Intelligent Automation & Soft Computing DOI:10.32604/iasc.2022.022643
Article

Modeling of Anthrax Disease via Efficient Computing Techniques

Ali Raza1,2, Dumitru Baleanu3,4, Muhammad Yousaf2, Naeem Akhter2, Syed Kashif Mahmood2 and Muhammad Rafiq5,*

1Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College Wazirabad, 52000, Punjab Higher Education Department (PHED), Lahore, 54000, Pakistan
2Department of Mathematics, National College of Business Administration and Economics, Lahore, 54660, Pakistan
3Department of Mathematics, Cankaya University, Ankara, 06530, Turkey
4Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
5Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, 54000, Pakistan
*Corresponding Author: Muhammad Rafiq. Email: m.rafiq@ucp.edu.pk
Received: 13 August 2021; Accepted: 14 September 2021

Abstract: Computer methods have a significant role in the scientific literature. Nowadays, development in computational methods for solving highly complex and nonlinear systems is a hot issue in different disciplines like engineering, physics, biology, and many more. Anthrax is primarily a zoonotic disease in herbivores caused by a bacterium called Bacillus anthracis. Humans generally acquire the disease directly or indirectly from infected animals, or through occupational exposure to infected or contaminated animal products. The outbreak of human anthrax is reported in the Eastern Mediterranean regions like Pakistan, Iran, Iraq, Afghanistan, Morocco, and Sudan. Almost ninety-five percent chances are the transmission of the bacteria from forming spores by the World Health Organization (WHO). The modeling of an anthrax disease is based on the four compartments along with two humans (susceptible and infected) and others are dead bodies and sporing agents. The mathematical analysis is studied along with the fundamental properties of deterministic modeling. The stability of the model along with equilibria is studied rigorously. The authentication of analytical results is examined through well-known computer methods like Euler, Runge Kutta, and Non-standard finite difference (NSFD) along with the feasible properties (positivity, boundedness, and dynamical consistency) of the model. In the end, comparison analysis of algorithms shows the effectiveness of the methods.

Keywords: Anthrax disease; deterministic modeling; stability analysis; computer methods

1  Introduction

The major cause of many diseases in the world is viruses and bacteria. Anthrax is a bacterial disease. Anthrax is a disease of herbivores such as sheep, goats, horses, cows and transmitted to humans. Infected Animals are the source to transmit the bacteria in humans. It does not spread from animal to animal or man to man like covid 19. Anthrax spores can enter the human body through via breathing and cuts on the skin. Its incubation period varies from one to several days. Insects are also the source of the transmission of anthrax. Bacteria are prokaryotes that contain a cell wall, capsule, ribosomes, pili, flagella, and DNA. This species of a bacterium can survive in a harsh environment. Bacteria are considered the eldest living organisms on the earth. The Greek word anthrax means coal. Egypt is the region of anthrax disease. Pieces of evidence show that many scholars of Greece and Rome are well known about anthrax. Anthrax disease is the cause of the downfall of Rome. Those animals which become the victims of anthrax do not show any proper symptoms. They die after infection without leaving diagnostics symptoms. Three forms of anthrax are cutaneous, inhalation, and gastrointestinal. Cutaneous anthrax is about 95% in society. It is a very less dangerous form of anthrax. This form of anthrax usually shows its symptoms for 1 to 7 days. A person can get cutaneous anthrax through cuts on the skin. The persons who are working in tanning and woolen industries can easily get cutaneous anthrax. This form of anthrax is most common in the hands, foot, and neck regions. The common symptoms of cutaneous anthrax are small blisters appear on the skin, looks like an insect bite, swelling in tissues around the sore, a black-centered painless sore appears on the skin such as on the face, hands, and neck regions. We can easily survive by using proper antibiotics. Inhalation anthrax is the most dangerous form of anthrax. It usually shows its symptoms within a week and can become chronic. Its symptoms are fever, feeling cold disorder in the chest, shortness of breath, coughing, vomiting, headache, pain in the stomach, and fatigue. No evidence in the world shows its propagation with infected milk. There is a disorder in the elementary canal on ingesting. sixty percent of patients can easily survive on the proper treatment of gastrointestinal anthrax. The death ratio of humans via anthrax is 20% (cutaneous), 75% (gastrointestinal), and 80% (lung infection). This disease is common in developing countries like the Sahara Desert in Africa, central and southwestern Asia (Turkey, Labnan, Syria, Iran, Egypt), and the Caribbean (Belarus Hungry, Romania, Slovakia). Dozens of cows are died due to anthrax in Pakistan. Mackey et al. studied a model in which describes the transmission of disease in zebras [1]. Baloba et al. investigated the control measures or mechanisms to overcome anthrax [2]. Stella et al. suggested the SIR model for anthrax disease in the animal community [3]. Rezapour et al. proposed a mathematical model for explaining the spread of anthrax with the help of the Caputo-Fabrizio technique [4]. Croicu suggested the IACN model of anthrax disease in herbivores [5]. Gomez et al. suggested the SIML model for transmission of anthrax [6]. Mushayabasa et al. proposed the SEIPC model with the well-known assumptions of mathematics to dispose of carcasses [7]. Roy et al. suggested the SIAC model for the spread of anthrax in animal populations [8]. Mushayabasa suggested the model and its reproduction number with the help of the Lyapunov method [9]. Wanying et al. suggested different factors for controlling the death rate during the bioterrorist attack of anthrax [10]. Pantha et al. studied optimal control of a mathematical model for the spread of anthrax [11]. Gutting studied the inhalation of anthrax in rabbits [12]. Toth et al. estimate the incubation period for the treatment of inhalation anthrax in humans [13]. Day et al. present a model for control of inhalation anthrax [14]. Wilkening proposed an incubation period and describe the importance of the incubation period for controlling inhalation anthrax [15]. Li et al. studied a dynamical mathematical model for the analysis of anthrax disease [16]. Pittman described the importance of the anthrax vaccine in humans [17]. Furniss et al. suggested a model for the control of anthrax in Kruger national park [18]. Pauline described the significance of modeling diseases [19]. Helikumi et al. described the importance of reproduction numbers in modeling various diseases like anthrax [20]. Webb proposed the risk of anthrax in biowarfare [21]. Afshar et al. discussed the detail of a clinical case of anthrax in Iran [22]. Hashemi et al. argued the gastrointestinal patient of anthrax in northeast Iran [23]. Osman et al. developed a model for Listerriosis and anthrax [24]. Brookmeyer et al. suggested the incubation period of anthrax, and they also discussed the incubation period depends on the age and amount of spore that enters in lungs [25]. Karginov et al. discussed that ciprofloxacin with antibodies and the growth of bacteria should be efficient for curing anthrax [26]. Loving et al. described that mice species were more efficient for the study of anthrax [27]. Radosavljevic et al. suggested the curing techniques for the bioterrorist attack of anthrax [28]. Mathematical techniques were studied to analyze the transmission of infectious diseases [29–42]. In this paper, we study the dynamics of anthrax disease via computational methods. We can observe that computational methods in literature have many problems like negativity, unboundedness, and inconsistency of solutions. These issues will resolve by our proposed idea that is a non-standard finite difference method (NSFD). Also, NSFD fulfills the properties of the biological problem. The rest of the paper is styled as follows: In Section 2 the modeling of anthrax is defined. In Section 3 the construction way of the anthrax epidemic model, equilibrium points and computer methods and their convergence are explained. In the last section conclusion and future problems are discussed.

2  Modelling of Anthrax

By using the theory of population dynamics of epidemiological type, susceptible and infected is the sum of the total population. The evolution process of the bacteria describes the effect of the epidemiological parameters based on the system of nonlinear coupled ordinary differential equations. Thus, a continuous model for populations regarding anthrax is described in Fig. 1.

Figure 1: Transmission map of anthrax

The vertical transmission of humans, birth and death rates of humans are considered approximately equal. The physical relationship of variables and constants are shown in Tab. 1 as follows:

The system of differential equations can be deriving from the above flow chart of population as follows:

dsdt=Λ−ηaas−ηccs−ηiis−μs+τi,t≥0 (1)

didt=ηaas+ηccs+ηiis−(γ+μ+τ)i,t≥0 (2)

dadt=−αa+βc,t≥0 (3)

dcdt=(γ+μ)i−δc−kc,t≥0 (4)

The system (1-4) is based on the feasible region of the model as follows:

Z={(s,i,a,c)ϵR+4:s≥0,i≥0,a≥0,c≥0:s+i≤Λμ,c≤μ(γ+μ)δΛ+μk,a≤μ(γ+μ)αΛ+μk} .

Lemma 1: The solutions (s,i,a,c)ϵR+4 of the system (1-4) are positive at any time t ≥0 , if the rate of change of state variable is non-negative at the trivial stage.

Proof. In this section, we prove the positivity of the first two compartments due to human populations. Forgiven an initial condition s≥0,i≥0,

From Eq. (1),

s′=Λ−ηaas−ηccs−ηiis−μs+τi,s′≥(−ηaa−ηcc−ηii−μ)s .

From Eq. (2),

lni≥[ηis−(γ+μ+τ)i]t+c2 ,

i(t)=i(0)×e[ηis−(γ+μ+τ)i]t≥0 .

Hence, the system (1-4) admits a positive solution.

Lemma 2: The solutions (s,i,a,c)ϵR+4 of the system (1-4) are bounded at any time t ≥0 , and limt→∞⁡SupN(t)≤Λμ .

Proof. Consider the population function as follows:

N=s+i .

dNdt≤Λ−μ(s+i) .

dNdt+μN≤Λ .

limt→∞⁡SupN(t)≤Λμ .

Hence, the system (1-4) admits bounded solution and lie in the feasible region Z.

2.1 Model Equilibria

Considering the system (1-4) in which state variables treated as constant and the change in variables are assumed to be zero as follows:

0=Λ−ηaas−ηccs−ηiis−μs+τi, (5)

0=ηaas+ηccs+ηiis−(γ+μ+τ)i (6)

0=−αa+βc (7)

0=(γ+μ)i−δc−kc (8)

The system (5-8), admitted two types of equilibria that are anthrax free equilibrium (AFE) and anthrax existing equilibrium (AEE). The anthrax existing equilibrium (AEE) is denoted by A1=(s∗,i∗,a∗,c∗)

s∗=γ+μ+τA∗+B∗+ηi,i∗=Λ−μs∗A∗s∗+B∗s∗+ηis∗−τ,a∗=βc∗α,c∗=(γ+μ)i∗δ+k .

where A∗=ηa(βα)(γ+μδ+k),B∗=ηc(γ+μδ+k) .

And the anthrax free equilibrium (AFE) is denoted by A2=(s1,i1,a1,c1)=(Λμ,0,0,0) .

2.2 Reproduction Number

The reproduction number has a significant role in disease dynamics. It presented the ratio of infectivity in the population. For the sake, in the system of anthrax disease, there is only a single compartment is related to infected ness. Without loss of generality, we consider

By assuming the anthrax free equilibrium A2=(s1,i1,a1,c1)=(Λμ,0,0,0) as follows:

Ληiμ>(γ+μ+τ),ηiΛμ(γ+μ+τ)>1 .

R0=ηiΛμ(γ+μ+τ)>1 .

Note that, R0 is called the reproduction number of the model.

2.3 Local Stability

In this section, we discussed two well-known theorems for stability. Considering the system (5-8) as functions of the equations.

The partial derivatives along with the states variables as follows:

where ∂F4∂s=0 , ∂F3∂s=0 , ∂F2∂s=ηaa+ηcc+ηii , ∂F1∂s=−ηaa−ηcc−ηii−μ , ∂F1∂i=−ηis+τ , ∂F2∂i=ηiis−(γ+μ+τ) , ∂F3∂i=0 , ∂F4∂i=(γ+μ) , ∂F1∂a=−ηas , ∂F2∂a=ηas , ∂F3∂a=−α , ∂F4∂a=0 , ∂F1∂c=−ηcs , ∂F2∂c=ηcs , ∂F3∂c=β , ∂F4∂c=−δ−k .

The general form of the Jacobin matrix is defined as

J=[−ηaa−ηcc−ηii−μ−ηis+τ−ηas−ηcsηaa+ηcc+ηiiηiis−(γ+μ+τ)ηasηcs00−αβ0(γ+μ)0−δ−k] (9)

Theorem 1: If R0<1 , then A2=(s1,i1,a1,c1)=(Λμ,0,0,0) is locally asymptotically stable in the system (1-4). Otherwise, unstable.

Proof. The Jacobean matrix of Eq. (9) at A2 is as follows:

|J(A2)−λI|=|−μ−λ−ηi(Λμ)+τ−ηa(Λμ)−ηc(Λμ)0ηi(Λμ)−(γ+μ+τ)−ληa(Λμ)ηc(Λμ)00−α−λβ0(γ+μ)0−δ−K−λ|=0 .

(−μ−λ)|ηi(Λμ)−(γ+μ+τ)−ληa(Λμ)ηc(Λμ)0−α−λβγ+μ0−δ−k−λ|=0 .

λ1=−μ<0 ,

(A−λ)(B+Cλ+λ2)+D+Eλ=0 .

where A=ηiΛμ−(γ+μ+ı`) , B=αδ+αk, C=δ+k , D=(γ+μ)(ηΛμβ+αηcΛμ),E=(γ+μ)ηcΛμ.

λ3−(A−C)λ2−(AC−B+E)λ−(AB+D)=0 .

λ3+a2λ2+a1λ+a0=0 .

Where a2=A−C, a1=E−AC+B , a0=−(AB+D) .

By using the Routh Hurwitz criterion of 3rd order, the following constraint is satisfied a2,a0>0, and a2a1>a0 when R0<1 . Hence the anthrax-free equilibrium A2 is locally asymptotically stable (LAS).

Theorem 2: If R0>1 , then A1=(s∗,i∗,a∗,c∗) is locally asymptotically stable in the system (1-4). Otherwise, unstable.

Proof: The Jacobean matrix Eq. (9) at A1 is as follows:

|J(A1)−λI|=[−ηaa∗−ηcc∗−ηii∗−μ−λ−ηis∗+τ−ηas∗−ηcs∗ηaa∗+ηcc∗+ηii∗ηis∗−(γ+μ+τ)−ληas∗ηcs∗00−α−λβ0(γ+μ)0−δ−k−λ]=0 .

where g=−ηaa∗−ηcc∗−ηii∗−μ , g1=ηis∗−(γ+μ+τ) , g2=ηas∗ , g3=ηcs∗ , g4=r+μ , g5=−δ−k , h=ηaa∗+ηcc∗+ηii∗ , h1=−ηis∗+τ , h2=ηas∗ , h3=−ηcs∗ , h4=r+μ , h5= −δ−k .

where G1=αg3g4−αg1g5+βg2g4 , G2=αg1+αg5+g3g4−g1g5 , G3=−α+g1+g5 , H1=αh3h4−αh1h5+βh2h4 , H2=αh1+h3h4−h1h5 ,

gG1+gG2λ+gG3λ2−gλ3−G1λ−G2λ2−G3λ3+λ4−hH1−hH2λ−hh1λ2=0 .

λ4+(−g−G3)λ3+(gG3−G2−hh1)λ2+(gG2−G1−hH2)λ+(gG1−hH1)=0 .

λ4+b0λ3+b1λ2+b2λ+b3=0 .

where b0=(−g−G3) , b1=gG3−G2−hh1 , b2=gG2−G1−hH2 , b3=gG1−hH1 ,

By using Routh Hurwitz criteria of order 4th, all constraints have been satisfied. So, anthrax existing equilibrium A1 is locally asymptotically stable (LAS).

3  Computer Methods

In this section, we discussed some well-known computer methods like Euler, Runge Kutta, and the non-standard finite difference method for the system (1-4).

3.1 Euler Method

This method could be applied in the system (1-4) as follows:

sn+1=sn+h[Λ−ηaansn−ηccnsn−ηiinsn−μsn+τin]. (10)

in+1=in+h[ηaansn+ηccnsn+ηiinsn−(γ+μ+τ)in]. (11)

an+1=an+h[−αan+βcn]. (12)

cn+1=cn+h[(γ+μ)in−(δ+K)cn]. (13)

where n = 0,1,2,3… and discretization gap is denoted by h.

3.2 Runge Kutta Method

This method could be applied in the system (1-4) as follows:

Stage 1

K1=h[Λ−ηaansn−ηccnsn−ηiinsn−μsn+τin] .

L1=h[ηaansn+ηccnsn+ηiinsn−(γ+μ+τ)in] .

M1=h[−αan+βcn] .

N1=h[(γ+μ)in−(δ+K)cn] .

Stage 2

K2=h[Λ−ηa(an+M12)(sn+K12)−ηc(cn+N12)(sn+K12)−ηi(in+L12)(sn+K12)−μ(sn+K12)+τ(in+L12)] .

L2=h[ηa(an+M12)(sn+K12)+ηc(cn+N12)(sn+K12)+ηi(in+L12)(sn+K12)−(γ+μ+τ)(in+L12)] .

M2=h[−α(an+M12)+β(cn+N12)] .

N2=h[(γ+μ)(in+L12)−(δ+K)(cn+N12)] .

Stage 3

K3=h[Λ−ηa(an+M22)(sn+K22)−ηc(cn+N22)(sn+K22)−ηi(in+L22)(sn+K22)−μ(sn+K22)+τ(in+L22)] .

L3=h[ηa(an+M22)(sn+K22)+ηc(cn+N22)(sn+K22)+ηi(in+L22)(sn+K22)−(γ+μ+τ)(in+L22)] .

M3=h[−α(an+M22)+β(cn+N22)] .

N3=h[(γ+μ)(in+L22)−(δ+K)(cn+N22)] .

Stage 4

K4=h[Λ−ηa(an+M1)(sn+K1)−ηc(cn+N1)(sn+K1)−ηi((in+L1)(sn+K1)+K1−μ(sn+K1) +τ(cn+N1)] .

L2=h[ηa(an+M1)(sn+K1)+ηc(cn+N1)(sn+K1)+ηi(in+L1)(sn+K1)−(γ+μ+τ)(in+L1)] .

M2=h[−α(an+M1)+β(cn+N1)] .

N2=h[(γ+μ)(in+L1)−(δ+K)(cn+N1)] .

Final stage

sn+1=sn+16[K1+2k2+2k3+k4]in+1=in+16[L1+2L2+2L3+L4]an+1=an+16[M1+2M2+2M3+M4]cn+1=cn+16[N1+2N2+2N3+N4]} (14)

where h is any discretization and n≥0 .

3.3 Non-Standard Finite Difference Method

This method could be applied in the system (1-4) as follows:

sn+1=sn+h(Λ+τin)1+hηaan+ηchcn+hηiin+hμ. (15)

in+1=in+hηasnan+ηcsnhcn+hηiinsn+hμ1+h(γ+μ+τ). (16)

an+1=an+hβcn1+αh. (17)

cn+1=cn+h(γ+μ)in1+h(δ+k). (18)

where h is any discretization and n≥0 .

3.4 Convergence Analysis

Theorem 3: For any n≥ 0, the proposed NSFD method is stable if the Eigenvalues of the system lie in the unit circle if R0 < 1.

Proof. Consider the right-hand sides of the system (15-18), as function W, X, Y, and Z as follows:

W=s+h(Λ+τi)1+hηaa+ηchc+hηii+hμ , X=i+hηasa+ηcshc+hηiis+hμ1+h(γ+μ+τ) , Y=a+hβc1+αh , Z=c+h(γ+μ)i1+h(δ+k) .

The partial derivatives with respect to the state variables as follows:

where ∂W∂s=11+hμ , ∂W∂i=hτ1+hμ−hηi(Λμ+hΛ)(1+hμ)2 , ∂W∂a=−hηa(Λμ+hΛ)(1+hμ)2 , ∂W∂c=−hηc(Λμ+hΛ)(1+hμ)2 , ∂X∂s=0 , ∂X∂i=1+hηi(Λμ)1+h(γ+μ+τ) , ∂X∂a=hηa(Λμ)1+h(γ+μ+τ) , ∂X∂c=hηc(Λμ)1+h(γ+μ+τ) , ∂Y∂s=0 , ∂Y∂i=0 , ∂Y∂a=11+hα , ∂Y∂c=hβ1+hα , ∂Z∂s=0 , ∂Z∂i=h(γ+μ)1+h(δ+k) , ∂Z∂a=0 , ∂Z∂c=11+h(δ+k) .

The Jacobean matrix at anthrax free equilibrium is as follows:

J(Λμ,0,0,0)=[11+hμ000 hτ1+hμ−hηi(Λμ+hΛ)(1+hμ)21+hηi(Λμ)1+h(γ+μ+τ)oh(γ+μ)1+h(δ+k) −hηa(Λμ+hΛ)(1+hμ)2hηa(Λμ)1+h(γ+μ+τ)11+hα0 −hηc(Λμ+hΛ)(1+hμ)2hηc(Λμ)1+h(γ+μ+τ)hβ1+hα11+h(δ+k) ] .

|J−λI|=|11+hμ−λ000 hτ1+hμ−hηi(Λμ+hΛ)(1+hμ)21+hηi(Λμ)1+h(γ+μ+τ)−λoh(γ+μ)1+h(δ+k) −hηa(Λμ+hΛ)(1+hμ)2hηa(Λμ)1+h(γ+μ+τ)11+hα−λ0 −hηc(Λμ+hΛ)(1+hμ)2hηc(Λμ)1+h(γ+μ+τ)hβ1+hα11+h(δ+k)−λ |=0 .