Nonlinear stochastic modelling plays an important character in the different fields of sciences such as environmental, material, engineering, chemistry, physics, biomedical engineering, and many more. In the current study, we studied the computational dynamics of the stochastic dengue model with the real material of the model. Positivity, boundedness, and dynamical consistency are essential features of stochastic modelling. Our focus is to design the computational method which preserves essential features of the model. The stochastic non-standard finite difference technique is most efficient as compared to other techniques used in literature. Analysis and comparison were explored in favour of convergence. Also, we address the comparison between the stochastic and deterministic models.

Dengue model stochastic ordinary differential equations numerical methods convergence of the proposed method
Introduction

Preliminaries

A GBM or (otherwise called exponential Brownian motion) is a ceaseless time stochastic procedure wherein the logarithm of the haphazardly fluctuating amount pursues a Brownian movement (additionally known as Wiener procedure) with the float. It is a significant cause of stochastic procedures fulfilling a stochastic differential condition (SDE). Specifically, it is utilized in the scientific fund to show stock costs operating at a profit Scholes model . A Geometric Brownian movement X(t) is the arrangement of an SDE with straight float and dispersion coefficients.

dX(t)=μX(t)dt+σX(t)dW(t).

with initial values X(0)=x0.

Deterministic Material

In this section, we considered the dynamics of the deterministic model, as presented in . For any time t, the susceptible humans are detailed as SH(t), the asymptomatic infected humans are detailed as AH(t), the symptomatic infected humans are detailed as IH(t), TH(t) individual treated humans, FH(t) represent the humans who fail treatment, SV(t) shows the amount of disposed of mosquitoes, EV(t) represent visible mosquitoes and IV(t) shows the quantity of quick-spreading mosquitoes. The transmission flow is modelled in Fig. 1.

Flow of material from the vector population to the human population

The transmission rates are explained as λH (denotes the rate at which the person infected), λV (the rate at which mosquitoes infected), μV (denotes natural death rate of mosquitoes), μH (denotes the natural death rate of persons), λV (represented the rate of infection of mosquitoes), ΠH (represented the new recruitment of the humans), ΠV (represented the new recruitment of the mosquitoes), σH (probability of transmission AH to IH classes), σV (probability of transmission IV to EV classes), δH (represented the mortality rate of humans), δV (represented the mortality rate of the mosquitoes), τH (represented the rate of vaccination of the humans), γH (represented the rate of unsuccessful vaccination). The nonlinear equations of the model as follows:

SH=ΠH-λHSHIV-μHSH,

AH=λHSHIV-σHAHIH-μHAH.

IH=σHAHIH-τHIHTH-μHIH, TH=τHIHTH-γHFHTH-μHTH, FH=γHFHTH-(μH+δH)FH, SV =ΠV-λVSVIH-μVSV, EV =λVSVIH-σVEVIV-μVEV, IV =σVEVIV-μVIV. whereNH(t)=SH(t)+AH(t)+IH(t)+TH(t)+FH(t). Nv(t)=SV(t)+EV(t)+IV(t).

therefore, dNTdt=0,dNvdt=0.

The stabilized form of model (2)(9) is as follows:

SH=ΠH-λHSHIV-μHSH,

AH=λHSHIV-σHAHIH-μHAH,

IH=σHAHIH-μHIH, SV =ΠV-λVSVIH-μVSV, EV =λVSVIH-σVEVIV-μVEV, IV =σVEVIV-μVIV.

There are two regions for the system (12)(17) like humans population and vector population are Γ1={(SH,AH,IH)ϵR+3:SH+AH+IHΠHμH,SH0,AH0,IH0} and Γ2={(SV,EV,IV)ϵR+3:SV+EV+IVΠVμV,SV0,EV0,IV0} respectively. These given regions are bounded and closed. The solution of system (12)(17) lie in the same regions. So, these given regions are also called the non-negative invariant region.

Equilibria of the Material

The equilibria of system (12)(17) can be classified into two ways under as Disease-free equilibrium is D1=(ΠHμH,0,0,ΠVμV,0,0). Endemic equilibrium is E1=(SH,AH,IH,SV,EV,IV)SH=ΠHσH-σHIH-(μH)2μHσH,AH=μHσH,IH=λHSHIV-μH2AHσHAH,EV=μVσV,SV=ΠVσV-σVIV-(μV)2μVσV,IV=λVSVIH-μV2AVσVAVandR2=ρ(F2V2-1)=λV(μH+σH)(μV+σV)ΠHμV2T1T2 where T1=(σH+μH),T2=(σV+μV). Note that R2 is the threshold number of the model (12)(17).

Stochastic Material

The construction of a stochastic material of the model from the deterministic model has presented in . For this, we shall substitute in the model (12)(17) as follows:

Put λHdt=λHdt+σdB and λVdt=λVdt+σ1dB.

dSH=(ΠH-λHSHIV-μHSH)dt-σSHIVdB,

dAH=(λHSHIV-σHAHIH-μHAH)dt+σSHIVdB,

dIH=(σHAHIH-μHIH)dt, dSV=(ΠV-λVSVIH-μVSV)dt-σ1SVIHdB, dEV=(λVSVIH-σVEVIV-μVEV)dt+σ1SVIHdB, dIV=(σVEVIV-μVIV)dt.

where “B” is the geometric Brownian motion.

Stochastic Euler Method

This scheme is constructed for the model (18)(23) as follows:

SHn+1=SHn+h(ΠH-λHSHnIVn-μHSHn-σSHnIVnΔBn),

AHn+1=AHn+h(λHSHnIVn-σHAHnIHn-μHAHn+σSHnIVnΔBn),

IHn+1=IHn+h(σHAHnIHn-μHIHn), SVn+1=SVn+h(ΠV-λVSVnIHn-μVSVn-σ1SVnIHnΔBn), EVn+1=EVn+h(λVSVnIHn-σVEVnIVn-μVEVn+σ1SVnIHnΔBn), IVn+1=IVn+h(σVEVnIVn-μVIVn).

where ‘h’ is the time parameter. The solution of the deterministic model for the DFE, i.e., D1=(ΠHμH,0,0,ΠVμV,0,0) and the reproduction number R2=0.8944<1 means help us these procedures to switch the dengue virus. The EE, i.e., E1=(0.07339,0.625,0.3016,0.08108,0.5556,0.3640) and the reproduction number $]]>R2=6.3875>1 means dengue is endemic. We are simulating the solution of this technique by using parameters values prearranged in  and see Tab. 1. Rates of the material of the model Rates DFE EE μH 0.5 0.5 λH 0.1 10.1 μV 0.5 0.5 λV 0.2 10.2 σV 0.9 0.9 σH 0.8 0.8 ΠV 0.5 0.5 ΠH 0.5 0.5 σ1 0.1 0.1 σ 0.5 0.5 Stochastic Runge–Kutta Method The pseudo-code for stochastic Runge Kutta for the system (18)(23) is as follows: Begin: Declare all constants Set the step size ‘h Declare arrays for SH,AH,IH,,SV,EV,IV. The arrays should be able to store 2000 values. Put initial values for SH,AH,IH,,SV,EV,IV at index 1 of the corresponding arrays. Index=2 For t from 0.1 till t < 200 Calculate stage 1 equations Calculate stage 2 equations Calculate stage 3 equations Calculate stage 4 equations Calculate final stage equations Index=Index+1 t = t + 0.1 End For Plot required data end program We make the simulation of discussed technique by using parameters values prearranged in  and see Tab. 1. Stochastic NSFD Method This scheme is constructed for the model (18)(23) as follows: SHn+1=(SHn+hΠH)/(1+h(λHIVn+μH+σIVnΔBn)), AHn+1=(AHn+hλHSHnIVn+hσSHnIVnΔBn)/(1+hσHIHn+hμH), IHn+1=(IHn+hσHAHnIHn)/(1+hμH), SVn+1=(SVn+hΠV)/(1+h(λVIHn+μV+σ1IHnΔBn)), EVn+1=(EVn+hλVSVnIHn+hσ1SVnIHnΔBn)/(1+h(σVIVn+μV)), IVn+1=(IVn+hσVEVnIVn)/(1+hμV). We make the simulation of the above-discussed technique by using parameters values prearranged in  and see Tab. 1. Convergence Analysis For this, we shall satisfy the following theorems as follows: Theorem 1: Forgiven initial values (SH0,AH0,IH0) R+3 and (SV 0,EV 0,IV 0) R+.3 for the system (30)(35) has a unique positive solutions (SHn,AHn,IHn) R+3 and (SV n,EV n,IV n) R+.3, respectively for all n 0. Proof: Since all rates and state variables of the system are non-negative. So, the proof is straightforward. Theorem 2: These regions Γ1={(SH,AH,IH)ϵR+3:SHn+AHn+IHnΠHμH,SH0,AH0,IH0} and Γ2={(SV,EV,IV)ϵR+3:SV n+EV n+IV nΠVμV,SV0,EV0,IV0} for all n0 are the positive invariant set for the system (31)(35). Proof: Now, we rewrite the system (30)(32) as follows: SHn+1-SHnh=ΠH-λHSHnIVn-μHSHn-σSHnIVnΔBn.AHn+1-AHnh=λHSHnIVn-σHAHnIHn-μHAHn+σSHnIVnΔBn.IHn+1-IHnh=σHAHnIHn-μHIHn. So,SHn+1+AHn+1+IHn+1ΠHμH. Also, we rewrite the system (33)(35) as follows: SV n+1-SV nh=ΠV-λVSV nIHn-μVSV n-σSV nIHnΔBn.EV n+1-EV nh=λVSV nIHn-σVEV nIV n-μVEV n+σSV nIHnΔBn.IV n+1-IV nh=σVEV nIV n-μVIV n. More precisely, Ev(n+1)+SV n+1+IV n+1ΠVμV. Theorem 3: For given n0, the eigenvalue of the system (30)(35) lies in the semi-circle. Proof: Considering the functions F, G, H, I, J and K from the system (30)(35) as follows: F=SH+hΠH1+hλHIV+hμH+hσIVΔBn,G=AH+hλHSHIV+hσSHIVΔBn1+h(σHIH+μH),H=IH+hσHAHIH1+hμH,I=SV+hΠV1+h(λVIH+μV+σ1IHΔBn),J=EV+hλVSVIH+hσ1SVIHΔBn1+h(σVIV+μV)andK=IV+hσVEVIV1+hμV The Jacobi matrix J as J=[FSHFAHFIHFSVFEVFIVGSHGAHGIHGSVGEVGIVHSHHAHHIHHSVHEVHIVISHIAHIIHISVIEVIIVJSHJAHJIHJSVJEVJIVKSHKAHKIHKSVKEVKIV ] By simplify, we get the following eigenvalues as follows: λ1=11+hμH<1,λ2=λH+σΔBnμH<1,if R2<1,λ3=λHμH<1,λ4=11+hμV<1,λ5=(λV+σ1ΔBn)1+hμV<1ifR2<1,λ6=λVμV<1. Thus, the system is stable at the rates of the material of the model. Comparison Section In this part, we are going to discuss the comparison of existing stochastic techniques and proposed technique as follows: Covariance of the Model The correlation coefficient has obtained among the compartments of the model, and results are described in Tab. 2. Covariance coefficient Sub-populations Correlation coefficient (ρ) Relationship (SH,IH) −0.3595 Inverse (IH,AH) −0.7083 Inverse (SH,AH) −0.3930 Inverse (SV,IV) −0.2906 Inverse (IV,EV) −0.8331 Inverse (SV,EV) −0.2824 Inverse Tab. 2 exhibits that susceptible humans have an inverse relation with other components of the model. It is concluded that the decreases in the other components of the model prove the disease-free state for both populations of the model. Results and Discussion Fig. 2, exhibits the solution of the stochastic Euler method for different discretization values of the parameters, and lose the essential features for h=1, as desired. In Fig. 3, we can observe that the stochastic Runge–Kutta method meets equilibria for step size h=0.1 and lose the essential features. In Fig. 4, we can observe that the stochastic NSFD technique meets for both equilibria for any value of the parameters. In a stochastic context, we have concluded that this method preserves all the essential features presented in . In Fig. 5, we have presented a comparative analysis of both types of modelling. In Fig. 6, the 3D phase plots have presented with the comparison of the computational methods. (a) Symptomatic humans at <inline-formula id="ieqn-57"><alternatives><inline-graphic xlink:href="ieqn-57.png"/><tex-math id="tex-ieqn-57"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-57"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (b) symptomatic humans at <inline-formula id="ieqn-58"><alternatives><inline-graphic xlink:href="ieqn-58.png"/><tex-math id="tex-ieqn-58"><![CDATA[$\text{h}=1$]]></tex-math><mml:math id="mml-ieqn-58"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (c) infected mosquitoes’ section at <inline-formula id="ieqn-59"><alternatives><inline-graphic xlink:href="ieqn-59.png"/><tex-math id="tex-ieqn-59"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-59"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (d) infected mosquitoes’ section at <inline-formula id="ieqn-60"><alternatives><inline-graphic xlink:href="ieqn-60.png"/><tex-math id="tex-ieqn-60"><![CDATA[$\text{h}=1$]]></tex-math><mml:math id="mml-ieqn-60"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (a) Symptomatic humans section at <inline-formula id="ieqn-61"><alternatives><inline-graphic xlink:href="ieqn-61.png"/><tex-math id="tex-ieqn-61"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-61"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (b) symptomatic humans section at <inline-formula id="ieqn-62"><alternatives><inline-graphic xlink:href="ieqn-62.png"/><tex-math id="tex-ieqn-62"><![CDATA[$\text{h}=2$]]></tex-math><mml:math id="mml-ieqn-62"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></alternatives></inline-formula> (c) infected mosquitoes section at <inline-formula id="ieqn-63"><alternatives><inline-graphic xlink:href="ieqn-63.png"/><tex-math id="tex-ieqn-63"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-63"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (d) infected mosquitoes section at <inline-formula id="ieqn-64"><alternatives><inline-graphic xlink:href="ieqn-64.png"/><tex-math id="tex-ieqn-64"><![CDATA[$\text{h}=2$]]></tex-math><mml:math id="mml-ieqn-64"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></alternatives></inline-formula> (a) Susceptible humans at <inline-formula id="ieqn-82"><alternatives><inline-graphic xlink:href="ieqn-82.png"/><tex-math id="tex-ieqn-82"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-82"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (b) susceptible humans at <inline-formula id="ieqn-83"><alternatives><inline-graphic xlink:href="ieqn-83.png"/><tex-math id="tex-ieqn-83"><![CDATA[$\text{h}=100$]]></tex-math><mml:math id="mml-ieqn-83"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:math></alternatives></inline-formula> (c) symptomatic humans at <inline-formula id="ieqn-84"><alternatives><inline-graphic xlink:href="ieqn-84.png"/><tex-math id="tex-ieqn-84"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-84"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (d) symptomatic humans at <inline-formula id="ieqn-85"><alternatives><inline-graphic xlink:href="ieqn-85.png"/><tex-math id="tex-ieqn-85"><![CDATA[$\text{h}=100$]]></tex-math><mml:math id="mml-ieqn-85"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:math></alternatives></inline-formula> (e) infected mosquitoes at <inline-formula id="ieqn-86"><alternatives><inline-graphic xlink:href="ieqn-86.png"/><tex-math id="tex-ieqn-86"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-86"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (f) infected mosquitoes at <inline-formula id="ieqn-87"><alternatives><inline-graphic xlink:href="ieqn-87.png"/><tex-math id="tex-ieqn-87"><![CDATA[$\text{h}=100$]]></tex-math><mml:math id="mml-ieqn-87"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:math></alternatives></inline-formula> (a) Symptomatic infected individuals’ section at <inline-formula id="ieqn-88"><alternatives><inline-graphic xlink:href="ieqn-88.png"/><tex-math id="tex-ieqn-88"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-88"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (b) symptomatic infected individuals’ section at <inline-formula id="ieqn-89"><alternatives><inline-graphic xlink:href="ieqn-89.png"/><tex-math id="tex-ieqn-89"><![CDATA[$\text{h}=0.4$]]></tex-math><mml:math id="mml-ieqn-89"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>4</mml:mn></mml:math></alternatives></inline-formula> (c) symptomatic infected individuals’ section at <inline-formula id="ieqn-90"><alternatives><inline-graphic xlink:href="ieqn-90.png"/><tex-math id="tex-ieqn-90"><![CDATA[$\text{h}=0.1$]]></tex-math><mml:math id="mml-ieqn-90"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula> (d) symptomatic infected individuals’ section at <inline-formula id="ieqn-91"><alternatives><inline-graphic xlink:href="ieqn-91.png"/><tex-math id="tex-ieqn-91"><![CDATA[$\text{h}=0.5\$]]></tex-math><mml:math id="mml-ieqn-91"><mml:mstyle class="text"><mml:mtext>h</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn></mml:math></alternatives></inline-formula> 3D phase plots using stochastic NSFD scheme in comparison with stochastic Euler and stochastic Runge–Kutta methods
Conclusion and Future Directions

In this study, we must claim the most effective and real stochastic analysis in comparison with the deterministic analysis of the model. Also, the construction of the stochastic model has presented with the rates of the material of the model. Unfortunately, numerical research is presented in all disciplines of science because of the non-differentiability of Brown’s motion. The standard stochastic methods employed here were the usual stochastic Euler method and the fourth-order stochastic Runge–Kutta method. Numerical properties of those methods are well known like positivity, boundedness and dynamical consistency. Various simulations were produced using different step sizes, and comparisons were shown graphically. The results showed that the methodology proposed in this work is capable of preserving the essential features of the relevant solutions of the mathematical model, as expected. On the other hand, the graphical results showed that the standard methods were not able to guarantee the preservation of some of those essential features. In particular, we proved computationally that those standard stochastic methods are incapable of preserving the essential features of the model, as desired. In that sense, the stochastic NSFD method designed in this work is a more reliable technique. Moreover, we shall extend our work in all discipline as bio-economics, biophysics, biochemistry, and many more sub-branches of material sciences. Also, the given essential features preserving analysis could be extended in the fractional-order models [23,24].

We always warmly thanks to anonymous referees. We are also grateful to Vice-Chancellor, Air University, Islamabad, for providing an excellent research environment and facilities. The first author is also thankful to Prince Sultan University for funding this work.

Funding Statement: This research project is funded by the Research and initiative centre RG-DES2017-01-17, Prince Sultan University.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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