Intelligent Automation & Soft Computing DOI:10.32604/iasc.2022.024993
Article

Computational Approximations for Real-World Application of Epidemic Model

Shami A. M. Alsallami1, Ali Raza2,*, Mona Elmahi3, Muhammad Rafiq4, Shamas Bilal5, Nauman Ahmed6 and Emad E. Mahmoud7

1Department of Mathematical Sciences, College of Applied Science, Umm Al-Qura University, Makkah, 21955, Saudi Arabia
2Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College Wazirabad, Punjab Higher Education Department (PHED), Lahore, 54000, Pakistan
3Mathematics Department, College of Science & Arts in Riyadh Alkkbra, Qassim University, Saudi Arabia
4Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, 54600, Pakistan
5Department of Mathematics, University of Sialkot, 51310, Sialkot, Pakistan
6Department of Mathematics and Statistics, The University of Lahore, 54590, Lahore, Pakistan
7Department of Mathematics, College of Science, Taif University, P. O. Box 11099, Taif, 21944, Saudi Arabia
*Corresponding Authors: Ali Raza. Email: Alimustasamcheema@gmail.com
Received: 07 November 2021; Accepted: 24 December 2021

Abstract: The real-world applications and analysis have a significant role in the scientific literature. For instance, mathematical modeling, computer graphics, camera, operating system, Java, disk encryption, web, streaming, and many more are the applications of real-world problems. In this case, we consider disease modeling and its computational treatment. Computational approximations have a significant role in different sciences such as behavioral, social, physical, and biological sciences. But the well-known techniques that are widely used in the literature have many problems. These methods are not consistent with the physical nature and even violate the actual behavior of the continuous model. The structural properties needed for such disciplines, as dynamical consistency, positivity, and boundedness, are the critical requirements of the models in these fields. We studied the transmission of Lassa fever dynamically and numerically. The model’s positivity, boundedness, reproduction number, equilibria, and local stability are investigated in dynamical analysis. In numerical analysis, we developed some explicit and implicit methods. Unfortunately, explicit methods like Euler and Runge Kutta are time-dependent and violate the physical properties of the disease. Then, the proposed implicit method for the said model, the non-standard finite difference, is independent of the time step, dynamically consistent, positive, and bounded. In the end, a comparison of methods is presented.

Keywords: Lassa fever disease; epidemic model; computational approximations; convergence analysis

1  Literature Survey

Lassa fever is an intense hemorrhagic illness caused by the Lassa virus (member of the arenavirus). Lassa virus carries a rat, which is very common in West Africa. It is also known as a zoonosis, which means that disease spreads from animal to human. People usually become infected with Lassa fever because of food and household items’ exposure to urine of infected Mastomys rats. Its symptoms are varied and include cardiac, neurological, and pulmonary problems. In some West Africa, this disease is endemic to the rodent population. It is most common in Liberia, Sierra Leone, Guinea, and Nigeria. Lassa fever is transmitted by playing, touching, and cutting up a rat’s dead body. In 2018, Usman et al. analyzed the Lassa fever virus infection for the transmission dynamics of qualitative and analytic activities of a mathematical model [1]. In 2020, Peter et al. studied Lassa fever’s dynamics, and the solution model stayed and verified boundedness and positivity of basic properties [2]. In 2017, Olayiwola et al. analyzed in the lowliest endemic countries people with an ultimate risk of infection and need for constant investigation develops commanding in the endemic region like Africa with Nigeria at the essential attention will help in no small processes in scheming the scourge [3]. Woyessa et al. investigated the private and public health facilities, control interventions, and prevent infection when feverish patients avoid nosocomial infections [4]. In 2013, Ajayi et al. have studied that Epidemic contained rejoinder schemes and testing to control exertions comprised fright between health staffs, insufficient/poor quality of defensive things, insufficient extra preparation, and poor local laboratory capability [5]. In 2019, Ilori et al. analyzed the activity supporting improving planned for patient care and LLC, emerging infectious diseases, and Medscape through applied epidemiological characteristics and clarifying factors associated with mortality [6]. Adewuyi et al. studied the disease have the probable of actuality organize an infectious menace that essential be controlled by way of biological weapon and currently, vaccine of Lassa fever no available and some natural problems occurs for development of vaccines so prevention the way by control the rodent [7]. Iroezindu et al. analyzed Lassa fever spread; challenges, letdown to use proper defensive tackle, stigmatization of associates, and absence of a purpose-built isolation facility [8]. In 2019, Amodu et al. have studied Lassa fever as an acute disease of scarceness, high endemicity by way of cooperated environmentally-friendly sanitation, and relics susceptible populations to community health problems in Nigeria. Public meeting defense for attractive prevention strategy remains particular sanitation [9]. In 2019, Kangbai et al. highlighted that seasonal epidemics use an effective treatment to make a stratagem, which can control procedures of Lassa fever prevention and control the connection between humans to rodents [10]. Makinde et al. investigated Lassa fever to identify when a nonconformity toward the prestige quo has happened [11]. In 2020, Zhao et al. analyzed quantify of this impact in Nigeria’s presence of Lassa fever significance measure the connotation among local precipitation and infection reproduction number which facts has probable elect applied as a bad sign for Lassa fever epidemics [12]. In 2017, Obabiyi et al. developed a mathematical model for transmission of Lassa fever dynamics with the behavior of susceptible humans, recovered humans divided the population into two parts such as human populations, and rodent population by using the positivity, bounded theorem, and suggested the stability hygiene of environment [13]. In 2018, Akpede et al. studied the necessities for achievement and enduring capacity of the control exertions in Nigeria and the sub-region. In wholly these, the Nigerian administration with NCDC necessity carries a huge responsibility for the organization, supply deployment, and support. If necessary, even persuade sub-regional administrations addicted to action. In addition, there should be expected through determined action [14]. In 2019, Mazzola et al. explored the Diagnostics necessary for acknowledging and controlling epidemics of LASV, unique prevailing and genetically various mediators of VHF, that use scenarios with different performance requirements for text complexity, sensitivity, specificity, and development time [15]. In 2019, Nwafor et al. examined the Lassa fever outbreak in Nigeria; the Health maintenance workers necessity, take a high index of doubt of the infection and follow IPC measures even though provided that maintenance for all patients. Explaining health maintenance workers by the new strategies mentioned above is also significant to reduce the menace of nosocomial transmission of Lassa fever [16]. In 2018, Shehu et al. studied that the Occurrence of rural to urban change of clinical and epidemiological reduced the Lassa fever cases during 2016 for morbidity and mortality [17]. In 2007, Ogbu et al. discussed the situation of Lassa fever in the sub-region of West Africa and suggested strategies for socioeconomic behavior that control the shortage of health care system [18]. In 2020, Tewogbola et al. analyzed the overview and discussed the main reasons it damaged the human population and recommended the control measure of Lassa fever [19]. In 2014, Ajayi et al. reported a case of 59 years that recovers without taking a vaccine such as ribavirin. The symptoms of this disease increase day by day because few people do not use the main precautions to control Lassa fever [20]. Some well-known numerical models related to diseases are studied [21–26].

2  Formulation of Lassa Fever Model

The variables and parameters are described of the lassa fever model as follows: SH(t) : denoted as the susceptible class at any time t, IH(t) : characterized as the infectious class at any time t, RH(t) : characterized as the recovered class at any time t, SR(t) : characterized as the susceptible rodent vectors at any time t, IR(t) : characterized as the infectious rodent vectors at any time t, NH(t) : characterized as whole humans’ population at any time t, m = NRNH : characterized as the number of infectious rodent vectors by the human host, α1 : described as the rate at which contagious rodent vectors and a susceptible class of humans interact with each other, α2 : defined as the force of infection, α3 : defined as the rate at which sensitive rodent vectors and an infectious class of humans interact with each other, τc : denoted the speed at which infectious human hosts comply with the drug, τnc : indicated the rate at which infectious human hosts do not comply with the drug, rc : denoted the rate at which infectious human hosts are educated to adhere to the medication, δ: indicated the rate of mortality of an infectious class, γ : indicated the rate at which humans may lose their immunity. The leading equations of the model are as follows:

dSH(t)dt=ΛH−α1α2SH(t)IR(t)NH+γRH(t)+τncIH(t)−μHSH(t),t≥0. (1)

dIH(t)dt=α1α2SH(t)IR(t)NH−τcIH(t)−rcIH(t)−τncIH(t)−δIH(t)−μHIH(t),t≥0. (2)

dRH(t)dt=τcIH(t)+rcIH(t)−γRH(t)−μHRH(t),t≥0. (3)

dSR(t)dt=ΛR−α1α3SR(t)IH(t)NH−μRSR(t),t≥0. (4)

dIR(t)dt=α1α3SR(t)IH(t)NH−μRIR(t),t≥0. (5)

where SH(0)≥0,IH(0)≥0,RH(0)≥0,SR(0)≥0,IR(0)≥0 .

2.1 Fundamental Properties of Model

We studied the feasible region, positivity, and boundedness of the model at any time, t ≥ 0, as follows:

Π={(SH,IH,RH,SR,IR)εR+5:SH+IH+RH≤ΛHμH,SR+IR≤ΛRμR,SH≥0,IH≥0,RH≥0,SR≥0,IR≥0} .

Lemma 1: The solutions (SH,IH,RH,SR,IR)εR+5 of Eqs. (1)–(5) is positive at any time t ≥ 0, with given non-negative initial conditions.

Proof: It is clear from Eqs. (1)–(5),

dSHdt|SH=0=ΛH+γRH+τncIH≥0 , dIHdt|IH=0=α1α2SHIRNH≥0 , dRHdt|RH=0=τcIH+rcIH≥0 , dSRdt|SR=0=ΛR≥0 , dIRdt|IR=0=α1α3SRIHNH≥0, as desired.

Lemma 2: Forgiven any non-negative initial conditions for the solution of the system (1)–(5) is bounded if limt→∞⁡SupNH(t)≤ΛHμH,limt→∞⁡SupNR(t)≤ΛRμR.

Proof: let us consider the population function as follows:

NH(t)=SH+IH+RH,dNHdt=dSHdt+dIHdt+dRHdt,dNHdt=ΛH−μHNH,

NH(t)=A+ΛHμH

By the Gronwall’s inequality, we get

NH(t)≤NH(0)+ΛHμH,t≥0,limt→∞⁡SupNR(t)≤ΛHμH

NR(t)=SR+IR,dNRdt=dSRdt+dIRdt,dNRdt=ΛR−μRNR

NR(t)=B+ΛRμR

By the Gronwall’s inequality, we get

NR(t)≤NR(0)+ΛRμR,t≥0

limt→∞⁡SupNR(t)≤ΛRμR , as desired.

2.2 Steady States of Lassa fever Model

There are two steady states of Eqs. (1) to (5), as follows: disease-free equilibrium (DFE)=(SH,IH,RH,SR.IR)=(ΛHμH,0,0,ΛRμR,0) and endemic equilibrium (EE)=(SH∗,IH∗,RH∗,SR∗,IR∗) ,

RH∗=(τc+rc)IH∗γ+μH=A1IH∗,A1=(τc+rc)γ+μH,SH∗=ΛH+γA1IH∗−A2IH∗μH,A2=τc+rc+δ+μH,IR∗=α1α3SR∗IH∗μR,

SR∗=ΛRα1α3IH∗+μR,IH∗=ΛH−A4μRA4α1α3−γA1+A2,A3=τc+rc+τnc+δ+μH,A4=A3μHμRα12α2α3ΛR.

3  Reproduction Number of Lassa Fever Model

In this section, we shall find the two types of matrices like transmission and transition by assuming the disease-free equilibria in the system (1)–(5) by using the next-generation matrix method, furthermore, considering the infected classes as follows:

F=[α1α2ΛHμHNH0000000α1α2ΛRμRNH], V−1=[μR(γ+μH)00μR(τc+rc)μR(τc+rc+τnc+δ+μH)000(τc+rc+τnc+δ+μH)(γ+μH)](τc+rc+τnc+δ+μH)(γ+μH)μR .

FV−1=1(τc+rc+τnc+δ+μH)(γ+μH)μR[α1α2ΛHμHNH(μR)(γ+μH)0000000α1α2ΛRμRNH(τc+rc+τnc+δ+μH)(γ+μH)].

Thus, the dominant eigenvalue of the matrix is called reproduction number and denoted as follows:

R0=α1α2ΛHμHNH(τc+rc+τnc+δ+μH) .

4  Local Stability

In this section, we present two well-known theorems for stability in the sense of local. Again, consider the system (1)–(5) as function of A,B,C,DandE as follows:

A=ΛH−α1α2SHIRNH+γRH+τncIH−μHSH. (6)

B=α1α2SHIRNH−τcIH−rcIH−τncIH−δIH−μHIH. (7)

C=τcIH+rcIH−γRH−μHRH. (8)

D=ΛR−α1α3SRIHNH−μRSR. (9)

E=α1α3SRIHNH−μRIR. (10)

Theorem 1:The disease-free equilibrium is locally asymptotically stable for the system (6)–(10). If R0<1 and otherwise unstable if R0>1 .

Proof:First, we take the partial derivates of the system (6)–(10) concerning state variables as follows:

∂A∂SH=−α1α2IRNH−μH,∂A∂IH=τnc,∂A∂RH=γ,∂A∂SR=0,∂A∂IR=−α1α2SHNH .

∂B∂SH=α1α2IRNH,∂B∂IH=−τc−rc−τnc−δ−μH,∂B∂RH=0,∂B∂SR=0.

∂B∂IR=α1α2SHNH,∂C∂SH=0∂C∂IH=τc+rc,∂C∂RH=−γ−μH,∂C∂SR=0,

∂C∂IR=0,∂D∂SH=0,∂D∂IH=−α1α3SRNH,∂D∂RH=0,∂D∂SR=−α1α3IHNH−μR,∂D∂IR=0,∂F∂SH=0,

∂F∂IH=α1α3SRNH,∂F∂RH=0,∂F∂SR=α1α3IHNH,∂F∂IR=−μR.

Here, the Jacobian matrix as follows:

J=[−α1α2IRNH−μHτncγ0−α1α2SHNHα1α2IRNH−τc−rc−τnc−δ−μH00α1α2SHNH0τc+rc−γ−μH000−α1α3SRNH0−α1α3IHNH−μR00α1α3SRNH0α1α3IHNH−μR]

The Jacobian matrix at the disease-free equilibria of the system (6)–(10) as follows:

J(E0)=J(ΛHμH,0,0,ΛRμR,0)=[−μHτncγ0−α1α2ΛHμHNH0−τc−rc−τnc−δ−μH00α1α2ΛHμHNH0τc+rc−γ−μH000−α1α3ΛRμRNH00−μR00α1α3ΛRμRNH00−μR] .

|J−λI|=|−μH−λτncγ0−α1α2ΛHμHNH0(−τc−rc−τnc−δ−μH)−λ00α1α2ΛHμHNH0τc+rc−γ−μH−λ000−α1α3ΛRμRNH00−μR−λ00α1α3ΛRμRNH00−μR−λ|=0.

λ1=−μH<0,λ2=−μR<0,λ3=−(γ+μH)<0.

|J−λI|=|(−τc−rc−τnc−δ−μH)−λα1α2ΛHμHNHα1α3ΛRμRNH−μR−λ|=0 .

λ2+λ(a1+μR)+a1μR−a2=0 .

where, a1=τc+rc+τnc+δ+μH,a2=(α1α2ΛHμHNH)(α1α3ΛRμRNH)

Since all the coefficients of the polynomial are positive, therefore, by using Routh Hurwitz Criteria for 2nd order, the disease-free equilibria are locally asymptotically stable.

Theorem 2:The endemic equilibrium is locally asymptotically stable for the system (6)–(10) if R0>1 .

Proof: The Jacobian matrix at the endemic equilibria of the system (6)–(10) is as follows:

J(E∗) = J(SH∗,IH∗,RH∗,SR∗,IR∗)=[−α1α2IR∗NH−μHτncγ0−α1α2SH∗NHα1α2IR∗NH−τc−rc−τnc−δ−μH00α1α2SH∗NH0τc+rc−γ−μH000−α1α3SR∗NH0−α1α3IH∗NH−μR00α1α3SR∗NH0α1α3IH∗NH−μR] .

|J−λI|=|−B1−μH−λτncγ0−B2B1−B3−λ00B20B4−B5−λ000−B60−B7−μR−λ00B60B7−μR−λ|=0 .

where, B1=α1α2IR∗NH,B2=α1α2SH∗NH,B3=τc+rc+τnc+δ+μH,B4=τc+rc,B5=γ+μH,B6=α1α3SR∗NH,B7=α1α3IH∗NH.

|J−λI|=(−B1−μH−λ)|−B3−λ00B2B4−B5−λ00−B60−B7−μR−λ0B60B7−μR−λ|−B1|τncγ0−B2B4−B5−λ00−B60−B7−μR−λ0B60B7−μR−λ|=0 .

|J−λI|=(−B1−μH−λ)(B5+λ)[(−B3−λ)|−B7−μR−λ0B7−μR−λ|+B2|−B6−B7−μR−λB6B7|]+(−B1)(−γ)[(B4)|−B6−B7−μR−λB6B7|]+(−B1)(B5+λ)[τnc|−B7−μR−λ0B7μR−λ|−B2|−B6−B7−μR−λB6B7|]=0

λ5+λ4[2μR+B3+μH+B7+B5+B1]+λ3[B7μR+μR2+2B3μR+2B5μR+2B1μR+B3B7+B5B7+B1B7+B3B5+B1B3−B2B6+B5μH+B3μH+B7μH+2μRμH+B1B5−B1τnc]+λ2[B3B7μR+B5B7μR+B1B7μR+B3μR2+B5μR2+B1μR2+2B3B5μR+2B1B3μR+2B1B5μR+B3B5B7+B1B3B7+B1B5B7+B1B3B5+B1B4γ−2B1μRτnc−B1B7τnc−B1B5τnc−B1B2B6+B3B7μH+B7B5μH−2B2B6μH+2B3μRμH+2B5μRμH+B7μRμH−B2B5B6]+λ[B3B5B7μR+B1B3B7μR+B1B5B7μR+B3B5μR2+B5μR2+B1B5μR2+2B1B3B5μR−B2B6B5μR−2B1B2B5B6+2B1B4γμR−B1B2B6γ−2B1B5τncμR−B1B5B7τnc+B7B3B5μH+2B3B5μRμH−2B1B2B7B6−B1B2B6γ−2B1B5τncμR−B1B5B7τnc+B7B3B5μH+2B3B5μRμH−2B1B2B7B6−2B1B2B6μR+B5B3B7μH+B7B5μRμH+B3B7μRμH−B2B6μRμH−B2B6B5μH−B1B2B6]+B1B3B5B7μR+B1B3B5μR2+B1B4B7γμR+B1B4γμR2−B1B5B7τncμR+B1B2B6γμR−B1B5τncμR2+B1B3B7B5−2B1B2B5B6μR−2B1B2B5B7B6−B2B5B6μRμH+B5B3B7μRμH+B3B5μR2μH=0.

λ5a1+λ4a2+λ3a3+λ2a4+λa5+a6=0 .

where, a1=[1],a2=−[2μR+B3+μH+B7+B5+B1].

a3=−[B7μR+μR2+2B3μR+2B5μR+2B1μR+B3B7+B5B7+B1B7+B3B5+B1B3−B2B6+B5μH+B3μH+B7μH+2μRμH+B1B5−B1τnc]

a4=−[B3B7μR+B5B7μR+B1B7μR+B3μR2+B5μR2+B1μR2+2B3B5μR+2B1B3μR+2B1B5μR+B3B5B7+B1B3B7+B1B5B7+B1B3B5+B1B4γ−2B1μRτnc−B1B7τnc−B1B5τnc−B1B2B6+B3B7μH+B7B5μH−2B2B6μH+2B3μRμH+2B5μRμH+B7μRμH−B2B5B6].

a5=−[B3B5B7μR+B1B3B7μR+B1B5B7μR+B3B5μR2+B5μR2+B1B5μR2+2B1B3B5μR−B2B6B5μR+2B1B2B5B6μR+2B1B4γμR−B1B2B6γ−2B1B5τncμR−B1B5B7τnc+B7B3B5μH+2B3B5μRμH−2B1B2B7B6−2B1B2B6μR+B5B3B7μH+B7B5μRμH+B3B7μRμH−B2B6μRμH−B2B6B5μH−B1B2B6]

a6=−[B1B3B5B7μR+B1B3B5μR2+B1B4B7γμR+B1B4γμR2−B1B5B7τncμR+B1B2B6γμR−B1B5τncμR2+B1B3B7B5−2B1B2B5B6μR−2B1B2B5B7B6−B2B5B6μRμh+B5B3B7μRμH+B3B5μR2μH]

The Routh Hurwitz criteria of the 5th order are satisfied. Hence, the endemic equilibria are locally asymptomatically stable.

5  Computational Approximations

In this section, we present the well-known approximations like Euler, Runge Kutta, and non-standard finite difference for the system (1)–(5) as follows:

5.1 Euler Approximation

The discretization of the system (1)–(5) under the rules of the Euler approximation is as follows:

SHn+1=SHn+h[ΛH−α1α2sHnIRnNH+γRHn+τncIHn−μHSHn]. (11)

IHn+1=IHn+h[α1α2SHnIRnNH−τcIHn−rcIHn−τncIHn−δIHn−μHIHn]. (12)