Modeling of Anthrax Disease via Efficient Computing Techniques

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.


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 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.

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.
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:  Growth rate of spore rate / day / carcasses. The system of differential equations can be deriving from the above flow chart of population as follows: The system (1-4) is based on the feasible region of the model as follows: 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 0 ¼ Ã À g a as À g c cs À g i is À ls þ si; s 0 ! Àg a a À g c c À g i i À l ð Þ s.  SupN t ð Þ Ã l . Proof. Consider the population function as follows: SupN t ð Þ Ã l . Hence, the system (1-4) admits bounded solution and lie in the feasible region Z:

Model Equilibria
Considering the system (1)(2)(3)(4) in which state variables treated as constant and the change in variables are assumed to be zero as follows: 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 And the anthrax free equilibrium (AFE) is denoted by

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 A 2 ¼ s 1 ; i 1 ; a 1 ; c 1 ð Þ¼ Ã l ; 0; 0; 0 as follows: 1. Note that, R 0 is called the reproduction number of the model.

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 @F 4 @s ¼ 0, The general form of the Jacobin matrix is defined as J ¼ @F 1 @s @F 1 @i @F 1 @a @F 1 @c @F 2 @s @F 2 @i @F 2 @a @F 2 @c @F 3 @s @F 3 @i @F 3 @a @F 3 @c @F 4 @s @F 4 @i @F 4 @a @F 4 @c Proof. The Jacobean matrix of Eq. (9) at A 2 is as follows: By using the Routh Hurwitz criterion of 3 rd order, the following constraint is satisfied a 2 ; a 0 . 0; and a 2 a 1 . a 0 when R 0 , 1. Hence the anthrax-free equilibrium A 2 is locally asymptotically stable (LAS).
Proof: The Jacobean matrix Eq. (9) at A 1 is as follows: By using Routh Hurwitz criteria of order 4 th , all constraints have been satisfied. So, anthrax existing equilibrium A 1 is locally asymptotically stable (LAS).

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

Euler Method
This method could be applied in the system (1-4) as follows: s nþ1 ¼ s n þ h Ã À g a a n s n À g c c n s n À g i i n s n À ls n þ si n ½ : (10) i nþ1 ¼ i n þ h g a a n s n þ g c c n s n þ g i i n s n À c þ l þ s ð Þ i n ½ : (11) a nþ1 ¼ a n þ h Àaa n þ bc n ½ : where n = 0,1,2,3… and discretization gap is denoted by h.

Runge Kutta Method
This method could be applied in the system (1-4) as follows: g a a n s n À g c c n s n À g i i n s n À ls n þ si n ½ .
L 1 ¼ h g a a n s n þ g c c n s n þ g i i n s n À c þ l þ s ð Þ i n ½ .
Stage 2 where h is any discretization and n ! 0.

Non-Standard Finite Difference Method
This method could be applied in the system (1-4) as follows: s nþ1 ¼ s n þ hðÃ þ si n Þ 1 þ hg a a n þ g c hc n þ hg i i n þ hl : i nþ1 ¼ i n þ hg a s n a n þ g c s n hc n þ hg i i n s n þ hl 1 þ hðc þ l þ sÞ : a nþ1 ¼ a n þ hbc n 1 þ ah : where h is any discretization and n ! 0.

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 R 0 < 1.

Results and Concluding Remarks
In Figs. 2a and 2b, we used the command-built software ODE-45 to simulate the behavior of the model at any time t. In Figs. 3a and 3b, the behavior of the computer method like Euler at different time step sizes are presented in which we can analyze the said method is converges for the small-time step size. In Figs. 4a and 4b, the second important method like Runge Kutta is also diverged and depends upon time step size. In meanwhile, Figs. 5a and 5b, is the true sense results simulates by non-standard finite difference method at any time step size. This computer method has the advantage over the other two methods like Euler and Runge Kutta. Figs. 6a and 6d is the ethnicity of the proposed computer method. Independent of time step size, low-cost and effective technique. Through the study, anthrax disease modeling is analyzed along with analytical approaches and computer techniques. The model is based on four compartments two humans (Susceptible and infected) and two related to dead bodies and spores' germs (bacteria). The human compartment along with positivity, boundedness, equilibria, reproduction number, and local stability is studied rigorously. In the future, we could extend this type of modeling to other complex epidemiological models and their branches.