The stability analysis of a nonlinear mathematical model for typhoid fever disease

Typhoid fever is a contagious disease that is generally caused by bacteria known as Salmonella typhi. This disease spreads through manure contamination of food or water and infects unprotected people. In this work, our focus is to numerically examine the dynamical behavior of a typhoid fever nonlinear mathematical model. To achieve our objective, we utilize a conditionally stable Runge–Kutta scheme of order 4 (RK-4) and an unconditionally stable non-standard finite difference (NSFD) scheme to better understand the dynamical behavior of the continuous model. The primary advantage of using the NSFD scheme to solve differential equations is its capacity to discretize the continuous model while upholding crucial dynamical properties like the solutions convergence to equilibria and its positivity for all finite step sizes. Additionally, the NSFD scheme does not only address the deficiencies of the RK-4 scheme, but also provides results that are consistent with the continuous system's solutions. Our numerical results demonstrate that RK-4 scheme is dynamically reliable only for lower step size and, consequently cannot exactly retain the important features of the original continuous model. The NSFD scheme, on the other hand, is a strong and efficient method that presents an accurate portrayal of the original model. The purpose of developing the NSFD scheme for differential equations is to make sure that it is dynamically consistent, which means to discretize the continuous model while keeping significant dynamical properties including the convergence of equilibria and positivity of solutions for all step sizes. The numerical simulation also indicates that all the dynamical characteristics of the continuous model are conserved by discrete NSFD scheme. The theoretical and numerical results in the current work can be engaged as a useful tool for tracking the occurrence of typhoid fever disease.


Mathematical model and parameters explanation
The present paper discusses and evaluates a deterministic model for the dynamics of typhoid disease.We presume that the whole population N(t) is separated into four sections: Susceptible ( S ), Exposed ( E ), Infected ( I ), and Recovered ( R ), i.e.N(t)=S(t) + E(t) + I(t) + R(t) .The model employs the subsequent procedure as: S → E → I → R .A fractional map for SEIR model of typhoid disease transmission among unprotected people compartments is shown in Fig. 1.
Figure1.The detailed description of epidemic model for typhoid fever disease transmission.

Parameters.
The following are descriptions of the parameters used in model (1). 1.
From Eq. (2), we can write and.
Therefore, the feasible area for the continuous system (1) becomes.

Equilibrium points and basic reproductive number
In this section, we establish the equilibrium points of the model (1) and basic reproductive number.
Equilibrium points.For system (1), there exist the following two nonnegative equilibria.
1. Disease-free equilibrium (DFE) point E 0 = S 0 , E 0 , I 0 , R 0 = ϕ ψ , 0, 0, 0 .( Vol:.(1234567890)To find the reproductive number, we utilize the idea of next generation matrix provided by the authors in 29 .For typhoid fever disease model (1), we can easily get In the following two sections, a comparison between NSFD scheme and RK-4 is provided.As our main concern is NSFD scheme, therefore the advantages and uses of NSFD are discussed in detail.Specifically, traditional issues concerned to the behavior of these schemes, i.e. equilibrium points, positivity and stability are discussed with respect to increment of the time step.

The RK-4 scheme
The RK-4 scheme 30 is widely used approach to solve a system of ordinary differential equations.In numerous situations, we frequently employ the RK-4 scheme, unless stated otherwise.Let S = H i , E = L i , I = P i , R = Q i for i = 1, 2, 3, 4 , then system (1) may be signified using the RK-4 scheme as given below.

Stage-4
Thus general form is .
Vol.:(0123456789) www.nature.com/scientificreports/ The RK-4 approach is graphically depicted in Fig. 2a-d with various step sizes.The RK-4 approach clearly yields stable and positive solutions for small step sizes, as seen in Fig. 2a,b.As the step size is increased, the equilibrium point stability is shattered for model (1), as illustrated in Fig. 2c,d.Hence, we conclude that the RK-4 technique cannot be employed for large step sizes.

The NSFD scheme
In this section, our main objective is to discuss the dynamics of NSFD scheme for model (1).The NSFD scheme concept was presented by Mickens in 1994 28 .The NSFD scheme is an iterative method in which we move closer to a solution through iterations 31 .The NSFD scheme is a valuable technique used to solve problems in epidemiology [32][33][34][35][36] , ecology 37,38 , and meta-population modeling 39 .The following will show that, despite the step size h , the discrete NSFD scheme sustains the dynamical properties of the corresponding continuous model (1).
Construction of NSFD scheme.For model (1), we use the notation S n , E n , I n and R n to indicate the numerical estimates of S(t), E(t), I(t) and R(t) at time step t = nh , where h is nonnegative time step size and n is a nonnegative integer 40 .www.nature.com/scientificreports/It is assumed that the starting quantities of NSFD SEIR model ( 4) are also nonnegative.From (4), we get In the same way like continuous model (1), we can determine a feasible region for the discrete scheme ( 5).If we indicate N n = S n + E n + I n + R n , then by combining all the four equations in system (4) we obtain and.
In the following, we provide the stability conditions of DFE and DEE points for the discrete NSFD scheme (5).We first describe the local stability of both equilibrium points in order to achieve this goal.

Local stability for NSFD Scheme. To prove that DFE and DEE points are locally asymptotically stable (LAS), we consider
To demonstrate that DFE and DEE points are LAS, as stated in Lemma 1, we shall utilize the Schur-Cohn condition 44,45 .ii.

Lemma 1 The roots of equation
(4) where T and D stand for the Jacobian matrix trace and determinant, respectively.
Theorem 1 For all h > 0 , the DFE point E 0 of the NSFD model ( 5) is LAS whenever R 0 < 1.
Proof Let us consider the Jacobian matrix.
In the following, we first find all the derivatives used in (8). .
By substituting the values of all the derivatives in Eq. ( 8), we get.
The above equation can be rewritten as.
All of the Schur-Cohn requirements described in Lemma 1 are consequently satisfied whenever R 0 < 1 .As a result, the DFE point E 0 of discrete NSFD scheme ( 5) is LAS, provided that R 0 < 1.
When a disease is prevalent in a population, it will continue to exist in that community.In the following theorem, we will use Routh-Hurwitz criterion 46,47 to examine that DEE point E * is LAS.
Proof The Jacobian matrix is obtained according to Theorem 1 as.
By putting DEE point E * , we obtain.
To determine the eigenvalues, we take into consideration.
From above informations, it is clear that if R 0 > 1 , then According to the Routh-Hurwitz criteria, all of the solutions to the Eq. ( 12) must have negative real parts.As a result, the DEE point E * of the discrete NSFD scheme ( 5) is LAS whenever R 0 > 1.
Global stability for NSFD Scheme.In the following, we now show that R 0 is a critical value the global stability.If R 0 ≤ 1 , then the DFE point E 0 is globally asymptotically stable (GAS) and when R 0 > 1 , then DEE point E * becomes GAS.To discuss the global stability of equilibria, we employ the same criterion used by Vaz et al. 48.

Theorem 3 For all
Proof From the feasible region (6) discussed for NSFD scheme (5), it is clear that Therefore, if we take η > 0 , then there exists an integer n 0 such that for any n ≥ n 0 , S n+1 < ϕ ψ + η .Consider the sequence {w(n)} ∞ n=0 such that.
where C 2 = (σ + ψ) and C 3 = (θ + δ + ψ) .For n ≥ n 0 , we have By applying the NSFD scheme (5), we obtain Vol:.(1234567890)We can choose β a very small positive number such that Therefore, we get Let C = ψ(τ +ψ)(θ +δ+ψ) αϕ , then we get www.nature.com/scientificreports/Since β is a very small number and η is imprecise Therefore, if R 0 ≤ 1 then we reach the conclusion that for any n ≥ 0 , 0 ≥ w(n + 1) − w(n) and lim n→∞ I n = 0 .The sequence {w(n)} ∞ n=0 is a monotonically decreasing and lim n→∞ S n = ϕ ψ .Therefore, the DFE point E 0 is GAS whenever R 0 ≤ 1.The numerical simulations shown in Figs.3a-d and 4a-d for R 0 < 1 and R 0 = 1 , respectively also exhibit that the solutions of NSFD scheme (5) converges to the DFE point E 0 independent of the step size.By combining the above two conditions, we conclude that if R 0 ≤ 1 then the DFE point E 0 is GAS for NSFD scheme (5).The NSFD scheme is hence convergent for model (1) for all finite step sizes.

Scientific
Theorem 4 For all h > 0 , the DEE point E * of the NSFD model (5) is GAS whenever R 0 > 1.
Proof Let us construct a sequence {w(n)} ∞ n=0 such that. Where It is evident that p(x) ≥ 0 and if x = 1 , the equality holds.From above, we can write Vol.:(0123456789)The numerical illustration in Fig. 5a-d additionally illustrates that, for any step size, the NSFD scheme (5) solutions converge to the DEE point if R 0 > 1 .This reveals the NSFD scheme's unconditional convergence.

Conclusions
In the present paper, a nonlinear epidemic model of a typhoid fever disease is numerically studied using two finite difference schemes.It was shown that the spread of the disease is mostly determined by the rate of contact with sick individuals within a community.The RK-4 and NSFD schemes are employed to discuss the dynamical characteristics of DFE and DEE points, including their local and global stabilities.The findings demonstrate that the NSFD method provides precise numerical solutions while eliminating drawback of RK-4 scheme.The convergence is demonstrated that presents that the NSFD scheme retains their stability and positivity characteristics.The key benefits of NSFD are demonstrated theoretically as well as numerically which reveal that this scheme have good dynamical behavior even for large time step size.At the same time, the RK-4 scheme cannot exactly sustain the fundamental properties of the original continues model and consequently, it can produce numerical solutions which are not quite the same as the solutions of the original model.The NSFD scheme is an easy approach that exhibits how discrete and continuous models act appropriately and produce results that are mathematically accurate.Figures 3, 4, and 5 depicts that the NSFD scheme (5) remains stable for each step size.This demonstrates that for each step size, the NSFD scheme is positive forever and unconditionally convergent.The RK-4 scheme, however, shows convergence only for lower step sizes, as shown in Fig. 3a-d.We can effectively monitor the spread of typhoid fever disease by utilizing the NSFD system.The results provided in this study are advantageous to humanity as well as in the sector of healthcare.Numerical simulations are included in each section to support our theoretical conclusions.
The fundamental reproductive number (R 0 ), which measures the aver- age rate of latest cases in a residents that is perfectly susceptible, is a critical threshold value in epidemiology.