucial to take into account the influence of preventive measures as well as preventive steady decline on the transmission of infectious diseases. In order to address the above problems, the author [31] plan to alter the SIR model by taking the nonlinear Monod type equation as incidence rate to study the effect of preventive reduction on the transmission of infectious diseases. e reason behind considering Monod type equation as incidence rate was to explore the effect of intervention decrease on the spread of infectious disease.
e Monod equation as incidence suggests that the incidence rate is low for little quantities of infected individuals due to rigorous intervention; however, it increases as the number of infected individuals increases until it becomes autonomous of the diseased subpopulation. e author explored the essential analytical results, containing the stability of disease-free and disease endemic equilibria for the continuous model. e aim of the present paper is to use a more advanced NSFD scheme to verify different characteristic of the model to display its sustainability and biological vitality. e purpose is to develop policies for preventing or regulating disease transmission among individuals and to better understand disease dynamics. e NSFD scheme constructed for the model is dynamically reliable with the original system for any step size. Our theoretical and numerical outcomes show that the NSFD scheme conserves the necessary qualitative properties of the continuous model. erefore, this scheme is not only reasonable for the model but also the outcomes acquired through this scheme are extremely proficient and precise. e rest of paper is arranged as follows: In Section 2, the SIR system is presented and the associated parameters are explained. In the same section, the expression for reproduction number, and disease-free and disease endemic equilibria are obtained for the system. e NSFD scheme is established for the system in Section 3. e local asymptotic stability (LAS) of disease-free and disease endemic equilibria for the discrete model obtained by the NSFD scheme are proved by the Jacobian method and Schur-Cohn conditions; however, the Enatsu criterion and Lyapunov function are employed to discuss the global asymptotic stability (GAS). Some important conclusions are given in the final section.

System of Differential Equations.
e dynamical system [31] with nonlinear recovery and Monod type equation as incidence rates including three differential equations is given as follows. e total population N(t) is distributed into three classes: susceptible S(t), infected I(t), and recovered R(t) where N(t) � S(t) + I(t) + R(t). e detailed description of the model can be seen in Figure 1.
e information about the variables and parameters used in model (1) is given in the following table (seeTable1).

Disease-Free and Endemic Equilibria of SIR Model.
e disease-free and endemic equilibria can be found by solving the following equations: To find the disease-free equilibrium (DFE) point, we take all other classes equal to zero except the susceptible class.

Basic Reproduction Number (R 0 ).
e fundamental reproduction number is one of the most significant threshold quantities used in epidemiology.
is is indicated by R 0 which define the number of secondary infected cells. By applying the next-generation matrix method [32], we can easily find R 0 for model (2). Let By putting DFE point into E 0 , we get As R 0 is the spectral radius of FV − 1 , we get

The NSFD Scheme for the Modified SIR Model
In 1994, the concept of NSFD was given by Mickens [33]. e NSFD schemes [34][35][36][37][38] is used to construct general method to find the numerical solution of ordinary and partial differential equations by generating discrete models. According to Shokri et al. [37], the exploration of NSFD schemes depend on two factors. First, how to estimate nonlinear terms in the most suitable way, and second is how to the discretized the derivative. One of the general methods for discretization is the forward finite difference approximation for derivative of the first order. In the standard form, the first order derivative dy/dx is represented as y(x + h) − y(x)/h, where h represents step size. According to Mickens, this term can be expressed as is an increasing continuous function known as denominator function. Different expressions for φ(h) can be seen in Refs [37][38][39]; however, we will consider φ(h) � exp(τh) − 1/τ in upcoming calculations. e quantity ϕ(h) represents the time step size and should be nonnegative. We will show that NSFD is not only positive forever but converge quickly to DFE point for any step size. To discuss all the above properties, we first develop the NSFD scheme for system (1) in the following subsection. (1), we indicate S n ,I n , and R n as the numerical approximations of S(t), I(t), and R(t) at t � nh, where n � 0, 1, 2, . . ., and h denotes the time step size which should be nonnegative. en, based on the above all information, we can construct the NSFD scheme for model (1) as follows: Minimum recovery rate c 1 Maximum recovery rate u Represents the mediation level and decide the figure of the frequency rate as an element of the tainted population I It is assumed that all parameters α, u, τ, c 0 , c 1 , δ, u, θ are positive constants. e first two equations of the given model (1) do not depend on the third equation. Hence, it is sufficient to take the first two equations; that is, the target in the upcoming discussion will be the following reduced model: S n+1 − S n ϕ � C − αI n S n+1 u + I n − τS n+1 ,

Construction of the Discrete NSFD Scheme. For model
We assume that the initial values S 0 , I 0 , and R 0 of the discrete NSFD scheme (7) are nonnegative. e total population P n from the discrete NSFD scheme (8) satisfy P n � S n + I n + R n such that If we take ϕ � exp(τh) − 1/τ, then the solution of (8) satisfies where P 0 � S 0 + I 0 + R 0 for any h > 0. When δ � 0, it can be expressed that the total population of the discrete NSFD scheme (7) is precisely the same as that of model (1). e scheme (7) is implicit; however, it can be expressed explicitly as follows: S n+1 � S n + ϕC 1 + ϕ Ψ n + τ , I n+1 � I n + ϕΨ n S n+1 1 + ϕ c 0 + c 1 − c 0 θ/θ + I n +(δ + τ) , where Ψ(z) � αz/u + z and Ψ n � Ψ(I n ). As all parameters in (10) are positive, therefore S n ≥ 0, I n ≥ 0 and R n ≥ 0 for all n > 0 and for any h > 0. ese findings confirm that the discrete NSFD scheme (10) maintains the nonnegative solutions for any step size h.
It is clear that R n is not included in first two equations of system (10). erefore, for further investigation, we only take the following reduced model: After simple computation, we can prove that the discrete NSFD model (11) always has a unique DFE point E 0 � (C/τ, 0) and DEE point E * � (S * , I * ). ese DFE and DEE points, and their continuation situation is totally the same as the continuous model (1) no matter what is h. In the upcoming subsection, we first examine the LAS of the above discussed equilibrium points.

Local Stability of the Discrete NSFD Scheme. To investigate the LAS of DFE and DEE points, we consider
� G(S, I).
Proof. We take the Jacobean matrix as After easy calculation and the putting E 0 , we get erefore, the Jacobean matrix (13) becomes e above matrix clearly gives the following eigenvalues: λ 1 � 1/1 + ϕτ < 1 and λ 2 � 1 + ϕC/1 + ϕc 1 + (δ + τ)τ. e eigenvalue λ 2 can also be expressed as In the following, we give the statement of Schur-Cohn criterion [20,40] which plays an important role in the investigation of LAS of DEE point E * .

Lemma 1.
e quadratic equation λ 2 − Tλ + D � 0 roots satisfy |λ i | < 1, i � 1, 2 if and only if the below three conditions are fulfilled: Mathematical Problems in Engineering Theorem 2. If R 0 > 1, then the DEE point E * of the NSFD scheme (12) is LAS for any step size h.
3 Direct calculation presents that if R 0 > 1, then Based on the Schur-Cohn criterion discussed in Lemma 1, we conclude that DEE point E * is LAS if R 0 > 1 for any step size h.

Global Stability of the Discrete NSFD Model.
In the following, we now explore the GAS of DFE and DEE points by applying an appropriate Lyapunov function. (12) is GAS, as shown in Figure 2(a)-2(d).
Proof. We can define the discrete Lyapunov function as follows: Mathematical Problems in Engineering 5   Mathematical Problems in Engineering where g(y) � y − 1 − ln y ≥ g(1) � 0. Depending on the first and second equations of system (7), from (21), we attain By using Entasu et al.'s [41] criterion ln S 2 /S 1 ≥ S 2 − S 1 /S 2 , we get Definitely, if R 0 ≤ 1 then U n+1 − U n ≤ 0 for all n ≥ 0. is presents that U n is monotonic decreasing sequences. As U n ≥ 0, then lim n⟶∞ U n ≥ 0 and lim n⟶∞ U n+1 − U n � 0 Accordingly, we get lim n⟶∞ S n+1 � S 0 and lim n⟶∞ (δ + τ)I n+1 � 0. It is clear that if R 0 ≤ 1, then lim n⟶∞ I n+1 � 0. Hence, we conclude that E 0 is GAS.  Hence, ΔW n is a monotonic decreasing sequence. By applying the same techniques as those in eorem 3, we can show that lim n⟶∞ (W n+1 − W n ) � 0. erefore, we get lim n⟶∞ S n+1 � S * and lim n⟶∞ I n+1 � I * for all h. Hence, the proof is completed.

Conclusions
In the present work, we have discussed the SIR epidemic model with nonlinear recovery and Monod type equation as incidence rates. We calculated the basic reproduction which plays an essential role in the investigation of local and global stability of DFE and DEE points. e NSFD scheme is constructed for the model which is not only unconditionally convergent but also gives more accurate results which are mathematically and biologically reasonable. By using different criteria and conditions, the LAS and GAS of both DFE and DEE points are proved for the NSFD scheme. It is confirmed that for all time step sizes, the discrete NSFD scheme is vigorously reliable with the related continuous model. e numerical simulations are important to study the complex dynamical behavior the nonlinear models [34][35][36][37][38][39][40][41][42][43].
erefore, numerical simulations have also been presented to verify the sustainability of the theoretical results. e advantages of NSFD scheme are provided, which explains that the outcomes of NSFD scheme are qualitatively precise and efficient. In the same manner, the NSFD can be constructed and analyzed for other generalized epidemic models.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.