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

Authors: Ihsan Ullah Khan; Muhammad Qasim; Amine El Koufi; Hafiz Ullah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used; however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.


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## Body

## 1. Introduction

Infectious diseases are key troubles for human beings [1, 2]. They 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 differ 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 [5–7]. COVID-19 is an infection that can cause lung problems such as pneumonia and, in the most rigorous cases, acute respiratory syndrome. The COVID-19 is an infectious sickness that can be spread through aerosols of droplets and by direct contact with the people [8–14]. This disease has extended rapidly over the world after it was displayed to have a more significant 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. The 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. The vaccinated people also need to be attentive of the additional defensive behaviors required to manage the disease.The mathematical modeling is a helpful device to concentrate on the procedure that how an irresistible infection can reach out into a population. The researchers are using fractional order [17, 18] as well as integer order [19–24] mathematical models to discuss epidemic infectious diseases; however, mathematical models of integer order are among the most studied problems in the world. The 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 [19–30]. For instance, the standard bilinear incidence rates βSI have been regularly utilized in epidemiological models [19–23]. 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+a1S+a2I is extensively presented in Refs. [27–29]. In Ref. [30], the author modified the above incidence rate to nonlinear recovery rate aSI/1+a1S+a2I2 The nonlinear recovery rate can display rich dynamics such as saddle-node, backward, Hopf, and Bogdanov–Takens bifurcations.The 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 structureaSI Therefore, 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. The reason behind considering Monod type equation as incidence rate was to explore the effect of intervention decrease on the spread of infectious disease. The 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. The author explored the essential analytical results, containing the stability of disease-free and disease endemic equilibria for the continuous model. The 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. The purpose is to develop policies for preventing or regulating disease transmission among individuals and to better understand disease dynamics. The 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. Therefore, this scheme is not only reasonable for the model but also the outcomes acquired through this scheme are extremely proficient and precise.The rest of paper is arranged as follows: In Section2, 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. The NSFD scheme is established for the system in Section 3. The 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.


## 2. Mathematical Model and Equilibria

### 2.1. System of Differential Equations

The dynamical system [31] with nonlinear recovery and Monod type equation as incidence rates including three differential equations is given as follows. The total population Nt is distributed into three classes: susceptible St, infected It, and recovered Rt where Nt=St+It+Rt. The detailed description of the model can be seen in Figure 1.(1)dSdt=C−αISu+I−τS,dIdt=αISu+I−γ0+γ1−γ0θθ+II−δ+τI,dRdt=γ0+γ1−γ0θθ+II−τR.Figure 1
The detailed illustration of SIR model with nonlinear recovery and Monod type equation as incidence rates.The information about the variables and parameters used in model (1) is given in the following table (seeTable1).Table 1
The detail descriptions of parameters given in system (1).
ParametersParameters descriptionSSusceptible populationIInfected populationRRecovered populationCConstant quantity addressing the rate of birthαIS/u+IRate of disease transmissionτRate of natural deaths at each compartmentδRate of deaths due to the diseaseθImpact of the quantity of clinic bedsαConstant that represents the diseases infection forceγ0Minimum recovery rateγ1Maximum recovery rateuRepresents the mediation level and decide the figure of the frequency rate as an element of the tainted populationIIt is assumed that all parametersα,u,τ,γ0, γ1,δ, u,θ are positive constants. The 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:(2)dSdt=C−αISu+I−τS,dIdt=αISu+I−γ0+γ1−γ0θθ+II−δ+τI.


### 2.2. Disease-Free and Endemic Equilibria of SIR Model

The disease-free and endemic equilibria can be found by solving the following equations:(3)C−αISu+I−τS=0,αISu+I−γ0+γ1−γ0θθ+II−δ+τI=0.To find the disease-free equilibrium (DFE) point, we take all other classes equal to zero except the susceptible class. Then, the DFE pointE0 becomes E0=C/τ,0. On the other hand, to find the disease endemic equilibrium (DEE) point, we simultaneously solve equation (3) for S and I. If we denote DEE point by E∗=S∗,I∗, then from equation (3) we get S∗t=I∗γ0+θγ1+θδ+δI∗+τθ+τI∗u+I∗/αθ+I∗ and I∗t=Cu−τuS∗/αS∗+τS∗−C.


### 2.3. Basic Reproduction Number (R0)

The fundamental reproduction number is one of the most significant threshold quantities used in epidemiology. This is indicated byR0 which define the number of secondary infected cells. By applying the next-generation matrix method [32], we can easily find R0 for model (2).Let(4)F=αISu+IαISu+I00,V=τS00γ0+γ1−γ0θ/θ+II+δ+τI.By putting DFE point intoE0, we get(5)F=0αCτu0αCτu,V=τ00γ1+δ+τ.AsR0 is the spectral radius of FV−1, we get(6)R0=αCτuγ1+δ+τ.


## 2.1. System of Differential Equations

The dynamical system [31] with nonlinear recovery and Monod type equation as incidence rates including three differential equations is given as follows. The total population Nt is distributed into three classes: susceptible St, infected It, and recovered Rt where Nt=St+It+Rt. The detailed description of the model can be seen in Figure 1.(1)dSdt=C−αISu+I−τS,dIdt=αISu+I−γ0+γ1−γ0θθ+II−δ+τI,dRdt=γ0+γ1−γ0θθ+II−τR.Figure 1
The detailed illustration of SIR model with nonlinear recovery and Monod type equation as incidence rates.The information about the variables and parameters used in model (1) is given in the following table (seeTable1).Table 1
The detail descriptions of parameters given in system (1).
ParametersParameters descriptionSSusceptible populationIInfected populationRRecovered populationCConstant quantity addressing the rate of birthαIS/u+IRate of disease transmissionτRate of natural deaths at each compartmentδRate of deaths due to the diseaseθImpact of the quantity of clinic bedsαConstant that represents the diseases infection forceγ0Minimum recovery rateγ1Maximum recovery rateuRepresents the mediation level and decide the figure of the frequency rate as an element of the tainted populationIIt is assumed that all parametersα,u,τ,γ0, γ1,δ, u,θ are positive constants. The 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:(2)dSdt=C−αISu+I−τS,dIdt=αISu+I−γ0+γ1−γ0θθ+II−δ+τI.


## 2.2. Disease-Free and Endemic Equilibria of SIR Model

The disease-free and endemic equilibria can be found by solving the following equations:(3)C−αISu+I−τS=0,αISu+I−γ0+γ1−γ0θθ+II−δ+τI=0.To find the disease-free equilibrium (DFE) point, we take all other classes equal to zero except the susceptible class. Then, the DFE pointE0 becomes E0=C/τ,0. On the other hand, to find the disease endemic equilibrium (DEE) point, we simultaneously solve equation (3) for S and I. If we denote DEE point by E∗=S∗,I∗, then from equation (3) we get S∗t=I∗γ0+θγ1+θδ+δI∗+τθ+τI∗u+I∗/αθ+I∗ and I∗t=Cu−τuS∗/αS∗+τS∗−C.


## 2.3. Basic Reproduction Number (R0)

The fundamental reproduction number is one of the most significant threshold quantities used in epidemiology. This is indicated byR0 which define the number of secondary infected cells. By applying the next-generation matrix method [32], we can easily find R0 for model (2).Let(4)F=αISu+IαISu+I00,V=τS00γ0+γ1−γ0θ/θ+II+δ+τI.By putting DFE point intoE0, we get(5)F=0αCτu0αCτu,V=τ00γ1+δ+τ.AsR0 is the spectral radius of FV−1, we get(6)R0=αCτuγ1+δ+τ.


## 3. The NSFD Scheme for the Modified SIR Model

In 1994, the concept of NSFD was given by Mickens [33]. The NSFD schemes [34–38] 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 yx+h−yx/h, where h represents step size. According to Mickens, this term can be expressed as yx+h−yx/φh, where φh is an increasing continuous function known as denominator function. Different expressions for φh can be seen in Refs [37–39]; however, we will consider φh=expτh−1/τ in upcoming calculations. The quantity ϕh 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.


### 3.1. Construction of the Discrete NSFD Scheme

For model (1), we indicate Sn,In, and Rn as the numerical approximations of St,It, and Rt at t=nh, where n = 0, 1, 2, …, and h denotes the time step size which should be nonnegative. Then, based on the above all information, we can construct the NSFD scheme for model (1) as follows:(7)Sn+1−Snϕ=C−αInSn+1u+In−τSn+1,In+1−Inϕ=αInSn+1u+In−γ0+γ1−γ0θθ+InIn+1−δ+τIn+1,Rn+1−Rnϕ=γ0+γ1−γ0θθ+InIn+1−τRn+1.We assume that the initial valuesS0,I0, and R0 of the discrete NSFD scheme (7) are nonnegative. The total population Pn from the discrete NSFD scheme (8) satisfy Pn=Sn+In+Rn such that(8)Pn+1−Pnϕ≤C−τPn+1−δIn+1≤C−τPn+1.If we takeϕ=expτh−1/τ, then the solution of (8) satisfies(9)Pn≤Cτ+P0−Cτexp−τh,where P0=S0+I0+R0 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). The scheme (7) is implicit; however, it can be expressed explicitly as follows:(10)Sn+1=Sn+ϕC1+ϕΨn+τ,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ,Rn+1=Rn+ϕγ0+γ1−γ0θ/θ+InIn+11+ϕτ,where Ψz=αz/u+z and Ψn=ΨIn. As all parameters in (10) are positive, therefore Sn≥0,In≥0 and Rn≥0 for all n>0 and for any h>0. These findings confirm that the discrete NSFD scheme (10) maintains the nonnegative solutions for any step size h.It is clear thatRn is not included in first two equations of system (10). Therefore, for further investigation, we only take the following reduced model:(11)Sn+1=Sn+ϕC1+ϕΨn+τ,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ.After simple computation, we can prove that the discrete NSFD model (11) always has a unique DFE point E0=C/τ,0 and DEE point E∗=S∗,I∗. These 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.


### 3.2. Local Stability of the Discrete NSFD Scheme

To investigate the LAS of DFE and DEE points, we consider(12)Sn+1=Sn+ϕC1+ϕΨn+τ=FS,I,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ=GS,I.Theorem 1.
IfR0<1, then DFE point E0 of the NSFD Scheme (12) is LAS for any step size h.Proof.
We take the Jacobean matrix as(13)JS,I=∂F∂SS,I∂F∂IS,I∂G∂SS,I∂G∂IS,I.
After easy calculation and the puttingE0, we get(14)∂F∂S=11+ϕτ,∂F∂I=−Cϕτ1+ϕτ,∂F∂S=0,∂F∂I=τ+ϕC1+ϕγ1+δ+ττ.
Therefore, the Jacobean matrix (13) becomes(15)JE0=11+ϕτ−Cϕτ1+ϕτ01+ϕC1+ϕγ1+δ+ττ.
The above matrix clearly gives the following eigenvalues:λ1=1/1+ϕτ< 1 and λ2=1+ϕC/1+ϕγ1+δ+ττ.
The eigenvalueλ2 can also be expressed as λ2=τu+ϕαC/αAϕ2γ1+δ+τ+ϕR0. Thus, it is clear that if R0<1then0<λ2<1, while if R0>1, then λ2>1. This verifies that if R0<1, then DFE point E0 is LAS, and it is unstable if R0>1.
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.
The quadratic equationλ2−Tλ+D=0 roots satisfy λi<1,i=1,2 if and only if the below three conditions are fulfilled:1D<121+T+D>031−T+D>0.Theorem 2.
IfR0>1, then the DEE point E∗ of the NSFD scheme (12) is LAS for any step size h.Proof.
The Jacobian matrix of the discrete NSFD scheme (12) is(16)JS,I=∂F∂SS,I∂F∂IS,I∂G∂SS,I∂G∂IS,I.
By replacingF and G, and then putting DEE point E∗ in equation (16), we get(17)JE∗=1p−sp2rq1+sq,where p=1+ (Ψ∗+τ) >1, q=1+ϕγ0+γ1−γ0θ/θ+In+δ+τ>1, r=ϕΨ∗, and s=ϕzS∗+ϕA/u+I∗. Obviously, p>1,q>1,r>0,s>0 and p>r.
The characteristic equation ofJE∗ is provided by(18)λ2−Tλ+D=0,where T= Trace JE∗=1/p+1+s/q>0 and D=p1+s+rs/p2q>0. Then, we get the following results:1 Whenp>1, we have(19)D=p1+s+rsp2q<1+spq+rsp2q=1+ϕΨ∗+τ+1+ϕΨ∗+τ.ϕS∗α+ϕ2Cα/u+I∗+ϕΨ∗ϕS∗α+ϕ2Cα/u+I∗1+ϕΨ∗+τ2.1+ϕγ0+γ1−γ0θ/θ+In+δ+τ<1.2 It is clear that1+T+D>0.3 Direct calculation presents that ifR0>1, then(20)1−T+D=1+ϕΨ∗+τ2.1+ϕγ0+γ1−γ1θθ+In+δ+τ1−ϕS∗α+ϕ2Cαu+I∗+1+ϕΨ∗+τ1+ϕS∗α+ϕ2Cαu+I∗+ϕΨ∗ϕS∗α+ϕ2Cαu+I∗>0.
Based on the Schur–Cohn criterion discussed in Lemma1, we conclude that DEE point E∗ is LAS if R0>1 for any step size h.


### 3.3. 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.Theorem 3.
IfR0≤1, then DFE point E0 of the NSFD scheme (12) is GAS, as shown in Figure 2(a)–2(d).Figure 2
Numerical solutions of the SIR model (1) attained from NSFD scheme with (a)h = 0.5, (b) h = 2, (c) h = 50, (d) h = 500. Other parameters are set as C=2.5,α=0.01,τ=0.3,δ=0.00197,θ=0.00768,γ0=0.00989,γ1=0.000007,u=0.7.
(a)(b)(c)(d)Proof.
We can define the discrete Lyapunov function as follows:(21)Un=1ϕS0gSnS0+In+S0Ψn,where gy=y−1−lny≥g1=0. Depending on the first and second equations of system (7), from (21), we attain(22)ΔUn=Un+1−Un=1ϕS0gSn+1S0−S0gSnS0+In+1−In+S0Ψn+1−Ψn=1ϕS0Sn+1−SnS0−S0lnSn+1Sn+In+1−In+S0Ψn+1−Ψn=1ϕSn+1−Sn−S0lnSn+1Sn+In+1−In+S0Ψn+1−Ψn.
By using Entasu et al.’s [41] criterion lnS2/S1≥S2−S1/S2, we get(23)≤1ϕSn+1−Sn−S0Sn+1−SnSn+1+In+1−In+S0Ψn+1−Ψn=1Sn+1Sn+1−S0Sn+1−Snϕ+In+1−Inϕ+S0Ψn+1−Ψn=1Sn+1Sn+1−S0C−ΨnSn+1−τSn+1+ΨnSn+1−γ0+γ1−γ0θθ+InIn+1−δ+τIn+1+S0Ψn+1−Ψn=−τSn+1Sn+1−S02−γ1+δ+τIn+1+CαIn+1τu+In=−τSn+1Sn+1−S02−γ1+δ+τ1−Cατuγ1+δ+τIn+1=−τSn+1Sn+1−S02−γ1+δ+τ1−R0In+1.
Definitely, ifR0≤1 then Un+1−Un≤0 for all n≥0. This presents that Un is monotonic decreasing sequences. As Un≥0, then limn⟶∞Un≥0 and limn⟶∞Un+1−Un=0 Accordingly, we get limn⟶∞Sn+1=S0 and limn⟶∞δ+τIn+1=0. It is clear that if R0≤1, then limn⟶∞In+1=0. Hence, we conclude that E0 is GAS.Theorem 4.
IfR0>1, then DEE point E∗ of the NSFD model 8 is GAS for any step size h, as shown in Figure 3(a)–3(d).Figure 3
Numerical solutions of SIR model (1) attained from NSFD scheme with (a)h = 0.5, (b) h = 1, (c) h = 50, (d) h = 500. Other parameters are set as C= 5.8 α=0.9,τ=0.3,δ=0.00197,θ=0.00768,γ0=0.00989,γ1=0.000007,u=0.7.
(a)(b)(c)(d)Proof.
To present sufficient conditions for the global stability ofE∗, we use Enatsu et al.’s [41] criterion. For this, we define the discrete Lyapunov function as follows:(24)Wn=S∗gSnS∗+I∗gInI∗+ϕΨ∗S∗gΨnΨ∗,where Ψ∗=ΨI∗ and gy=y−1−lny≥g1=0.
By replacing the first and second equations of (7) into (24), we acquire(25)ΔWn=Wn+1−Wn=S∗gSn+1S∗−gSnS∗+I∗gIn+1I∗n−gInI∗+ϕΨ∗S∗gΨn+1Ψ∗−gΨnΨ∗=Sn+1−Sn−S∗lnSn+1S∗+In+1−In−I∗lnIn+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=S∗Sn+1−SnS∗−S∗lnSn+1S∗+I∗In+1−InI∗−I∗lnIn+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗,≤1Sn+1Sn+1−S∗Sn+1−Sn+1In+1In+1−I∗In+1−In+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=ϕSn+1Sn+1−S∗Sn+1−Snϕ+1In+1In+1−I∗In+1−Inϕ+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=ϕSn+1Sn+1−S∗C−ΨnSn+1−τSn+1+1In+1In+1−I∗ΨnSn+1−γ0+γ1−γ0θIn+1–δ+τIn+1+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=τϕSn+1Sn+1−S∗2+ϕΨ∗S∗1−S∗Sn+11−ΨnSn+1Ψ∗S∗+ϕΨ∗S∗1−I∗In+1ΨnSn+1Ψ∗S∗−In+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗.
Let us denoteΥn = Sn/S∗, ηn=In/I∗, and ξn=Ψn/Ψ∗, then (25) can be expressed as(26)ΔWn≤−τϕS∗χn+1Υn+1−12+hΨ∗S∗1−1Υn+11−ξnΥn+1+ϕΨ∗S∗1−1ηn+1ξnΥn+1−ηn+1+ϕΨ∗S∗ξn+1−ξn−lnξn+1−lnξn=−τϕS∗χn+1Υn+1−12−g1Υn+1−gΥn+1ξnηn+1+ϕΨ∗S∗gξn+1−gηn+1.
Using the definition ofgy, we get(27)gξn+1−gηn+1=Ψn+1Ψ∗−In+1I∗+lnIn+1I∗Ψ∗Ψn+1≤Ψn+1Ψ∗−In+1I∗+In+1I∗Ψ∗Ψn+1−1=−αIn+1−I∗2I∗1+αI∗1+αIn+1≤0.
Hence,ΔWn is a monotonic decreasing sequence. By applying the same techniques as those in Theorem 3, we can show that limn⟶∞Wn+1−Wn=0. Therefore, we get limn⟶∞Sn+1=S∗ and limn⟶∞In+1=I∗ for all h. Hence, the proof is completed.


## 3.1. Construction of the Discrete NSFD Scheme

For model (1), we indicate Sn,In, and Rn as the numerical approximations of St,It, and Rt at t=nh, where n = 0, 1, 2, …, and h denotes the time step size which should be nonnegative. Then, based on the above all information, we can construct the NSFD scheme for model (1) as follows:(7)Sn+1−Snϕ=C−αInSn+1u+In−τSn+1,In+1−Inϕ=αInSn+1u+In−γ0+γ1−γ0θθ+InIn+1−δ+τIn+1,Rn+1−Rnϕ=γ0+γ1−γ0θθ+InIn+1−τRn+1.We assume that the initial valuesS0,I0, and R0 of the discrete NSFD scheme (7) are nonnegative. The total population Pn from the discrete NSFD scheme (8) satisfy Pn=Sn+In+Rn such that(8)Pn+1−Pnϕ≤C−τPn+1−δIn+1≤C−τPn+1.If we takeϕ=expτh−1/τ, then the solution of (8) satisfies(9)Pn≤Cτ+P0−Cτexp−τh,where P0=S0+I0+R0 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). The scheme (7) is implicit; however, it can be expressed explicitly as follows:(10)Sn+1=Sn+ϕC1+ϕΨn+τ,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ,Rn+1=Rn+ϕγ0+γ1−γ0θ/θ+InIn+11+ϕτ,where Ψz=αz/u+z and Ψn=ΨIn. As all parameters in (10) are positive, therefore Sn≥0,In≥0 and Rn≥0 for all n>0 and for any h>0. These findings confirm that the discrete NSFD scheme (10) maintains the nonnegative solutions for any step size h.It is clear thatRn is not included in first two equations of system (10). Therefore, for further investigation, we only take the following reduced model:(11)Sn+1=Sn+ϕC1+ϕΨn+τ,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ.After simple computation, we can prove that the discrete NSFD model (11) always has a unique DFE point E0=C/τ,0 and DEE point E∗=S∗,I∗. These 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.


## 3.2. Local Stability of the Discrete NSFD Scheme

To investigate the LAS of DFE and DEE points, we consider(12)Sn+1=Sn+ϕC1+ϕΨn+τ=FS,I,In+1=In+ϕΨnSn+11+ϕγ0+γ1−γ0θ/θ+In+δ+τ=GS,I.Theorem 1.
IfR0<1, then DFE point E0 of the NSFD Scheme (12) is LAS for any step size h.Proof.
We take the Jacobean matrix as(13)JS,I=∂F∂SS,I∂F∂IS,I∂G∂SS,I∂G∂IS,I.
After easy calculation and the puttingE0, we get(14)∂F∂S=11+ϕτ,∂F∂I=−Cϕτ1+ϕτ,∂F∂S=0,∂F∂I=τ+ϕC1+ϕγ1+δ+ττ.
Therefore, the Jacobean matrix (13) becomes(15)JE0=11+ϕτ−Cϕτ1+ϕτ01+ϕC1+ϕγ1+δ+ττ.
The above matrix clearly gives the following eigenvalues:λ1=1/1+ϕτ< 1 and λ2=1+ϕC/1+ϕγ1+δ+ττ.
The eigenvalueλ2 can also be expressed as λ2=τu+ϕαC/αAϕ2γ1+δ+τ+ϕR0. Thus, it is clear that if R0<1then0<λ2<1, while if R0>1, then λ2>1. This verifies that if R0<1, then DFE point E0 is LAS, and it is unstable if R0>1.
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.
The quadratic equationλ2−Tλ+D=0 roots satisfy λi<1,i=1,2 if and only if the below three conditions are fulfilled:1D<121+T+D>031−T+D>0.Theorem 2.
IfR0>1, then the DEE point E∗ of the NSFD scheme (12) is LAS for any step size h.Proof.
The Jacobian matrix of the discrete NSFD scheme (12) is(16)JS,I=∂F∂SS,I∂F∂IS,I∂G∂SS,I∂G∂IS,I.
By replacingF and G, and then putting DEE point E∗ in equation (16), we get(17)JE∗=1p−sp2rq1+sq,where p=1+ (Ψ∗+τ) >1, q=1+ϕγ0+γ1−γ0θ/θ+In+δ+τ>1, r=ϕΨ∗, and s=ϕzS∗+ϕA/u+I∗. Obviously, p>1,q>1,r>0,s>0 and p>r.
The characteristic equation ofJE∗ is provided by(18)λ2−Tλ+D=0,where T= Trace JE∗=1/p+1+s/q>0 and D=p1+s+rs/p2q>0. Then, we get the following results:1 Whenp>1, we have(19)D=p1+s+rsp2q<1+spq+rsp2q=1+ϕΨ∗+τ+1+ϕΨ∗+τ.ϕS∗α+ϕ2Cα/u+I∗+ϕΨ∗ϕS∗α+ϕ2Cα/u+I∗1+ϕΨ∗+τ2.1+ϕγ0+γ1−γ0θ/θ+In+δ+τ<1.2 It is clear that1+T+D>0.3 Direct calculation presents that ifR0>1, then(20)1−T+D=1+ϕΨ∗+τ2.1+ϕγ0+γ1−γ1θθ+In+δ+τ1−ϕS∗α+ϕ2Cαu+I∗+1+ϕΨ∗+τ1+ϕS∗α+ϕ2Cαu+I∗+ϕΨ∗ϕS∗α+ϕ2Cαu+I∗>0.
Based on the Schur–Cohn criterion discussed in Lemma1, we conclude that DEE point E∗ is LAS if R0>1 for any step size h.


## 3.3. 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.Theorem 3.
IfR0≤1, then DFE point E0 of the NSFD scheme (12) is GAS, as shown in Figure 2(a)–2(d).Figure 2
Numerical solutions of the SIR model (1) attained from NSFD scheme with (a)h = 0.5, (b) h = 2, (c) h = 50, (d) h = 500. Other parameters are set as C=2.5,α=0.01,τ=0.3,δ=0.00197,θ=0.00768,γ0=0.00989,γ1=0.000007,u=0.7.
(a)(b)(c)(d)Proof.
We can define the discrete Lyapunov function as follows:(21)Un=1ϕS0gSnS0+In+S0Ψn,where gy=y−1−lny≥g1=0. Depending on the first and second equations of system (7), from (21), we attain(22)ΔUn=Un+1−Un=1ϕS0gSn+1S0−S0gSnS0+In+1−In+S0Ψn+1−Ψn=1ϕS0Sn+1−SnS0−S0lnSn+1Sn+In+1−In+S0Ψn+1−Ψn=1ϕSn+1−Sn−S0lnSn+1Sn+In+1−In+S0Ψn+1−Ψn.
By using Entasu et al.’s [41] criterion lnS2/S1≥S2−S1/S2, we get(23)≤1ϕSn+1−Sn−S0Sn+1−SnSn+1+In+1−In+S0Ψn+1−Ψn=1Sn+1Sn+1−S0Sn+1−Snϕ+In+1−Inϕ+S0Ψn+1−Ψn=1Sn+1Sn+1−S0C−ΨnSn+1−τSn+1+ΨnSn+1−γ0+γ1−γ0θθ+InIn+1−δ+τIn+1+S0Ψn+1−Ψn=−τSn+1Sn+1−S02−γ1+δ+τIn+1+CαIn+1τu+In=−τSn+1Sn+1−S02−γ1+δ+τ1−Cατuγ1+δ+τIn+1=−τSn+1Sn+1−S02−γ1+δ+τ1−R0In+1.
Definitely, ifR0≤1 then Un+1−Un≤0 for all n≥0. This presents that Un is monotonic decreasing sequences. As Un≥0, then limn⟶∞Un≥0 and limn⟶∞Un+1−Un=0 Accordingly, we get limn⟶∞Sn+1=S0 and limn⟶∞δ+τIn+1=0. It is clear that if R0≤1, then limn⟶∞In+1=0. Hence, we conclude that E0 is GAS.Theorem 4.
IfR0>1, then DEE point E∗ of the NSFD model 8 is GAS for any step size h, as shown in Figure 3(a)–3(d).Figure 3
Numerical solutions of SIR model (1) attained from NSFD scheme with (a)h = 0.5, (b) h = 1, (c) h = 50, (d) h = 500. Other parameters are set as C= 5.8 α=0.9,τ=0.3,δ=0.00197,θ=0.00768,γ0=0.00989,γ1=0.000007,u=0.7.
(a)(b)(c)(d)Proof.
To present sufficient conditions for the global stability ofE∗, we use Enatsu et al.’s [41] criterion. For this, we define the discrete Lyapunov function as follows:(24)Wn=S∗gSnS∗+I∗gInI∗+ϕΨ∗S∗gΨnΨ∗,where Ψ∗=ΨI∗ and gy=y−1−lny≥g1=0.
By replacing the first and second equations of (7) into (24), we acquire(25)ΔWn=Wn+1−Wn=S∗gSn+1S∗−gSnS∗+I∗gIn+1I∗n−gInI∗+ϕΨ∗S∗gΨn+1Ψ∗−gΨnΨ∗=Sn+1−Sn−S∗lnSn+1S∗+In+1−In−I∗lnIn+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=S∗Sn+1−SnS∗−S∗lnSn+1S∗+I∗In+1−InI∗−I∗lnIn+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗,≤1Sn+1Sn+1−S∗Sn+1−Sn+1In+1In+1−I∗In+1−In+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=ϕSn+1Sn+1−S∗Sn+1−Snϕ+1In+1In+1−I∗In+1−Inϕ+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=ϕSn+1Sn+1−S∗C−ΨnSn+1−τSn+1+1In+1In+1−I∗ΨnSn+1−γ0+γ1−γ0θIn+1–δ+τIn+1+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗=τϕSn+1Sn+1−S∗2+ϕΨ∗S∗1−S∗Sn+11−ΨnSn+1Ψ∗S∗+ϕΨ∗S∗1−I∗In+1ΨnSn+1Ψ∗S∗−In+1I∗+ϕΨ∗S∗Ψn+1Ψ∗−ΨnΨ∗+lnΨnΨ∗−lnΨn+1Ψ∗.
Let us denoteΥn = Sn/S∗, ηn=In/I∗, and ξn=Ψn/Ψ∗, then (25) can be expressed as(26)ΔWn≤−τϕS∗χn+1Υn+1−12+hΨ∗S∗1−1Υn+11−ξnΥn+1+ϕΨ∗S∗1−1ηn+1ξnΥn+1−ηn+1+ϕΨ∗S∗ξn+1−ξn−lnξn+1−lnξn=−τϕS∗χn+1Υn+1−12−g1Υn+1−gΥn+1ξnηn+1+ϕΨ∗S∗gξn+1−gηn+1.
Using the definition ofgy, we get(27)gξn+1−gηn+1=Ψn+1Ψ∗−In+1I∗+lnIn+1I∗Ψ∗Ψn+1≤Ψn+1Ψ∗−In+1I∗+In+1I∗Ψ∗Ψn+1−1=−αIn+1−I∗2I∗1+αI∗1+αIn+1≤0.
Hence,ΔWn is a monotonic decreasing sequence. By applying the same techniques as those in Theorem 3, we can show that limn⟶∞Wn+1−Wn=0. Therefore, we get limn⟶∞Sn+1=S∗ and limn⟶∞In+1=I∗ for all h. Hence, the proof is completed.


## 4. 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. The 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. The numerical simulations are important to study the complex dynamical behavior the nonlinear models [34–43]. Therefore, numerical simulations have also been presented to verify the sustainability of the theoretical results. The 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.



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