A Dynamical Study of Modeling the Transmission of Typhoid Fever through Delayed Strategies

1  Introduction

Infectious diseases can be transmitted directly or indirectly between individuals. These illnesses emerge when microorganisms enter the body, increase, and trigger a resistant reaction. Typhoid fever, whenever left untreated, represents a severe gamble, possibly harming various organs and prompting deadly entanglements. It is brought about by Salmonella Typhi [1], frequently present in sullied food or water. Ingesting sullied substances presents the microorganisms, causing sickness. Notwithstanding upgrades in water neatness, typhoid fever remains a critical worry in many non-industrial countries [2]. Normal side effects incorporate migraines, queasiness, heaving, looseness of the bowels, rashes, fever, joint and muscle torment, and loss of hunger. Untreated, it can rise, influencing the bloodstream and causing stomach tissue contamination [3]. Numerical models have played a crucial role in disentangling typhoid fever’s transmission components and assessing the viability of preventive measures. Although some explicitly tackle direct human-to-human transmission [4–6], others additionally consider the job of debased food and water in aberrant transmission [7–9].

Far-reaching models have provided a more profound comprehension by investigating both immediate and roundabout transmission methods [10–12]. These models have fundamentally supported evaluating intercessions, for example, sickness therapy [13,14], sterilization rehearsal [15,16], instructive missions [17], and media drives [18] pointed toward forestalling typhoid flare-ups. Their bits of knowledge are instrumental in contriving effective control measures for typhoid fever. For example, Musa et al. [19] fostered a plague model to investigate typhoid fever transmission and evaluate the effect of general Wellbeing instruction crusades on bringing down infection rates. Using information from Taiwan and China, they adjusted the model, laying out security measures for endemic and illness-free harmony. Utilizing wavelet examination, they distinguished urgent occasional examples in episodes and decided critical boundaries for typhoid disease the executives. Peter et al. [20] introduced a numerical model to investigate typhoid transmission elements and the impacts of immunization, incorporating immediate and circuitous effects, like group invulnerability. They showed that while inoculation could briefly offer backhanded assurance and decrease sickness occurrence, destroying Typhoid exclusively through immunization appeared to be unrealistic. Edward et al. [21] created a numerical model thinking about both immediate and backhanded transmissions of typhoid fever. They assessed medications such as treatment, inoculation inclusion, and water disinfection. They focused on the significance of restricting contact with typhoid patients and forestalling water pollution with fertilizer as the essential strategies to stop typhoid scourges. Inside the range of enteritis problems, typhoid fever remains a common irresistible sickness set off by Salmonella Typhi microorganisms in the human body.

This sickness normally spreads through sullied food and water corrupted by a contaminated person’s dung or pee [22]. It is remarkably more pervasive in immature locales because of unfortunate food disinfection, debased water sources, and deficient ecological cleanliness. Side effects manifest during the seven to fourteen-day hatching period and incorporate migraines, stomach inconvenience, joint and back torments, muscle throbs, decreased craving, retching, runs, rashes, and fever [23–25]. According to the World Wellbeing Association’s assessments, Typhoid fever generally represents 0.6 million passings and 17 million cases annually [26]. Over the past two decades, numerous studies have resulted in the development of diverse mathematical models. For instance, a model is formulated featuring four compartments: susceptible, infected, asymptomatic carriers, and recovered individuals [27,28].

Delay in mathematical epidemic models refers to the time it takes for an individual to become infectious after being infected. This delay is crucial in understanding the spread of diseases and is incorporated into models through time-delay differential equations. Based on the most recent research, time delays favorably impact the dynamics of epidemic models by influencing time delay pattern creation and the emergence of variational patterns, as well as guiding the spectrum of dynamical phenomena of epidemic models. Time delay plays an important part in modeling numerous aspects of disease transmission, such as the quarantine period in an infectious condition, which provides a more valuable portrayal of epidemic scenarios [29–31]. Time delays are advantageous in epidemics by emphasizing patterns and their dynamic analysis, executing model findings and regulating techniques for the distinct management of illness, and providing a thorough grasp of disease propagation dynamics compared to older models. Many new researchers have a remarkable insight that favors the application of delay differential equations in epidemics modeling. Delay dynamics in transmission rates can aid in anticipating the transmission rates of disease spread and control measures. In addition, delays can be considered for the intervention and majorly impact regulating tactics. These findings highlighted the importance of including delays in mathematical epidemic models to improve prediction behavior and assist public health decision-making [32]. Goel et al. [33] studied a new nonlinear time delay SIR (Susceptible-Infectious-Recovered) model with a Beddington-DeAngelis incidence rate, and Hill’s functional-type saturated recovery rate was developed to study the epidemic control, which further contributed to the development of a two-strain epidemic model considered the distributed recovery and death rates and involving delay equations.

Compared to standard models, epidemic models with delays provide a more practical knowledge of disease dynamics by encompassing the temporal components of transmission and control methods. Ghosh et al. [34] highlighted that time delay can improve time lags on epidemic outcomes, evaluate the effectiveness of control strategies, and optimize resource allocation in mathematical epidemic models present infectious disease, allowing for more accurate examination of epidemic risk and assisting in taking precautions for public health. The NSFD scheme is employed to obtain reliable solutions in delayed stochastic and fuzzy extensions [35–39], to mention a few. Cheng et al. [40] developed and examined a network-based SIQS (susceptible-infected-quarantined-susceptible) infectious disease model featuring a nonmonotone incidence rate. The basic reproduction number and the equilibria of the model are determined analytically, global stability is explored, and numerical simulations are conducted to support the theoretical findings.

This study aims to increase the reliability of the mathematical models of typhoid fever transmission by including time delays in the equation because they strongly affect the trends of the infection. This kind of modeling can enhance prediction and public health efforts to tackle typhoid fever outbreaks. The SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) model and the SEIRS delay model are made up of the same structural parts, but the time delay function distinguishes them. In its baseline model, SEIRS makes no provision for delays; rather, compartments Susceptible, Exposed, Infectious, Recovered, and Susceptible are described by ODEs (ordinary differential equation) that do not include time delays when an individual shifts from the former compartment to the latter one or vice versa. However, the SEIRS delay model introduces the concept of delay using DDEs (delayed differential equation) to capture a given incubation period and other temporal delays better than the basic SEIRS model. This addition enhances projection performance for disease type with a long latency time and enhances computational calculation. The delay Susceptible-Exposed-Infectious-Recovered (SEIR) model is a better version that demonstrates how the diseases progress over time and can deliver more specifics for implementing a plan to prevent the spread of diseases. The novelty of the proposed approach is the construction, implementation, and analysis of a first-order explicit numerical scheme in non-standard finite difference settings for Typhoid fever with time delay.

This research looks at several aspects of Typhoid virus transmission among communities. Section 2 offers a mathematical model that depicts the dynamics of the propagation. This section addresses the virus’s reproductive characteristics and immediate transmission without delay. It also investigates the existence of endemic equilibrium and evaluates its local stability. Section 3 contains the delayed epidemic model and conducts a stability analysis considering delays. Sections 4 and 5 are dedicated to presenting mathematical modeling and numerical simulations that validate the outcomes of our proposed system to corroborate the results obtained. Sections 6 and 7 contain parameter estimation analysis and conclusions, respectively.

2  Modified SEIR Epidemic Model

The system of differential equations derived from [41] is

{dSdt=∝−βSI+ωR−μS,dEdt=βSI−φE−μE,dIdt=φE−σI−δI−μI,dRdt=σI−ωR−μR.(1)

These factors are inherent in an epidemiological model, frequently associated with the SEIR model used in disease transmission research. Within this approach, S symbolizes the susceptible population that has not yet been impacted by the disease, whereas E represents the exposed population of individuals who have been exposed to the disease but are not yet contagious. I indicates people actively infected and spreading the disease, whereas R represents those who have recovered and gained immunity. The symbol ∝ reflects the recruiting rate, indicating new people joining the vulnerable group. μ represents the post-infection mortality rate, whereas β represents infection rates among vulnerable individuals. φ denotes the rate of transition from exposed to infected, σ denotes the rate of transition from infected to recovered, δ represents post-infection mortality rate. Finally, ω reflects the rate of change from recovered to susceptible, which may indicate diminishing immunity or susceptibility to reinfection. These factors are critical in mathematical models for simulating disease transmission patterns and forecasting population outbreaks. This study added the concept of susceptibility to the improved SEIRS epidemic model. The rate indicates that individuals who recovered will transition back to the susceptible group after being infectious. This explains why they are less likely to develop the same infection again. The flowchart of the model is shown in Fig. 1.

Figure 1: Flowchart of the model

2.1 The Basic Reproductive Number (BRN)

Let X=(E, I)t then dxdt=ℱ(x)−Δ(x) and ℱ(x)=[βSI0], Δ(x)=[(φ+μ)E−φE+(σ+δ+μ)I]. Further, ℱ and Δ are the Jacobians of ℱ(x) and Δ(x) respectively at disease-free equilibrium point Ed1(S0, E0, I0, R0)=(∝μ, 0, 0, 0) and are given as follows:

ℱ=[0β∝μ00],Δ=[(φ+μ)0−φ(σ+δ+μ)].

Now

T =ℱΔ−1=[0β∝μ(φ+μ)0βφ∝μ(φ+μ)(σ+δ+μ)].

The spectral radius of ℱΔ−1 is

ϱ(ℱΔ−1)=βφ∝μ(φ+μ)(σ+δ+μ)

The BRN ℛo is given by

ℛo=βφ∝μ(φ+μ)(σ+δ+μ).(2)

2.2 Equilibrium Analysis

The system (1) has a disease-free equilibrium (DFE) point E1(So, Eo, Io, Ro)=(∝μ, 0, 0, 0) and an endemic equilibrium point E2(S∗, E∗, I∗, R∗), where

S∗=∝μℛo,

E∗=k3(μ∝(1−ℛo)k3)φ(μωℛo−β∝k3),

I∗=μ∝(1−ℛo)k3μωℛo−β∝k3,

R∗=σ(μ∝(1−ℛo))μωℛo−β∝k3, where k1=(φ+μ),k2=(σ+δ+μ) and k3=(μ+ω).

2.3 Stability of Modified SEIR Model

The local stability of the modified SIER model is performed at E1(So, Eo, Io, Ro)=(∝μ, 0, 0, 0). The eigenvalues of the system are λ1=−0.9000, λ2=−0.9500, λ3=−2.8248, and λ4=−0.4252. As all the eigenvalues are less than zero, it concluded that the system of differential equations of the SEIR model is locally asymptotically stable.

3  SEIR Model Modified with Time Delay

This section introduces a time delay parameter τ.

3.1 Delay Differential Equation Model

Incorporating the impact of time delay in infected populations, model (1) assumes the following structure:

{dSdt=∝−βSI+ωR−μS,dEdt=βS(t−τ)I(t−τ)e−μτ−φE−μE,dIdt=φE−σI−δI−μI,dRdt=σI−ωR−μR.(3)

The symbol τ is the latency period, indicating how long it takes for infected individuals to reach a contagious state, thus contributing to the ongoing spread of the virus. Within system (3), the expression βS(t−τ)I(t−τ)e−μτ at time ‘t’ reflects the transition of susceptible individuals to the infected group. The presence of e−μτ stems from presuming that population decline follows a linear pattern governed by μτ, signifying the decrease in population during that specific duration due to factors like mortality or recovery.

3.2 The Basic Reproductive Number of Delayed Epidemic Model

Let X=(E, I)t then dxdt=ℱ(x)−Δ(x) and ℱ(x)=[βSIe−μτ0], Δ(x)=[(φ+μ)E−φE+(σ+δ+μ)I]. Further, ℱ and Δ are now the Jacobians of ℱ(x) andΔ(x) respectively at disease-free equilibrium point Ed1(S′, E′, I′, R′)=(∝μ, 0, 0, 0) such that

ℱ=[0β∝e−μτμ00], Δ=[(φ+μ)0−φ(σ+δ+μ)].

Now,

T =ℱΔ−1=[0β∝e−μτμ(φ+μ)0βφ∝e−μτμ(φ+μ)(σ+δ+μ)].

The spectral radius of ℱΔ−1 is

ϱ(ℱΔ−1)=βφ∝e−μτμ(φ+μ)(σ+δ+μ)

The BRN of the delayed epidemic model ℛdo is given by

ℛdo=φ∝βe−μτμ(φ+μ)(σ+δ+μ)(4)

3.3 Equilibrium Analysis

The system (3) has a disease-free equilibrium point Ed1(S′, E′, I′, R′)=(∝μ, 0, 0, 0) and an endemic equilibrium point Ed2(S″,E″,I″,R″), where

S′′=∝μℛdo,

E′′=k3(μ∝(1−ℛdo)k3)φ(μωℛdo−β∝k3),

I′′=μ∝(1−ℛdo)k3μωℛdo−β∝k3,

R′′=σ(μ∝(1−ℛdo))μωℛdo−β∝k3, where k1=(φ+μ), k2=(σ+δ+μ) and k3=(μ+ω).

3.4 Stability of Modified SEIR Delayed Model

The local stability of the modified SIER model is performed at Ed1(S′, E′, I′, R′)=(∝μ, 0, 0, 0). The eigenvalues of the system are; λ1=−0.9000, λ2=−0.9500, λ3=−2.2412 and λ4=−1.0088. As all the eigenvalues are less than zero, it concluded that the system of differential equations of the SEIR model is locally asymptotically stable.

4  Numerical Computation of Delayed Epidemic Model

4.1 Forward Euler’s Method

The forward Euler scheme utilized for the model under study method involves making the following supposition can be expressed as follows:

Suppositions: S(t)≈Sn, S(t−τ)≈Sn−m, I(t)≈In andI(t−τ)≈In−m.

Sn+1=Sn+h[∝−βSnIn+ωRn−μSn],(5)

En+1=En+h[βSn−mIn−me−μτ−(φ+μ)En],(6)

In+1=In+h[φEn−(σ+δ+μ)In],(7)

Rn+1=Rn+h[σIn−(μ+ω)Rn.(8)

4.2 Fourth Order Runge Kutta Scheme (RK-4)

Considering the equations of system (3), the study develops an explicit RK-4 method. A numerical scheme for the RK-4 method is constructed as follows:

Step 1

k1=h[∝−βSnIn+ωRn−μSn],(9)

l1=h[βSn−mIn−me−μτ−(φ+μ)En],(10)

p1=h[φEn−(σ+δ+μ)In],(11)

m1=h[σIn−(μ+ω)Rn].(12)

Step 2

k2=h[∝−β(Sn+k12)(In+p12)+ω(Rn+m12)−μ(Sn+k12)],(13)

l2=h[β(Sn−m+k12)(In−m+p12)e−μτ−(φ+μ)(En+l12)],(14)

p2=h[φ(En+l12)−(σ+δ+μ)(In+p12)],(15)

m2=h[σ(In+p12)−(μ+ω)(Rn+m12)].(16)

Step 3

k3=h[∝−β(Sn+k22)(In+p22)+ω(Rn+m22)−μ(Sn+k22)],(17)

l3=h[β(Sn−m+k22)(In−m+p22)e−μτ−(φ+μ)(En+l22)],(18)

p3=h[φ(En+l22)−(σ+δ+μ)(In+p22)],(19)

m3=h[σ(In+p22)−(μ+ω)(Rn+m22)].(20)

Step 4

k4=h[∝−β(Sn+k3)(In+p3)+ω(Rn+m3)−μ(Sn+k3)],(21)

l4=h[β(Sn−m+k3)(In−m+p3)e−μτ−(φ+μ)(En+l3)],(22)

p4=h[φ(En+l3)−(σ+δ+μ)(In+p3)],(23)

m4=h[σ(In+p3)−(μ+ω)(Rn+m3)].(24)

Final Step

Sn+1=Sn+16[k1+2k2+2k3+k4],(25)

En+1=En+16[l1+2l2+2l3+l4],(26)

In+1=In+16[p1+2p2+2p3+p4],(27)

Rn+1=Rn+16[m1+2m2+2m3+m4].(28)

4.3 Non-Standard Finite Difference (NSFD) Scheme

The NSFD approach corresponding to (3) is

{Sn+1=Sn+h∝+hωRn((1+h(βIn+μ)),En+1=En+hβSn−m+1(t−τ)In−m(t−τ)e−μτ1+h(φ+μ),In+1=In+hφEn1+h(σ+δ+μ),Rn+1=Rn+hσIn1+h(μ+ω).(29)

4.4 Positivity of the NSFD Scheme

Since all state variables within the model represent proportions of a population, it follows that something like one of them should be positive, while the others should remain non-negative consistently. In this context, the theorem presented below provides valuable insight.

Theorem: Assume that S≥0, E≥0, I≥0, and R≥0 are all positive at t=0; furthermore, ∝ ≥0, δ≥0, μ≥0, ω≥0, σ≥0, φ≥0, τ≥0, and β≥0 then Sn+1≥0, En+1≥0, In+1≥0, and Rn+1≥0. ∀nϵZ+={0, 1, 2, 3,…}.

Proof: Now for n=0, system (29) becomes

{S1=S0+h∝+hωR0((1+h(βI0+μ))≥0,E1=E0+hβS0I0e−μτ1+h(φ+μ)≥0,I1=I0+hφE01+h(σ+δ+μ)≥0,R1=R0+hσI01+h(μ+ω)≥0.(30)

Now for n=1, system (29) becomes

{S2=S1+h∝+hωR1((1+h(βI1+μ))≥0,E2=E1+hβS1I1e−μτ1+h(φ+μ)≥0,I2=I1+hφE11+h(σ+δ+μ)≥0,R2=R1+hσI11+h(μ+ω)≥0.(31)

The study assumes that the preceding set of equations guarantees that the values of S, E, I and R have the property of positivity for n = 2, 3, 4,…,n−1. In other words, for n = 2, 3, 4,…,n−1, Sn+1≥0, En+1≥0, In+1≥0 and Rn+1≥0. The positivity will now be investigated for a random positive integer n, and the system (29) becomes

{Sn+1=Sn+h∝+hωRn((1+h(βIn+μ))≥0,En+1=En+hβSnIne−μτ1+h(φ+μ)≥0,In+1=In+hφEn1+h(σ+δ+μ)≥0,Rn+1=Rn+hσIn1+h(μ+ω)≥0.(32)

Obviously, Sn+1≥0, En+1≥0, In+1≥0, and Rn+1≥0, hence the proof.

4.5 Boundedness of NSFD Scheme

Given that the model is about the human population, it is critical to ensure that at any given time ‘t’, the sum of populations in all compartments does not exceed the overall population. The following theorem addresses this condition effectively:

Theorem: Let S≥0,E≥0,I≥0 and R≥0 are finite, such that S+E+I+R≤N; furthermore, ∝ ≥0, δ≥0,μ≥0,ω≥0,σ≥0,φ≥0,τ≥0, and β≥0 then there is a constant such that Sn+1+En+1+In+1+Rn+1≤Nn for all n∀∈Z+.

Proof: For bondedness of the proposed NSFD scheme from above, the following can be obtained:

Sn+1(1+h(βIn+μ)+En+1(1+h(φ+μ)+In+1(1+h(σ+δ+μ)+Rn+1(1+h(μ+ω)=Sn+h∝+hωRn+En+hβSnIne−μτ+In+hφEn+Rn+hσIn,(33)

(Sn+1+In+1+En+1+Rn+1)(1+hμ)+hβIn+h(φ+σ+δ+ω)=(Sn+In+En+Rn)+h(∝+hωRn)+h(βSnIne−μτ+δIn+φEn+σIn),(34)

(Sn+1+In+1+En+1+Rn+1)+C=(Sn+In+En+Rn)+h(∝+hωRn)+h(βSnIne−μτ+δIn+φEn+σIn)(1+hμ),

Now for n=0 system, the above equation becomes

(S1+I1+E1+R1)+C=(S0+I0+E0+R0)+h(∝+hωR0)+h(βS0I0e−μτ+δI0+φE0+σI0)(1+hμ),

Since N1=S0+E0+I0+R0, and S1+E1+I1+R1≤N1. Similarly, for nϵz+, the above equation becomes

(Sn+1+In+1+En+1+Rn+1)+C=(Sn+In+En+Rn)+h(∝+hωRn)+h(βSnIne−μτ+δIn+φEn+σIn)(1+hμ).

Consequently Sn+1≤Nn,En+1≤Nn,In+1≤NnandRn+1≤Nn, hence the proof.

4.6 Convergence Analysis of NSFD Scheme

Let

{f∗=S+h∝+hωR((1+h(βI+μ)),g∗=E+hβSIe−μτ1+h(φ+μ),q∗=I+hφE1+h(σ+δ+μ),r∗=R+hσI1+h(μ+ω).(35)

The Jacobian matrix of the system (35) at the DFE point is given below:

J∗=[1(1+hμ)0−hβ∝μ(1+hμ)hω1+hμ0μ+hβ∝e−μτμ(1+hk1)11+hk100hφ1+hk211+hk2000hσ1+hk311+hk3].

For the above Jacobian matrix, it has eigenvalues λd1=0.7385, λd2=0.7279, λd3=0.4003 and λd4=0.9902. Clearly, λd1<1, λd2<1, λd3<1 and λd4<1. Hence, the proposed NSFD scheme is convergent.

4.7 Consistency of NSFD Scheme

Beginning with the first equation in the system (29), then

Sn+1(1+h(βIn+μ)=Sn+h∝+hωRn,(36)

The Taylor’s series expansion of the Sn+1 is as follows:

Sn+1=(Sn+hdSdt+h22!d2Sdt2+h33!d3Sdt3+…).(37)

The following can be obtained from Eq. (36):

(Sn+hdSdt+h22!d2Sdt2+h33!d3Sdt3+…)(1+h(βIn+μ)=Sn+h∝+hωRn,

After some simplification and applying h→0, then

Sn(βIn+μ)+dSdt=∝+ωRn,

dSdt=∝−βSnIn+ωRn−μSn,

⟹dSdt=∝−βSnIn+ωRn−μSn.

From the second equation of the NSFD scheme, it can have

En+1(1+h(φ+μ))=En+hβSnIne−μτ,(38)

The Taylor’s series expansion of the En+1 is as follows:

En+1=(En+hdEdt+h22!d2Edt2+h33!d3Edt3+…).(39)

Substituting the value of En+1 in (38), it can obtain

(En+hdEdt+h22!d2Edt2+h33!d3Edt3+…)(1+h(φ+μ))=En+hβSnIne−μτ,

Apply h→0, then

En(φ+μ)+dEdt=βSnIne−μτ,

dEdt=βSnIne−μτ−En(φ+μ),

⟹dEdt=βSnIne−μτ−En(φ+μ)

From the third equation of the NSFD scheme, the following can be obtained:

In+1(1+h(σ+δ+μ))=In+hφEn,(40)

The Taylor’s series expansion of the In+1 is as follows:

In+1=(In+hdIdt+h22!d2Idt2+h33!d3Idt3+…),(41)

(In+hdIdt+h22!d2Idt2+h33!d3Idt3+…)(1+h(σ+δ+μ))=In+hφEn,

By applying h→0, the following can be obtained:

In(σ+δ+μ)+dIdt=φEn,

dIdt=φEn−In(σ+δ+μ).

⟹dIdt=φEn−In(σ+δ+μ).

From the fourth equation of the NSFD scheme, it can have

Rn+1(1+h(μ+ω))=Rn+hσIn,(42)

The Taylor’s series expansion of the Rn+1 is as follows:

Rn+1=(Rn+hdRdt+h22!d2Rdt2+h33!d3Rdt3+…),(43)

Similarly,

dRdt=δIn−Rn(μ+ω)

by substituting (42) in (43) and simplifying it. Hence, the proposed NSFD scheme with the delay effect is consistent with the first order.

4.8 Global Stability

By demonstrating global stability, the study provides strong theoretical support for using time-delayed models to understand and manage typhoid fever outbreaks.

Case 1: Trivial Case

Consider a candidate Lyapunov function as follows:

L1(t)=S(t)+E(t)+I(t)+R(t),(44)

L1(t)=dSdt+dEdt+dIdt+dRdt,(45)

L1(t)=∝−μ(S+E+I+R)−δI,(46)

L1(t)=∝−μN−δI,(47)

L1(t)≤ 0.

Case 2: Disease-Free Equilibrium

Consider Volterra type Lyapunov function it can have

dΔdt=ddt(S−S∗lnS)+dEdt+dIdt+dRdt,(48)

dΔdt≤(1−S∗S)dSdt+dEdt+dIdt+dRdt,

dΔdt≤(−S∗S)dSdt+dEdt+dIdt+dRdt,

dΔdt≤(−S∗S)dSdt+0,

dΔdt≤0.

Case 3: Endemic Equilibrium

Δ(S, E, I, R)=K1(S−S∗lnS)+K2(E−E∗lnE)+K3(I−I∗lnI)+K4(R−R∗lnR),(49)

ddtΔ(S,E,I,R)=K1ddt(S−S∗lnS)+K2ddt(E−E∗lnE)+K3ddt+K4ddt(R−R∗lnR)+K4ddt(R−R∗lnR),(50)

dΔdt≤K1(1−S∗S)dSdt+K2(1−E∗E)dEdt+K3(1−I∗I)dIdt+K4(1−R∗R)dRdt,

dΔdt≤K1(S−S∗S)dSdt+K2(E−E∗E)dEdt+K3(I−I∗I)dIdt+K4(R−R∗R)dRdt,

dΔdt≤K1(S−S∗)(∝S+ωRS−(βI+μ))+K2(E−E∗)(βSIe−μτE−(φ+μ))+

K3(I−I∗)(φEI−(σ+δ+μ))+K4(R−R∗)(σIR−(ω+μ)).

From the first equations of system (3)

∝−βS∗I+ωR−μS∗=0,

βI+μ=∝S∗+ωRS∗.

From the second equation of system (3)

βSIe−μτ−φE∗−μE∗=0,

φ+μ=βSIe−μτE∗.

From the third equation of system (3)

φE−σI∗−δI∗−μI∗=0,

σ+δ+μ=φEI∗.

From the forth equation of system (3)

σI−ωR∗−μR∗=0,

ω+μ=σIR∗.

Now, it can have

dΔdt≤K1(S−S∗)(∝S+ωRS−∝S∗+ωRS∗)+K2(E−E∗)(βSIe−μτE−βSIe−μτE∗)+K3(I−I∗)(φEI−φEI∗)+K4(R−R∗)(σIR−σIR∗),

dΔdt≤−K1(S−S∗)2(∝+ωRSS∗)−K2(E−E∗)2(βSIe−μτEE∗)−K3(I−I∗)2(φEII∗)−K4(R−R∗)2(σIRR∗).

Hence, set K1=K2=K3=K4=1.