1. Introduction

2. Materials and Methods

3. Results

3.2. The Anthrax Model’s Equations

The set of time fractional differential equations designed to describe the dynamics of the disease is basically on the descriptions and assumptions of the above Model (Figure 1).

$$\begin{array}{l}{D}^{\alpha }{S}_H(t)={\mathsf{\Lambda }}_H-{\mu }_H{S}_H-{\mathsf{\Phi }}_1{S}_H+\rho R_H\\ D^{\alpha }{E}_H(t)={\mathsf{\Phi }}_1{S}_H-{\mu }_H{E}_H-\tau E_H\\ D^{\alpha }{I}_H(t)=\tau E_H-{\mu }_H{I}_H-\delta I_H-{\eta }_1{I}_H\\ D^{\alpha }{R}_H(t)={\eta }_1{I}_H-{\mu }_H{R}_H-\rho R_H\\ D^{\alpha }{S}_V(t)={\mathsf{\Lambda }}_V-{\mu }_V{S}_V-{\mathsf{\Phi }}_2{S}_V\\ D^{\alpha }{I}_V(t)={\mathsf{\Phi }}_2{S}_V-{\mu }_V{I}_V-{\eta }_2{I}_V\\ D^{\alpha }{R}_V(t)={\eta }_2{I}_V-{\mu }_V{R}_V\end{array}\}$$

(11)

Here $D^{\alpha }=\frac{d^{\alpha }}{d{t}^{\alpha }}$, for 0 < α ≤ 1 in the sense of Caputo Fractional derivative of order α

Table 1.  .

Table 1.  .

Parameters

Description

SH EH IH RH SV IV RV ${\mathsf{\Lambda }}_H$ ${\mathsf{\Lambda }}_V$ ${\mathsf{\Phi }}_1$ ${\mathsf{\Phi }}_2$ $\rho$ ${\mu }_H$ ${\mu }_V$ $\tau$ ${\eta }_1$ ${\eta }_2$ ${\beta }_1$ ${\beta }_2$ ${\beta }_3$ $\delta$

human Susceptible human Exposed human Infected human Recover vector Susceptible Infected vector Recovered vector Immigration/recruitment rate for human Immigration/recruitment rate for vector Force of infection for human(saturation function) Force of infection for vector (saturation function) Loss of immunity human Natural death rate for human Natural death rate for vector Rate of Progression from exposed to infected class of human Rate of Progression from infected to recovery class of human Rate of Progression from infected to recovery class of vector The rate of effective contact between vector and humans rate of effective contact between domestic vector and Bacillus anthracis Rate of contact leading to infection for the vector disease induced death

Mini-proof

$\frac{dK}{dt}=({\mathsf{\Lambda }}_H+\rho R_H)-{\mu }_H{S}_H$, using the method of $I.F=e^{{\displaystyle \int \mu dt}}$

$\frac{d(S_H{e}^{{\mu }_{H}t})}{dt}=({\mathsf{\Lambda }}_H+\rho R_H)e^{{\mu }_{H}t}$

$S_H=\frac{{\mathsf{\Lambda }}_H}{{\mu }_H}+\frac{1}{e^{{\mu }_{H}t}}{\displaystyle \int \rho R_H{e}^{{\mu }_{H}t}dt}$, as $t\to \infty$ we have that $S_H(t)\to \frac{{\mathsf{\Lambda }}_H}{{\mu }_H}$, this also goes for vector population

$S_V(t)\to \frac{{\mathsf{\Lambda }}_V}{{\mu }_V}$as $t\to \infty$

Conveying the global convergence of $F(K,0)\in R_{+}^7$

This complete the mini-proof

Proof of Second condition

We have that

$$G(K,J)=\frac{dY}{dt}G(K,J)=\begin{array}{l}{\mathsf{\Phi }}_1{S}_H-{\mu }_H{E}_H-\tau E_H\\ \tau E_H-{\mu }_H{I}_H-\delta I_H-{\eta }_1{I}_H\\ {\mathsf{\Phi }}_2{S}_V-{\mu }_V{I}_V-{\eta }_2{I}_V\end{array}\}$$

(36)

$G(K,J)$could also be written in the form

$$G(K,J)=\left[\begin{array}{c}{\mathsf{\Phi }}_1{S}_H\\ 0\\ {\mathsf{\Phi }}_1{S}_V\end{array}\right]+\left[\begin{array}{ccc}-({\mu }_H+\tau )& 0& 0\\ \tau & -({\mu }_H+\delta +{\eta }_1)& 0\\ 0& 0& -({\mu }_V+{\eta }_2)\end{array}\right]\left[\begin{array}{c}{E}_H\\ I_H\\ I_V\end{array}\right]$$

(37)

We need to show that $G(K,J)=BC-G^0(K,J)$ in (37) the vector Matrix C is given as

$C=(E_H,I_H,I_V)$, where $G(K,J)=G^0(K,J)$ is the stable state matrix which establish the second condition of the GAS

$$B=\left[\begin{array}{ccc}-({\mu }_H+\tau )& 0& 0\\ \tau & -({\mu }_H+\delta +{\eta }_1)& 0\\ 0& 0& -({\mu }_V+{\eta }_2)\end{array}\right]$$

$$-G^0(K,J)=\left[\begin{array}{c}{\mathsf{\Phi }}_1{S}_H\\ 0\\ {\mathsf{\Phi }}_1{S}_V\end{array}\right]$$

(38)

Equation (38) becomes

$$G^0(K,J)=\left[\begin{array}{c}-{\mathsf{\Phi }}_1{S}_H\\ 0\\ -{\mathsf{\Phi }}_2{S}_V\end{array}\right]\le 0$$

(39)

Which completes the proof. □

3.8. Analysis of the Sensitivity of Parameters of Anthrax-Vector Model

The main goal of sensitivity analysis is to ascertain how each variable affects the rate of reproduction. This aids in determining the parameters that have a significant influence on R_HV. In order to determine whether or not immunization, mortality, and infection treatment could aid in the management or eradication of the disease in the system, the basic reproduction rate is often examined [38]. This technique is widely employed to test the resilience of model assumptions to variable values. The examination of the model's reproduction number R_HV, which looks for variance and the impact of changing a parameter on R_HV, will be carried out.

Definition: The normalised forward sensitivity index of a variable a, is given by:

$$M_b^a=\frac{\partial a}{\partial b}x\frac{b}{a}$$

(40)

Which depends on the differentiability of parameter b

We would now have to carry out the analysis of the fundamental reproduction number specifically on some key parameters that makes up the reproduction number $R_{HV}$and with regard to parameters that makes up our model

The sensitivity index for ${\beta }_1,{\beta }_2,{\beta }_3,\mathrm{.....}$ and every other parameters will be computed for

$$R_{HV}=\sqrt{\frac{{\beta }_1{\beta }_3{\mathsf{\Lambda }}_V{\mathsf{\Lambda }}_H\tau }{{\mu }_V{\mu }_H({\mu }_V+{\eta }_2)({\mu }_H+\delta +{\eta }_1)({\mu }_H+\tau )}}$$

(41)

By analyzing the derivatives in (41)

$$\begin{gathered} M_{{\beta }_1}^{R_{HV}}=\frac{\partial R_{HV}}{\partial {\beta }_1}.\frac{{\beta }_1}{R_{HV}}, \frac{\partial R_{HV}}{\partial {\beta }_1}=0.5\frac{R_{HV}}{{\beta }_1},\mathrm{hence} \\ {M}_{{\beta }_1}^{R_{HV}}=\frac{\partial R_{HV}}{\partial {\beta }_1}.\frac{{\beta }_1}{R_{HV}}=0.5\frac{R_{HV}}{{\beta }_1}.\frac{{\beta }_1}{R_{HV}} \end{gathered}$$

Conclusively

$$M_{{\beta }_1}^{R_{HV}}=+0.5$$

By this we obtain the sensitivity index for ${\beta }_1$

In the same vein we will use this method for all other parameters that makes up the reproduction number in other get there sensitivity index, though not all the parameters have positive values according to Table 2

3.8.1. Interpretation of R_HV Sensitivity Indices

The value of R_hv in epidemiological models indicates how easily a disease may spread throughout the community. By calculating the sensitivity indices of R_hv with regard to the model's parameter values, we will be able to calculate the decrease in infection caused by the diseases. The contribution of each parameter to the dynamics of the diseases is determined by the sensitivity indices. We determine the sensitivity of R_HV to each parameter in the model (11) taking into account all of the system's properties.

Table 2 lists the sensitivity indices of Rhv for each system parameter in model (11).

A thorough sensitivity analysis of R_HV revealed that a rise in $\eta$ would result in a reduction in RHV. Furthermore, if $\eta$ were to decrease, RHV would rise. Moreover, for a rise in RHV implies that there is an increase in the values of ${\mathsf{\Lambda }}_H,{\mathsf{\Lambda }}_V,{\beta }_1,{\beta }_2,and{\beta }_3,$, and a fall in R_HV would result from a decrease in those values respectively

3.9. Numerical Simulation

We have reviewed, investigated, and evaluated the analytical/simulated outcomes of the Anthrax-vector interactive model in this section. This has been done in order to confirm the results of our sensitivity analysis, to validate our theoretical conclusions, and to investigate, evaluate, and elucidate the dynamic behavior of the population being considered. We have simulated the full model and individual compartments using the MATLAB (software) program, taking into account the initial circumstances (state variable values) and changing a few significant model parameters. Table 3 presents the values, dimensions, and origins of these parameters.

While some of these parameters were based on reasonable estimations and assumptions, the majority of them were clinical facts that were taken from reputable literature.

Figure 1. The human population dynamical behavior with${\mathsf{\Lambda }}_H=0.2$, ${\mathsf{\Phi }}_1=0.091193$, ${\mu }_H=0.0003466$, $\rho =0.0085$. $\tau =0.10$, $\delta =0.01710615$, ${\eta }_1$= 0.30, demonstrating the coexistence of the compartment and driving the system's tendency toward an endemic condition.

Figure 1. The human population dynamical behavior with${\mathsf{\Lambda }}_H=0.2$, ${\mathsf{\Phi }}_1=0.091193$, ${\mu }_H=0.0003466$, $\rho =0.0085$. $\tau =0.10$, $\delta =0.01710615$, ${\eta }_1$= 0.30, demonstrating the coexistence of the compartment and driving the system's tendency toward an endemic condition.

4. Discussion

Figure 2. Here the recruitment was increased ${\mathsf{\Lambda }}_H=12$ and the force of infection ${\mathsf{\Phi }}_1=0.5$ while other parameters remain the same which also shows that all compartment co-exist causing the population to approach endemic state, but one thing that is noticeable here is that the susceptible shows a sharp decline compared to Figure 1.

Figure 2. Here the recruitment was increased ${\mathsf{\Lambda }}_H=12$ and the force of infection ${\mathsf{\Phi }}_1=0.5$ while other parameters remain the same which also shows that all compartment co-exist causing the population to approach endemic state, but one thing that is noticeable here is that the susceptible shows a sharp decline compared to Figure 1.

Figure 3. Similarly the recruitment rate was increased the third time i.e ${\mathsf{\Lambda }}_H=25,and{\mathsf{\Phi }}_1=1$, again we notice a sharp decline in the susceptible region which will hence affect the exposed and the infected causing the whole population to tends to toward endemic state.

Figure 3. Similarly the recruitment rate was increased the third time i.e ${\mathsf{\Lambda }}_H=25,and{\mathsf{\Phi }}_1=1$, again we notice a sharp decline in the susceptible region which will hence affect the exposed and the infected causing the whole population to tends to toward endemic state.

Figure 4. The susceptible human's changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of human vulnerability to anthrax.

Figure 4. The susceptible human's changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of human vulnerability to anthrax.

Figure 5. The Exposed human's changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of exposed compartment to anthrax.

Figure 5. The Exposed human's changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of exposed compartment to anthrax.

Figure 6. The effect of increasing the rate of treatment for infected individuals by vaccination administration (along with other essential treatments) demonstrates how early antiviral medication administration and implementation may significantly reduce the population inside the infected compartment.

Figure 6. The effect of increasing the rate of treatment for infected individuals by vaccination administration (along with other essential treatments) demonstrates how early antiviral medication administration and implementation may significantly reduce the population inside the infected compartment.

Figure 7. The dynamical behavior of vector population with ${\mathsf{\Lambda }}_V=0.005$, ${\mathsf{\Phi }}_2=0.1$, $\delta =0.00261076$, ${\eta }_2=0.057$showing how the compartment co-exist and which in essence the vector population tends to endemic state.

Figure 7. The dynamical behavior of vector population with ${\mathsf{\Lambda }}_V=0.005$, ${\mathsf{\Phi }}_2=0.1$, $\delta =0.00261076$, ${\eta }_2=0.057$showing how the compartment co-exist and which in essence the vector population tends to endemic state.

Figure 8. In the above figure above ${\mathsf{\Phi }}_2=0.2$, that is we doubled the incidence figure, while other parameter remain same, we notice a sharp decline in the susceptible class which changes the behavior of the population as whole.

Figure 8. In the above figure above ${\mathsf{\Phi }}_2=0.2$, that is we doubled the incidence figure, while other parameter remain same, we notice a sharp decline in the susceptible class which changes the behavior of the population as whole.

Figure 9. In the above figure above the incidence figure was tripled in other to see how the population, ${\mathsf{\Phi }}_2=0.3$, while other parameter remain same, we notice a sharp decline in the susceptible class which changes the behavior of the population as whole.

=10, while other parameters remain the same, we noticed that irrespective of the susceptible number to the system with effective control strategies the vector population will still attain DFE.

Figure 9. In the above figure above the incidence figure was tripled in other to see how the population, ${\mathsf{\Phi }}_2=0.3$, while other parameter remain same, we notice a sharp decline in the susceptible class which changes the behavior of the population as whole.

=10, while other parameters remain the same, we noticed that irrespective of the susceptible number to the system with effective control strategies the vector population will still attain DFE.

Figure 10. Here we try to introduce effective control plan to vector population with ${\mathsf{\Lambda }}_V$=0.005, ${\mathsf{\Phi }}_2$=0.3 the figure above showing that the vector population can approach disease free equilibrium (DFE), especially the susceptible and infected population.

Figure 10. Here we try to introduce effective control plan to vector population with ${\mathsf{\Lambda }}_V$=0.005, ${\mathsf{\Phi }}_2$=0.3 the figure above showing that the vector population can approach disease free equilibrium (DFE), especially the susceptible and infected population.

Figure 11. In the figure above the vector recruitment rate was increased to ${\mathsf{\Lambda }}_V$

Figure 11. In the figure above the vector recruitment rate was increased to ${\mathsf{\Lambda }}_V$

Figure 12. The susceptible vector changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of vector vulnerability to anthrax.

Figure 12. The susceptible vector changing effects both with and without the "saturation factor" This indicates that the introduction of saturation lowers the incidence of vector vulnerability to anthrax.

5. Conclusions

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