
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39256433
70089
10.1038/s41598-024-70089-z
Article
Stochastic delayed analysis of coronavirus model through efficient computational method
Shahid Naveed 1
Raza Ali 2
Iqbal Sana 1
Ahmed Nauman 13
Fadhal Emad efadhal@kfu.edu.sa

4
Ceesay Baboucarr bceesay@utg.edu.gm

5
1 https://ror.org/051jrjw38 grid.440564.7 0000 0001 0415 4232 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
2 https://ror.org/01b009v28 Department of Physical Sciences, The University of Chenab, Gujrat, Pakistan
3 https://ror.org/00hqkan37 grid.411323.6 0000 0001 2324 5973 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
4 https://ror.org/00dn43547 grid.412140.2 0000 0004 1755 9687 Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, 31982 Al-Ahsa, Saudi Arabia
5 https://ror.org/038tkkk06 grid.442863.f 0000 0000 9692 3993 Mathematics Unit, The University of The Gambia, Sere Kunda, The Gambia
10 9 2024
10 9 2024
2024
14 211709 5 2024
13 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Stochastic delayed modeling has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. The covid-19 has been controlled globally but there is still a threat and appears in different variants like omicron and SARS-CoV-2 etc. globally. This article, considered pattern a mathematical model based on Susceptible, Infected, and recovered populations with highly nonlinear incidence rates. we studied the dynamics of the coronavirus model; a newly proposed version is a stochastic delayed model that is based on nonlinear stochastic delayed differential equations (SDDEs). Transition probabilities and parametric perturbation methods were used for the construction of the stochastic delayed model. The fundamental properties like positivity, boundedness, existence and uniqueness, and stability results of equilibria of the model with certain conditions of reproduction number are studied regularly. Also, the extinction and persistence of disease are studied with the help of well-known theorems. The numerical methods used to find a visualization of results due to the complexity of stochastic delayed differential equations. Furthermore, for computational analysis, we implemented existing methods in the literature and compared their results with the proposed method like nonstandard finite difference for stochastic delayed model. The proposed method restores all dynamical properties of the model with a free choice of time steps.

Keywords

Covid-19 disease model
Stochastic delayed differential equations (SDDEs)
Existence and uniqueness
Lyapunov function
Stability results
Reproduction number
Computational methods
Subject terms

Mathematics and computing
Nanoscience and technology
issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Noted that coronavirus attacks the respiratory system of children, which was initially identified in the 1960s. The severe acute respiratory infection significantly increased illness and death, discovered in 20031. Acute respiratory syndrome was a major outcome of coronavirus which was designated by the International Committee on Virus Taxonomy2. In3 a novel coronavirus-related respiratory sickness began in Wuhan, China in December 2019, and very fast it spread to other regions of China and other countries of the world. From 1 to 15 January, recorded some cases of COVID-19 and also noted the unreported cases and reproduction number R0. As of 5 April 2020, there had been 3819 cases of coronavirus, in including 106 death cases, since 30 January, the first case of 2019-nCoV was reported in India. The researchers have examined a Bats–Hosts–Reservoir–People transmission fractional-order COVID-19 model in this work to simulate possible transmission while accounting for individual responses and government control measures4. The symptoms of COVID-2019 vary according to the severity but to an extent, symptoms are the same as those caused by other family members of SARS like the common cold, respiratory disease, etc5. In6 employs artificial intelligence to establish the frame work for data driven forecast of 2019-CoV and mathematics to predict some confinement strategies. In7 Mathematics is blessed with a unique field that relates physical phenomena to calculations, named 'Modeling'. Health agencies collect information which is related to the death rate and recovered persons, as well as the number of persons who tested, symptomatic, or asymptomatic due to COVID-2019, and publish it daily. For accuracy of data compilation, the modeling process gives appropriate value. In8 the world economy has been severely impacted by the infectious and fatal coronavirus or COVID-19. This fear will probably reverberate across the world of education. There are several debates surrounding e-learning. Some of the arguments surrounding online pedagogy like life-long learning, accessibility, etc. It's said that Internet education is widely available and may even reach outlying and rural locations. During the COVID-19 pandemic remaining at home is no longer a safe option for a large number of mothers and children worldwide. During the lockdown, there is an increase in the number of reports of assaulting kids and domestic violence. Raising awareness of domestic abuse is so essential9. The research which relied on a large number of instructors and students from institutions in the Arab world, attempts to highlight the challenges to attaining quality in distance learning during COVID-19. The different ways that COVID-19-related institutional suspensions caused students to continue their education at home10. Even though several studies have been conducted, the study on the COVID-19 pandemic's effect on teaching and learning worldwide concludes that more research is necessary to determine appropriate pedagogy and platforms for various class levels in higher secondary, middle, and primary education in developing nations11. The coronavirus crisis put the educational system under strain and compelled teachers to switch entirely to online instructions. Modern teaching technologies have expanded the scope of education, and its practical orientation, intensified students, and increased their cognitive activity as a result of remote learning of foreign languages during the COVID-19 shutdown12. The actual threat posed by COVID-19 is still unknown, but due to the large number of cases and the virus's quick spread, it has brought the virus into the public eye and forced organizations to emergency plans and take preventative measures13. Many models released their work during COVID-19 in an effort to deliver the best data with highest level of precision. The SIR model was selected in order to obtain the highest possible ratio of COVID-19 patient; nevertheless, this model also required time to reach the pandemic level. Researchers investigate in a variety of methods in an effort to develop a cure for the virus and guarantee public safety as quick as feasible. There are dynamic symptoms of COVID-2019 which include mild, respiratory disorders, aches, fever, sore throat, etc.14. In 2020, Hellewell et al.15 worked on the isolation technique for COVID-2019 and were successful in this case. The National Center for disease control of Georgia recommended that the education process be discontinued nationwide in colleges and schools on March 02, 2020, citing the COVID-19 outbreak as the reason for the suspension of the current system16. According to its human-to-human transmission, SARS-CoV-2 is becoming to more widespread and posing a serious threat to public health. It is still unknown who SARS-CoV-2 intermediate hosts. It is essential to identify the potential intermediate host of SARS-CoV-2 to stop17. Numerous interconnected attitudinal elements impact student satisfaction in distant online learning. One such element, particularly for students who propose their education in a second language is self-efficacy in academic L2 use18. As a nosocomial illness that was acquired in the community, MERS-CoV-2 is transmitted from person to person. The study determined which patients were at high risk and required specialized cases to enhance their prognosis19. In southern China, the severe acute respiratory syndrome (SARS) first appeared in 2002–2003. Its etiological agent, the SARS coronavirus, has yet to be identified. We show here that bat species are natural hosts of coronavirus closely related to the ones causing the SARS pandemic20. In21, the authors studied the dynamics of a fractional-order delayed model of COVID-19 with vaccination efficacy. In22, the author gave a study to delay differential equations and applications to biology. In23–25, the authors studied the dynamics of a stochastic epidemic model with time delays for COVID‐19 including different mathematical aspects.

Moving away from a mathematical model that may explain the spread of infections such as COVID-19, we will focus on the current study. Concerning the existence of several distinct groups of people and their complex conversations, the mathematical model is a deterministic system. We study the generic expansion of this system in the form of a stochastic differential equation to introduce a delay element. We present the extension in its most generic version, considering states and probabilities. One of the more probabilistic, realistic, and nature-inspired modeling methods is stochastic modeling. Put another way, stochastic delayed modeling is the true means of understanding the dynamics of infectious diseases and the effect they have on physical processes. Here a stochastic model was used to represent the coronavirus in the human population. Our main goal is to provide an analysis of the stochastic coronavirus model that preserves dynamical structure. This motivates us to investigate the stochastic COVID-19 model's non-standard computational analysis due to the unavailability of analytical solutions. The model is an SIR system, that divides the human population into several smaller groups such as S for susceptible (who are not affected by the virus), I for infected (who are affected by the virus), and R for recovered (who recovered their virus due to hospitalization or vaccination).

The paper is organized into six sections. "Model formulation", model formulation and its feasible properties. "Reproduction number", stochastic model with delay formulation (stochastic delay differential equations (SDDEs) and their properties like positivity, boundedness, extinction, and persistence of disease. "Euler Maruyama method", for the implementation of standard and non-standard finite difference methods, and "Results" and "Conclusion", for results and conclusion.

Model formulation

In this section, we talk about the dynamical properties of COVID-19 which remained common in the past few years (See Fig. 1). Let us consider, Nt represents as total population and Nt,t≥0 is as defined N:0,∞→R. Similarly, S,I,R:0,∞→R represented the relation of population components, which are non-negative differential functions. Where S(t) denotes the susceptible humans, I(t) denotes the infected persons, andR(t) represents recovered persons.Figure 1 Flow dynamics of coronavirus26.

The delay differential equations of the given model as shown:1 S′(t)=a-cS(t-τ)I(t-τ)(1+γI(t-τ))e-μτ-μS(t)+αR(t),∀t≥0,

2 I′t=cIt-τSt-τ1+γIt-τe-μτ-β+μ+δ-bIt,∀t≥0,

3 R′t=βIt-α+μRt,∀t≥0,

Nt=St+It+R(t) which is satisfied at time t≥0andτ<t.4 S0≥0,I0≥0,T0≥0,R0≥0.

Existence and uniqueness of the model

To explore the fundamental properties of the model, firstly we verify that the system (1–3) has bounded, unique, and global positive solutions. In system (1–3), variables display the negative free solution ∀ times t≥0 with non-negative initial conditions.

Theorem 1

The system (1–3) is bounded at any time t, if limt→∞SupN(t)≤aμ.

Proof

The population function is N(t).So,N(t)=S(t)+I(t)+R(t).

Therefore,dNdt=dSdt+dIdt+dRdt

=a-δ-bIt-μN=0

5 Nt=N0e-μt+aμ,

limt→∞supN(t)≤aμ, as desired.

Theorem 2

(Uniqueness) For any time t, the system (1–3) shows the uniqueness and existence of the solutions.

Proof

Firstly, we will explain the term norm.

‖λ‖∞=suptϵDλλ(t), and also considered the Banach Space. To determine the results of the given problems, we need to verify the growth and Lipschitz condition. Let us consider three positive constants M1,M2 and M3<∞ such that S∞<M1 , I∞<M2 and R∞<M3

We have,6 S′=f1(S,I,R,t)I′=f2S,I,R,tR′=f3(S,I,R,t),∀t≥0

We verify that,7 fi(S,t)2<kiSi2+1,wherei=1,2,3

8 fiS1,t-fiS2,t2<ki¯S1-S22.

For proof, we consider the function f1S,I,R,t, and the following assumptions are hold.9 f12=a-cSI1+γIe-μτ-μS+αR2.

f12≤a2+3cSI1+γIe-μτ2+3μS+αR2.

f12≤(a2+suptϵDλcSI1+γIe-μτ2+suptϵDλμS+αR2).

f12≤(a2+cSIe-μτ(1+γ‖S‖∞)+μ2‖S‖∞+α2‖R‖∞).

f12≤a2+μ2‖S‖∞+α2R∞1+c‖S‖∞‖I‖∞e-μτ1+γ‖I‖∞a2+μ2‖S‖∞+α2‖R‖∞.

wherec‖S‖∞‖I‖∞e-μτ1+γ‖I‖∞a2+μ2‖S‖∞+α2‖R‖∞<1.

10 So,f1(S,I,R,t)2<k11+S2

Similarly, for function f2S,I,R,t, we get11 f22=cSIe-μτ1+γI-β+μ+δ-bI2.

f22≤cSIe-μτ1+γI2+β+μ+δ-bI2.

f22≤(suptϵDλcSIe-μτ1+γI2+suptϵDλβ+μ+δ-bI2)

f22≤(c‖S‖∞‖I‖∞e-μτ1+γ‖I‖∞+β+μ+δ-b2‖I‖∞)

f22≤β+μ+δ-b2‖I‖∞1+c‖S‖∞‖I‖∞e-μτ1+γ‖I‖∞β+μ+δ-b2‖I‖∞.

wherec‖S‖∞‖I‖∞e-μτ1+γ‖I‖∞β+μ+δ-b2‖I‖∞<1.

12 f2S,I,R,t2<k21+I2.

For f3, we have13 f32=βI-α+μR2.

f32≤βI2+α+μR2.

f32≤suptϵDλβI2+suptϵDλα+μR2.

f32≤β2‖I‖∞+α+μ2‖R‖∞.

f32≤α+μ2‖R‖∞(1+β2‖I‖∞α+μ2‖R‖∞)

Where(β2‖I‖∞α+μ2‖R‖∞)<1.

14 So,f32<k31+R2.

Therefore, the condition of linear growth is verified if

maxcSIe-μτ1+γ‖I‖∞a2+μ2‖S‖∞+α2‖R‖∞,cSIe-μτ1+γ‖I‖∞β+μ+δ-b2‖I‖∞,β2‖I‖∞α+μ2‖R‖∞<1, is desired.

Theorem 3

(Positivity) For given data in Eq. (4) (S0,I0,R(0))ϵR+3, then the solution of the system (1–3) is positive for all time t>0.

Proof

Let us consider a given system is as follows:S′=a-cSI1+γe-μτ-μS+αR.

S′≥-cI1+γIe-μτ+μS.

⇒St=S0×e-cI1+γIe-μτ+μt≥0.

Similarly, It≥I0×eβ+μ+δ-bt≥0,Rt≥R0×e-α+μt≥0, as desired.

Equilibrium points

For calculating the equilibrium points of the system, we consider that the state variables are constant. Let put the right side of the system (1–3) equal to zero. The system (1–3) is in equilibrium condition as shown below:(i) Corona virus free equilibrium = E1=S1,I1,R1=(aμ,0,0)

(ii) Corona virus existing equilibrium = E2=S∗,I∗,R∗,

Where:S∗=a+αR∗1+(cI∗1+γI∗-μ), I∗=(β+μ+δ-bcS∗γe-μτ-cS∗), R∗=βI∗α+μ.

Reproduction number

For finding the reproduction number, we use the next-generation method which is presented in27. This method gives an idea about transmission matrices represented as F and transition matrices represented as V respectively. In this way, we ignored the susceptible part and we took the remaining two parts infectious denoted by It, and recovered denoted by Rt. To obtain the result we use corona-free equilibrium points. Then we get,F=caμe-μτ000andV=β+μ+δ-b00α+μ

Then find FV-1.FV-1=caμ(β+μ+δ-b)e-μτ000

The spectral radius of ρ(FV-1) is denoted by R0=acμ(β+μ+δ-b)e-μτ.

Stability analysis

In this part, we check the local and global stability of the system (1–3), by using the equilibrium points.

Theorem 4

The system (1–3) at C1=S1,I1,R1=aμ,0,0 is locally asymptotically stable if R0<1.

Proof

By using the Jacobian Matrix of the system (1–3), as follows:J=∂F∂S∂F∂I∂F∂R∂G∂S∂G∂I∂G∂R∂H∂S∂H∂I∂H∂R

J=-cI1+γIe-μτ-μcSγ(1+γI)e-μταcI1+γIe-μτcSγ(1+γI)e-μτ-β+μ+δ-b00β-α+μ

J(C1)=-μcaμγe-μτα0caμγe-μτ-β+μ+δ-b00β-α+μ

Consider J-λI=0.

So,J=-μ-λcaμγe-μτα0caμγe-μτ-β+μ+δ-b-λ00β-α+μ-λ=0

λ1=-μ<0,λ2=caμγe-μτ-β+μ+δ-b<0.

λ3=-α+μ<0.

It shows that C1 is asymptotically stable, R0<1.

Theorem 5

The system (1–3) at C2=S∗,I∗,R∗ is locally asymptotically stable if R0>1.

Proof

By using the Jacobian Matrix of system (1–3) at C2, we obtainedJC2=-cI∗1+γI∗e-μτ-μcS∗γ(1+γI∗)e-μταcI∗1+γI∗e-μτcS∗γ(1+γI∗)e-μτ-β+μ+δ-b00β-α+μ

Consider J(C2)-λI=0.-cI∗1+γI∗e-μτ-μ-λcS∗γ(1+γI∗)e-μταcI∗1+γI∗e-μτcS∗γ(1+γI∗)e-μτ-β+μ+δ-b-λ00β-α+μ-λ=0.

λ3+λ2A2+λA1+A0=0

where A2= cI∗1+γI∗e-μτ-μ+cS∗γ1+γI∗e-μτ-A-α-μ,A1=[c2I∗S∗γ1+γI∗e-μτ-AcI∗1+γI∗e-μτ-αcI∗1+γI∗e-μτ-μcI∗1+γI∗e-μτ+μcS∗γ1+γI∗e-μτ-μA-αA-μ2+αcS∗γ1+γI∗e-μτ+μcS∗γ1+γI∗e-μτ-αA-μA]

A0=[αc2I∗S∗γ1+γI∗e-μτ+μc2I∗S∗γ1+γI∗e-μτ-αAcI∗1+γI∗e-μτ-μAcI∗1+γI∗e-μτ-μαcS∗γ1+γI∗e-μτ+μ2cS∗γ1+γI∗e-μτ-μαA-μ2A+αc2I∗S∗γ1+γI∗e-μτ+μc2I∗S∗γ1+γI∗e-μτ+αβcI∗1+γI∗e-μτ.

By Routh Hurwitz's criterion when R0>1,hereA2,A1,andA0 are positive. So, C2 is locally asymptotically stable.

Theorem 6

The system (1–3) at C1=S1,I1,R1=aμ,0,0 is globally asymptotically stable if R0<1.

Proof

To prove this theorem, we let the Lyapunov function U:Ω→R defined asUI=lnII0,

dUIdt=I′I×1I′dIdt,

dUdt≤β+μ+δ-bcae-μτμβ+μ+δ-b-1.

dUdt≤β+μ+δ-bR0-1.

dUdt≤0,ifR0<1.

Hence, the system is globally asymptotically stable.

Theorem 7

The system (1–3) is globally asymptotically stable at C2=(S∗,I∗,R∗), if R0>1.

Proof

By Lyapunov function, X:Ω→R defined as:X=S-S∗-S∗logSS∗+I-I∗-I∗logII∗+R-R∗-R∗logRR∗.

dXdt=-aSS∗S-S∗2-αRSS∗S-S∗2-(βIRR∗R-R∗2)≤0.

So, dxdt≤0, Hence, C2, is globally asymptotically stable.

Transition probabilities

We employed the terminology Ut=[St,It,Rt]T to give the stochastic extension of this model. The quantity of possibilities for an occurrence is shown in Table 1 as shown below:Table 1 Representation of changes in parameters.

Transitions (Ti)	Probabilities (Pi)	
T1=[100]T	P1=aΔt	
T2=[-100]T	P2=cSI(1+γI)e-μτΔt	
T3=[-100]T	P3=μSΔt	
T4=[10-1]T	P4=αRΔt	
T5=[0-10]T	P5=(μ+δ-b)Δt	
T6=[0-11]T	P6=βIΔt	
T7=[00-1]T	P7=μRΔt	

Next, we calculated the drift and diffusion of Eq. (1–3) as shown below:Expectation=E∗ΔU=∑i=17PiTi=a-cSI1+γIe-μτ-μS+αRcIS1+γIe-μτ-(β+μ+δ-b)IβI-α+μRΔt.

Variance=E∗ΔUΔUT=∑i=17Pi[Ti][Ti]T.

Drift =HUt,t=E∗ΔUΔt, Diffusion =KU(t),t=E∗[ΔUΔUT]Δt, so15 dUt=HU(t),tdt+KUt,tdBt.

The above Eq. (15) is called the stochastic delay differential equation. Where B(t) is Brownian motion.

Euler Maruyama method

The standard numerical methodology will be discussed in the proceeding section for approximation of stochastic model solutions. This is agreed by us that In={0,1,2,3,⋯n}. Consider N∈N, as well as the consistent division of the temporal interval [0,T] with uniform partition equal to τ=TN, and their respective nodes are presented as 0=t0<t1<t2<⋯tN=T.

For each n∈IN. Further, this will be agreed by us Un=U(tn), however n∈IN and U(t)=(S,I,R)t. ΔWn=Wtn+1-W(tn).

Stochastic delayed model

The phrase "time delay" is used in transmission terms to denote the duration of viral infection's incubation or the interval between infection and the start of symptoms. The addition of time delay in the model could result in many periodic solutions for varying time delay τ values. Considered that cdt=cdt+σdB(t) with parametric perturbation technique for the Eqs. (1–3) as follows28,16 dSt=a-cStIt1+γIte-μτ-μSt+αRtdt-σcStIt1+γItdB(t),dIt=[cI(t)S(t)1+γIte-μτ-(β+μ+δ-b)I(t)]dt+σcStIt1+γItdB(t),dRt=βIt-α+μRtdt,

In the above equations, τ is the delay effect, σ is the randomness, ∀t≥0 and B(t) is Brownian motion. These equations are non-integrable due to Brownian motion.

Positivity and boundedness

Let's introduce a probability space which is represented as (Ω,F,P) and {Ft}tϵR represented as a filtration. For satisfying the right continuous conditions and increasing29, F0 contains P-null sets. Symbolically represented as Ut=(St,It,Rt) and norm as U(t)=S2t+I2t+R2(t).

Theorem 8

For model (16) and any given initial value S0,I0,R0∈R+3, there is a unique solution St,It,Rt, which is defined on t∈[0,∞] and the solution of the system will remain in R+3 with probability one.

Proof

The initial value S0,I0,R0ϵR+3 of the model (16) satisfies the local Lipschitz condition, and has a unique local solution existing on t∈0,te,te shows the explosion time. Next to show that the solution of the system is global, te=∞.

Suppose d0≥1 for initial values, they all lie within the interval [1d0,d0]. For each integer, d≥d0, this sequence is known as stopping time.17 τd=inft∈0,te:minSt,It,Rt≤1dormaxSt,It,Rt≥d.

where inf ∅ = ∞ (∅ is an empty set) .

By the definition of stopping time, τd is increasing function as d→∞.18 So,τ∞=limd→∞τd

τ∞≤τe

If τ∞=∞, then τe=∞.

If this argument is false, then there exists a constant T>0 and εϵ(0,1) such that,19 P{τ∞≤T}>ε.

There exists an integer d1≥d0.20 Pτd≤T>ε,∀d≥d1.

For t≤τd, we getdS+I+R=a-cSI1+γIe-μτ-μS+αR+cSI1+γIe-μτ-β+μ+δ-bI+βI-α+μR.

=a-μS+I+R-δI+bIdt.

≤a-μS+I+Rdt.

After calculation, we see that St+It+Rt≤aμifS0+I0+R0≤aμotherwiseS0+I0+R0≥aμ

Define a C3-function V:R+3→R+ by21 VS,I,R=S-1-lnS+I-1-lnI+R-1-lnR.

By using Ito’s formula, we can obtaindVS,I,R=1-1SdS+1-1IdI+1-1RdR+σ22dt.

Putting values of dS,dI,dR.=a+2μ+β+δ+b+σ22dt+σcS-I1+γIe-μτdBt.

Let, M=a+2μ+β+δ+b+σ22; where M is a constant. Then above equation as22 dVS,I,R≤Mdt+σcS-I1+γIe-μτdBt.

Integrating above equation from limit 0toτd∧τ. Now,23 ∫0τd∧τdVS,I,R≤∫0τd∧τMdS+∫oτd∧τσcS-I1+γIe-μτdBt

where τd∧τ=min(τd,T). Now, taking expectations.24 EVSτd∧τ,Iτd∧τ,Rτd∧τ≤VS0,I0,R0+MT.

Let, Ωd={τd≤τ}P=Ωd>ε.

For every, uϵΩd. For some i,vi(τd,u) equals either dor1d for i = 1, 2, 3.

Hence,

V(Sτd,u,Iτd,u,Rτd,u) is less mind-1-lnd,1d-1-ln1d, we obtain25 VS0,I0,R0+MT≥EIΩduVSτd,u,Iτd,u,Rτd,u≥ε[(d-1-lnd)∧(1d-1-ln1d)]

Letting d→∞, leads to the contradiction,

∞=VS0,I0,R0+MT<∞ as desired.

Extinction and persistence

Consider B(t) represents the Brownian motion and I(t) be the Ito drift–diffusion process that satisfies the stochastic model as:dIt=μIt,tdt+σIt,tdBt.

If k(I,t)∈C2(R2,R) then k(I,t) is also Ito drift diffusion. So,dkI,t=dkdtIt,tdt+dkdtIt,tdBt+12d2kdt2It,tdBt2.

Theorem 9

The model (16) has contained (S(t),I(t),R(t)) be the solution with the initial value (S(0),I(0),R(0)), then we have two conditions

iIfσ2>cae-μτ2μβ+μ+δ-b-β+μ+δ-b holds, then limt→∞lnI(t)t>0.

(ii)Ifσ2<cae-μτ2μβ+μ+δ-b-(β+μ+δ-bholds, then limt→∞suplnI(t)t≤0.

Proof

Consider theI component of SDDE’s as:dIt=[cI(t)S(t)1+γIte-μτ-(β+μ+δ-b)I(t)]dt+σcStIt1+γItdB(t)

Applying Ito’s formula, we getdlnI=k′IdI+12k″II2σ2dt.

dlnI=cS1+γIe-μτ-β+μ+δ-b-12σ2dt+σcS1+γIe-μτdB.

Integrate with respect to I on both sides from 0tot, we getlnI=ln0+∫0tcS1+γIe-μτ-β+μ+δ-b-12σ2dt+∫0tσcS1+γIe-μτdB.

Ifσ2>cae-μτ2μβ+μ+δ-b-β+μ+δ-b, thenln(I)t>cae-μτ2μβ+μ+δ-b-β+μ+δ-b+Mtt+lnI(0)t

limt→∞ln(I)t>cae-μτ2μβ+μ+δ-b-β+μ+δ-b>0

Ifσ2<cae-μτ2μβ+μ+δ-b-(β+μ+δ-b , thenln(I)t<β+μ+δ-bcae-μτμβ+μ+δ-b2-σ22β+μ+δ-b-1+Mtt+lnI(0)t.

limt→∞supln(I)t<β+μ+δ-bR0s-1,

Therefore, R0s<1, R0s=R0d-σ22β+μ+δ-b<1limt→∞supI(t)t≤0.

limt→∞It=0 as desired.

Numerical analysis

In this section, we will analyze some standard and non-standard numerical methodologies of the solution of the stochastic model and present some numerical results. It is important to note that this method has the benefits of offering limited computational cost and extremely precise estimates. Furthermore, it is important to note that applying the non-standard method may ensure the preservation of the number of intriguing aspects of appropriate solutions.

Stochastic non-standard computational method

Our parametric perturbation model of Eq. (16) can be expressed in the non-standard computational method which is named as stochastic non-standard finite difference method.26 dSdt=a-cSI1+γIe-μτ-μS+αR-σc[SI1+γIdBdt.

Equation (16) decomposed in the stochastic non-standard finite difference method as shown:27 Sn+1-Snh=a-cSn+1In1+γIne-μτ-μSn+1+αRndt-σcSn+1In1+γIne-μτΔBn.

Same as Eq. (27), which is in the stochastic NSFD process, we write Eq. (16) in this way accordingly.28 Sn+1=Sn+ha+hαRn1+hcIn1+γIne-μτ+hμ+σhcIn1+γIne-μτΔBn,

29 In+1=In+hcSnIn1+γIne-μτ+hbIn+σhcSnIn1+γIne-μτΔBn1+h(β+μ+δ-b),

30 Rn+1=Rn+hβIn1+h(α+μ),

where, n=0,1,2,⋯ and ΔBn=ΔBtn+1-ΔBtn is standardized distribution. i.e. ΔBn∼N(0,1).

Convergence analysis

In this section, we present some theorems related to convergence analysis.

Theorem 10

The region Γ={Sn≥0,In≥0,Rn≥0;Sn+In+Rn≤aμ} is feasible for Eq. (30–32), ∀n≥0, where n is the positive function.

Proof

The Eq. (27–30) decayed as shown below.Sn+1-Snh=(a-cSnIn1+γIne-μτ-μSn+αRndt-σc[SnIn1+γIne-μτ]ΔBn

Same as for IandR,In+1-Inh=(cInSn1+γIne-μτ-(β+μ+δ-b)Indt-σc[SnIn1+γIne-μτ]ΔBn

Rn+1-Rnh=(βIn-α+μRn).

These equations are added up then we get,(Sn+1+In+1+Rn+1)-(Sn+In+Rn)h=a-μSn+In+Rn-(δ-b)In

(Sn+1+In+1+Rn+1)-(Sn+In+Rn)h≤a-μSn+In+Rn

(Sn+1+In+1+Rn+1)≤aμ.

The result is bounded ∀n≥0.

Theorem 11

(for stability) Our computational analysis method is stable if eigenvalues lie in the unit circle for any n≥0 and ΔBn=0.30

Proof

Let, A=S+ha+hαR1+hcSI1+γIe-μτ+kμ,B=1+hcSI(1+γI)e-μτ1+hβ+δ+μ-bI,C=R+hβI1+h(α+μ).

The system (27–30) converges to equilibrium points of the model, it satisfies the condition ρ(J)<1, where ρ(J) is spectral radius of the Jacobean. If ρ(J)>1, it is unstable then ρJ=1 is neutrally stable. The Jacobian matrix asJ=∂A∂S∂A∂I∂A∂R∂B∂S∂B∂I∂B∂R∂C∂S∂C∂I∂C∂R

∂A∂S=∂∂SS+ha+hαR1+hcSI1+γIe-μτ+hμ,∂A∂S=11+hcI1+γIe-μτ+kμ

∂A∂I=∂∂IS+ha+hαR1+hcSI1+γIe-μτ+hμ,∂A∂I=(S+ha+hαR)(hc+2hcγIe-μτ)(1+hcI1+γIe-μτ)2

∂A∂R=∂∂RS+ha+hαR1+hcSI1+γIe-μτ+hμ,∂A∂R=hα1+hcI(1+γI)

∂B∂S=hcI(1+γI)e-μτ1+h(β+μ+δ-b),∂B∂I=1+hcS+2hcSγI+hb1+h(β+μ+δ-b),∂B∂R=0

∂C∂S=0,∂C∂I=hβ1+h(α+β),∂C∂R=11+h(α+β).

At corona free equilibrium point E1=(aμ,0,0). The Jacobian matrix isJ(E1)=11+hμ-aμ+hahchα01+hacμ+hb1+h(β+μ+δ-b)00hβ1+h(α+β)11+h(α+β)

The eigenvalues are λ1=11+hμ<1, λ2=μ+hacμ+hμbμ(1+hβ+μ+δ-b)<1, λ3=11+h(α+β)<1.

At endemic equilibrium point E2=(S∗,I∗,R∗). The Jacobian matrix isJ(E2)=11+hcI∗1+γI∗e-μτ+kμ(S∗+ha+hαR∗)(hc+2hcγI∗e-μτ)(1+hcI∗1+γI∗e-μτ)2hα1+hcI∗(1+γI∗)hcI∗(1+γI∗)e-μτ1+h(β+μ+δ-b)1+hcS∗+2hcS∗γI∗+hb1+h(β+μ+δ-b)00hβ1+h(α+β)11+h(α+β)

By using MATLAB and the values of parameters given in Table 2, verified that the maximum eigenvalues for both Jacobian J(E1) and J(E2) are less than one, as desired.Table 2 Description of model parameters.

Parameters	Description	Values26	
c	Shows convex incidence rate	0.580 (CVEE)

0.380 (CVFE)

	
δ	Shows the rate of infected persons dying due to COVID-19	0.05	
α	Shows the rate of recovered persons	0.854302	
μ	Shows the rate of those humans who die due to other diseases or natural death	0.5	
β	Represent the rate at which infected persons are recovered due to vaccination, hospitalization, etc	0.09871	
a	Newly recruited rate	0.5	
b	the rate of infected human's immigrant to one location to another	0.205	
γ	The rate of infection	0.0003	
σ	Randomness of the model	≥0	

Results

In this section, we concentrate on recognizing the essential computational characteristics of the nonlinear computational model (27–30). Specifically, we aim to demonstrate the dynamical properties like positivity, boundedness, and dynamical consistency through computational methods around the steady state of the stochastic delayed coronavirus model. Furthermore, by analyzing the moduli of the discrete system's eigenvalues, we can show the stability of the discrete model. We show some results with the help of graphs of coronavirus existing equilibrium points. These comparison graphs are plotted with computational methods like Stochastic Euler, Stochastic Runge Kutta, and stochastic NSFD method under tiny changes of time step size. Also, the effect of delay on susceptible and infected compartments has been observed. The infected human rate converges to zero, when the value τ=0.7, τ is the delay effect, and the susceptible rate increases. Figure 2a, c, e observed that the results of the system merge at the desired equilibrium points with a small time step size through the numerical methods like stochastic Euler, stochastic Runge Kutta and stochastic NSFD. Time is defined in days and in Fig. 2b, d, we obtain unbounded and negative results with very small changes in time step size. Its means that stochastic Euler and stochastic Runge Kutta violate the dynamical properties of the model and depend on time step size. But Fig. 2f, shows that the results from the stochastic NSFD is consistent with biological nature of the model and independent of time size method. In Fig. 3a, b, (delay effect on infected person), we conclude that for the different values of τ, the rate of infected persons converges to zero. It means that coronavirus-controlled, susceptible rate increases. Figure 4, admits a comparison of the delay effect and reproduction number as desired.Figure 2 Comparison between computational methods of corona existing equilibrium (a) the behavior of sub-populations for coronavirus existing equilibrium by using stochastic Euler method at h=0.01. (b) The stochastic Euler method shows divergent behavior at h=1, (c) represent the behaviour of CVEE by using stochastic Rung Kutta method at h=0.01 (d). The stochastic Rung-Kutta shows unbounded result at h=2, (e) the behaviour of sub-populations for coronavirus equlibrium by using stochastic NSFD method at h=0.01 (f). The stochastic NSFD method shows the convergent behavior in at h=100.

Figure 3 Graph of delay effect on susceptible and infected humans. (a) Infected results for different values of delay. (b) Susceptible results for different values of delay.

Figure 4 Comparison of reproduction number and delay effect with time step size.

Conclusion

In the present study, a stochastic delayed mathematical model examined in this work may be used to simulate the spread of infectious illness. Specifically, the stochastic delayed mathematical model examined in this paper might be able to explain how the coronavirus spreads. In a deterministic system, the mathematical model takes into consideration the existence of several subpopulations and their nonlinear interactions. A generic expansion of this system in the form of a stochastic delay differential equation was developed. This study examines an extension that takes into consideration different states and transition probabilities with parametric perturbation; the most generic version of the extension is provided here. The positivity, boundedness, extinction, and persistence of the stochastic delayed mathematical model have been studied rigorously. Also, some standard numerical methods are used to simulate its results like Euler Maruyama, stochastic Euler, and stochastic Runge Kutta. Unfortunately, these methods did not restore the dynamical properties of the model. Then the newly developed method like nonstandard finite difference in the sense of stochastic with delay for the particular model has been designed. The proposed approach restores the feasible region of the model without any constraints. Also, the delay effect has been observed for both susceptible and infected classes. The increase in delay means that the susceptibility of humans increases in other words the infectivity of humans decreases and eventually moves to zero and vice versa. In the future, we shall extend our results to other types of modeling of real-world problems.

Acknowledgements

All authors are grateful for the valuable comments of anonymous reviewers. Furthermore, all authors reviewed the results and approved the final version of the manuscript. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [KFU241399].

Author contributions

Ali Raza, Sana Iqbal and Emad Fadhal initiated, conceptualized, designed the study, and interpreted the results along with their implications. Nauman Ahmed, Navid Shahid, and Baboucarr Ceesay conducted comprehensive analyses, managed all data and concluded remarks. All authors made substantial contributions to the study, participated in writing, and approved the manuscript. Also, read and approved a manuscript with the given study.

Data availability

The datasets analyzed during the current study are available from the corresponding author upon reasonable request. No pharmaceutical study is based on the coronavirus disease. Our focus is a non-pharmaceutical strategy to study the transmission dynamics of coronavirus.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Kahn JS McIntosh K History and recent advances in coronavirus discovery Pediatric Infect. Dis. J. 2005 24 11 S223 S227 10.1097/01.inf.0000188166.17324.60
Kahn, J. S. & McIntosh, K. History and recent advances in coronavirus discovery. Pediatric Infect. Dis. J. 24(11), S223–S227. 10.1097/01.inf.0000188166.17324.60 (2005).10.1097/01.inf.0000188166.17324.60
2. Yousaf M Zahir S Riaz M Hussain SM Shah K Statistical analysis of forecasting COVID-19 for the upcoming month in Pakistan Chaos Solitons Fractals 2020 138 109926 10.1016/j.chaos.2020.109926 32501377
Yousaf, M., Zahir, S., Riaz, M., Hussain, S. M. & Shah, K. Statistical analysis of forecasting COVID-19 for the upcoming month in Pakistan. Chaos Solitons Fractals 138, 109926 (2020).32501377 10.1016/j.chaos.2020.109926
3. Zhao S Musa SS Lin Q Ran J Yang G Wang W Lou Y Yang L Gao D He D Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: A data-driven modelling analysis of the early outbreak J. Clin. Med. 2020 9 388 10.3390/jcm9020388 32024089
Zhao, S. et al. Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: A data-driven modelling analysis of the early outbreak. J. Clin. Med. 9, 388. 10.3390/jcm9020388 (2020).32024089 10.3390/jcm9020388
4. Shaikh AS Shaikh IN Nisar KS A mathematical model of COVID-19 using fractional derivative: Outbreak in India with dynamics of transmission and control Adv. Differ. Equ. 2020 2020 373 10.1186/s13662-020-02834-3 32834815
Shaikh, A. S., Shaikh, I. N. & Nisar, K. S. A mathematical model of COVID-19 using fractional derivative: Outbreak in India with dynamics of transmission and control. Adv. Differ. Equ. 2020, 373. 10.1186/s13662-020-02834-3 (2020).32834815 10.1186/s13662-020-02834-3
5. Saif L Animal coronavirus vaccines: Lessons for SARS Dev. Biol. 2004 119 129140
Saif, L. Animal coronavirus vaccines: Lessons for SARS. Dev. Biol. 119, 129140 (2004).
6. Chen T Rui J Wang Q Zhao Z Cui JA Yin L A mathematical model for simulating the transmission of Wuhan novel coronavirus Infect. Dis. Poverty 2020 9 24 10.1186/s40249-020-00640-3 32111262
Chen, T. et al. A mathematical model for simulating the transmission of Wuhan novel coronavirus. Infect. Dis. Poverty 9, 24 (2020).32111262 10.1186/s40249-020-00640-3
7. Wu JT Leung K Leung GM Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study Lancet 2020 395 10225 689 697 10.1016/S0140-6736(20)30260-9 32014114
Wu, J. T., Leung, K. & Leung, G. M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study. Lancet 395(10225), 689–697 (2020).32014114 10.1016/S0140-6736(20)30260-9
8. Dhawan S Online learning: A panacea in the time of COVID-19 crises J. Educ. Technol. 2020 49 1 5 22 10.1177/0047239520934018
Dhawan, S. Online learning: A panacea in the time of COVID-19 crises. J. Educ. Technol. 49(1), 5–22. 10.1177/0047239520934018 (2020).10.1177/0047239520934018
9. Ravichandran P Shah AK Shadow pandemic: Domestic violence and child abuse during the COVID-19 lockdown in India Int. J. Res. Med. Sci. 2020 08 08 3118 10.18203/2320-6012.ijrms20203477
Ravichandran, P. & Shah, A. K. Shadow pandemic: Domestic violence and child abuse during the COVID-19 lockdown in India. Int. J. Res. Med. Sci. 08(08), 3118. 10.18203/2320-6012.ijrms20203477 (2020).10.18203/2320-6012.ijrms20203477
10. Lassoued Z Alhendawi M Bashitialshaaer R An exploratory study of the obstacles for achieving quality in distance learning during the COVID-19 pandemic Educ. Sci. 2020 10 9 232 10.3390/educsci10090232
Lassoued, Z., Alhendawi, M. & Bashitialshaaer, R. An exploratory study of the obstacles for achieving quality in distance learning during the COVID-19 pandemic. Educ. Sci. 10(9), 232 (2020).10.3390/educsci10090232
11. Pokhrel S Chhetri R A literature review on impact of COVID-19 pandemic on teaching and learning Higher Educ. Future 2021 8 1 133 141 10.1177/2347631120983481
Pokhrel, S. & Chhetri, R. A literature review on impact of COVID-19 pandemic on teaching and learning. Higher Educ. Future 8(1), 133–141. 10.1177/2347631120983481 (2021).10.1177/2347631120983481
12. Misirli O Ergulec F Emergency remote teaching during the COVID-19 pandemic: Parents experiences and perspectives Educ. Inf. Technol. 2021 26 6 6699 6718 10.1007/s10639-021-10520-4
Misirli, O. & Ergulec, F. Emergency remote teaching during the COVID-19 pandemic: Parents experiences and perspectives. Educ. Inf. Technol. 26(6), 6699–6718 (2021).10.1007/s10639-021-10520-4
13. Sun C Role of the eye in transmitting human coronavirus: What we know and what we do not know Front. Public Health 2020 8 542760 10.3389/fpubh.2020.00155
Sun, C. et al. Role of the eye in transmitting human coronavirus: What we know and what we do not know. Front. Public Health 8, 542760 (2020).10.3389/fpubh.2020.00155
14. Kucharski AJ Russell TW Diamond C Liu Y Edmunds J Early dynamics of transmission and control of COVID-19: A mathematical modelling study Lancet Infect. Dis. 2020 11 2 1 17
Kucharski, A. J. et al. Early dynamics of transmission and control of COVID-19: A mathematical modelling study. Lancet Infect. Dis. 11(2), 1–17 (2020).
15. Hellewell J Abbott S Gimma A Bosse NI Jarvis CI Russell TW Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts Lancet Glob. Health 2020 2020 488 496 10.1016/S2214-109X(20)30074-7
Hellewell, J. et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob. Health 2020, 488–496 (2020).10.1016/S2214-109X(20)30074-7
16. Basilaia G Replacing the classic learning form at universities as an immediate response to the COVID-19 virus infection in Georgia Int. J. Res. Appl. Sci. Eng. Technol. 2020 8 3 101 108 10.22214/ijraset.2020.3021
Basilaia, G. et al. Replacing the classic learning form at universities as an immediate response to the COVID-19 virus infection in Georgia. Int. J. Res. Appl. Sci. Eng. Technol. 8(3), 101–108 (2020).10.22214/ijraset.2020.3021
17. Liu Z Composition and divergence of coronavirus spike proteins and host ACE2 receptors predict potential intermediate hosts of SARS-CoV-2 J. Med. Virol. 2020 92 6 595 601 10.1002/jmv.25726 32100877
Liu, Z. et al. Composition and divergence of coronavirus spike proteins and host ACE2 receptors predict potential intermediate hosts of SARS-CoV-2. J. Med. Virol. 92(6), 595–601 (2020).32100877 10.1002/jmv.25726
18. Yüksel H Remote learning during COVID-19: Cognitive appraisals and perceptions of English medium of instruction (EMI) students Educ. Inf. Technol. 2022 27 347 363 10.1007/s10639-021-10678-x
Yüksel, H. Remote learning during COVID-19: Cognitive appraisals and perceptions of English medium of instruction (EMI) students. Educ. Inf. Technol. 27, 347–363. 10.1007/s10639-021-10678-x (2022).10.1007/s10639-021-10678-x
19. Al-Jasser FS Nouh RM Youssef RM Epidemiology and predictors of survival of MERS-CoV infections in Riyadh region, 2014–2015 J. Infect. Public Health 2019 12 2 171 177 10.1016/j.jiph.2018.09.008 30340964
Al-Jasser, F. S., Nouh, R. M. & Youssef, R. M. Epidemiology and predictors of survival of MERS-CoV infections in Riyadh region, 2014–2015. J. Infect. Public Health 12(2), 171–177 (2019).30340964 10.1016/j.jiph.2018.09.008
20. Li W Bats are natural reservoirs of SARS-like coronaviruses Science 2005 310 676 679 10.1126/science.1118391 16195424
Li, W. et al. Bats are natural reservoirs of SARS-like coronaviruses. Science 310, 676–679. 10.1126/science.1118391 (2005).16195424 10.1126/science.1118391
21. Rihan FA Kandasamy U Alsakaji HJ Sottocornola N Dynamics of a fractional-order delayed model of COVID-19 with vaccination efficacy Vaccines 2023 11 4 758 10.3390/vaccines11040758 37112670
Rihan, F. A., Kandasamy, U., Alsakaji, H. J. & Sottocornola, N. Dynamics of a fractional-order delayed model of COVID-19 with vaccination efficacy. Vaccines 11(4), 758 (2023).37112670 10.3390/vaccines11040758
22. Rihan FA Delay Differential Equations and Applications to Biology 2021 Springer 123 141
Rihan, F. A. Delay Differential Equations and Applications to Biology 123–141 (Springer, 2021).
23. Alsakaji HJ Rihan FA Hashish A Dynamics of a stochastic epidemic model with vaccination and multiple time-delays for COVID-19 in the UAE Complexity 2022 2022 1 4247800 10.1155/2022/4247800
Alsakaji, H. J., Rihan, F. A. & Hashish, A. Dynamics of a stochastic epidemic model with vaccination and multiple time-delays for COVID-19 in the UAE. Complexity 2022(1), 4247800 (2022).10.1155/2022/4247800
24. Khan T Rihan FA Riaz MB Altanji M Zaagan AA Ahmad H Stochastic epidemic model for the dynamics of novel coronavirus transmission AIMS Math. 2024 9 5 12433 12457 10.3934/math.2024608
Khan, T. et al. Stochastic epidemic model for the dynamics of novel coronavirus transmission. AIMS Math. 9(5), 12433–12457 (2024).10.3934/math.2024608
25. Rihan FA Alsakaji HJ Rajivganthi C Stochastic SIRC epidemic model with time-delay for COVID-19 Adv. Differ. Equ. 2020 2020 1 502 10.1186/s13662-020-02964-8 32963509
Rihan, F. A., Alsakaji, H. J. & Rajivganthi, C. Stochastic SIRC epidemic model with time-delay for COVID-19. Adv. Differ. Equ. 2020(1), 502 (2020).32963509 10.1186/s13662-020-02964-8
26. Ud Din R Shah K Ahmad I Study of transmission dynamics of novel COVID-19 by using mathematical model Adv. Differ. Equ. 2020 2020 323 10.1186/s13662-020-02783-x 32834812
Ud Din, R. et al. Study of transmission dynamics of novel COVID-19 by using mathematical model. Adv. Differ. Equ. 2020, 323. 10.1186/s13662-020-02783-x (2020).32834812 10.1186/s13662-020-02783-x
27. Sene N Analysis of the stochastic model for predicting the novel coronavirus disease Adv. Differ. Equ. 2020 568 1 01 19
Sene, N. Analysis of the stochastic model for predicting the novel coronavirus disease. Adv. Differ. Equ. 568(1), 01–19 (2020).
28. Allen EJ Construction of equivalent stochastic differential equation models Stoch. Anal. Appl. 2008 26 2 274 297 10.1080/07362990701857129
Allen, E. J. et al. Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl. 26(2), 274–297 (2008).10.1080/07362990701857129
29. Driekmann O Heesterbeek JAP Roberts MG The construction of next-generation matrices for compartmental epidemic models J. R. Soc. Interface 2009 07 47 873 885 10.1098/rsif.2009.0386
Driekmann, O., Heesterbeek, J. A. P. & Roberts, M. G. The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface 07(47), 873–885 (2009).10.1098/rsif.2009.0386
30. Jansen H Twizell EH An unconditionally convergent discretization of the SEIR model Math. Comput. Simul. 2002 58 01 147 158 10.1016/S0378-4754(01)00356-1
Jansen, H. & Twizell, E. H. An unconditionally convergent discretization of the SEIR model. Math. Comput. Simul. 58(01), 147–158 (2002).10.1016/S0378-4754(01)00356-1
