Fractional-Order Model for Evolution of Bovine Tuberculosis with Vaccination and Contaminated Environment

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Introduction
There has been progress in the fight against tuberculosis (TB) in Africa, but several obstacles are hampering efforts to end this preventable and curable disease.Global targets to eradicate the illness by 2030 are increasingly improbable at the current rate [1].Contact is the primary means of transmission for tuberculosis (TB), a chronic infectious disease mostly affecting the respiratory system.Reports [2] state that Africa has the highest prevalence of instances.India, China, and Indonesia followed with 72%, 27%, and 9%, respectively.The reorientation of funds to the COVID-19 response has hampered the provision of basic services in a number of countries.Due to the lockdowns, many who have tuberculosis had found it difficult to receive treat-ment.The ability to detect drug-resistant tuberculosis has been influenced by COVID-19 [1].In comparison to 2019, the number of cases reported in the WHO African Region declined by 28% in 2020 [1].
Cattle tuberculosis is a zoonotic infectious disease that is classified as a class B animal epidemic by the OIE (Office International des Epizooties).Humans and other animals may contract an infection mostly from an infected animal.The respiratory and digestive systems are the primary pathways of transmission.Contact with ill animals or consumption of their raw milk can result in infection in both healthy humans and animals [3][4][5].Since bTB-infected animals are put down as soon as they get ill, the disease has a significant negative economic impact [5].In addition, bTB deteriorates health and can occasionally be lethal.Some people may lose their self-employment as a result of it, especially those whose primary source of income is cattle rearing [6].Bovine tuberculosis can transmit from cattle to humans through three primary routes: consuming raw meat, inhaling aerosols, and drinking unpasteurized milk [7].Other ways that bTB spreads among animals include the consumption of contaminated milk, particularly during lactation, and breathing in of aerosols [8].It can also spread through close contact between infected and uninfected animals.
The intradermal skin test is the most established and widely used approach for bTB diagnosis [9].The fluctuating sensitivity and specificity are its principal drawbacks, as demonstrated in a number of papers.Furthermore, the use of tuberculosis vaccination procedures makes this test difficult because sensitised animals yield false-positive results [8].A deterministic mathematical model is developed in [5] to explore the dynamics of bTB transmission in infected individuals as well as animals.To determine the disease's behaviour, the basic reproduction number R 0 is computed.According to the sensitivity analysis, the frequency at which dairy products are generated, animals infect other animals with bTB, and humans contract bTB from contaminated dairy products is what causes bTB to spread.
A meta-analysis by specialists from India, the USA, the UK, the Netherlands, and Ethiopia that assesses the impact of the BCG vaccination on cattle is an intriguing article [10].According to their analyses, BCG immunization may hasten the control of bTB in endemic areas.Publications pertaining to the immunology of Mycobacterium bovis (Mb) infections have been written.The pathophysiology of tuberculosis in cattle is typified by lesions in the lymph nodes and lungs, which eventually lead to the development of granulomas.The immunopathology and chronic evolution of bTB are comparable to that of human TB in many aspects [8].
In [11], Ahmad et al. devised the reaction-diffusion model and applied the fractional differential equation to obtain traditional solutions to the nonlinear partial differential equation.The fraction differential calculus is a useful tool that may be used to explain the dynamics of many life events in the form of fractional orders.In [12], mathematical models for the evolution of potato leaf roll virus propagation are created using the differential equations with both integer and fractional orders.The models considered the combination of vectors and potato populations.The potato leaf roll virus (PLRV) model initially was created in integer order; however, the model was later extended into fractional order since fractional order gives memory and other benefits for modelling real-life occurrences.
In [3], the study looks at the evolution of bovine tuberculosis transmission in both human and animal populations, utilising a fractional-order model with the Caputo sense.The threshold quantity R 0 was also calculated using the Lyapunov functions of the Volterra type.In their study titled Review of Fractional Epidemic Models [13], concentrated on summarising several variants of the fractional epidemic model and assessing the outcomes of epidemiological modelling, specifically the fractional epidemic model.They developed simple, effective analytical techniques for solving fractional epidemic models that are easy to adapt and use.
These techniques can assist the relevant organisations in managing, preventing, and even forecasting the spread of infectious diseases.

TB Model Description and Formulation
At any given time (t), the model divides total populations of humans and animals into seven (7) subpopulations (compartments), with contaminated environment C e .
The following are bTB model assumptions: immigration and birth rates are within the susceptible human population, there exists a continuous interaction between human and animal populations, and because the model lacks a recovery class, it is presumed that there is no natural recovery.Consuming dairy and meat from sick animals can spread the illness to people.
The animal population size, Ω A , is classified into three categories: susceptible (S A ), exposed (E A ), and infectious (I A ), where The individual population size, denoted by Ω H , is classified into vaccinated humans V H , exposed E H , infected I H , and susceptible S H .
The bTB vulnerable population is recruited at an average rate of Λ H . Furthermore, individuals catch the latent illness at a Λ H rate by eating uncooked meat and milk and other dairy products from animals that are infected, along with regular coming into contact with contaminated people and livestock.
Effective immunizations are administered to a selection of people at an average rate of κ, where κ ∈ 0, 1 .As the infectious stage advances, the graph shows that the passive infection of susceptible individuals S H drops at a rate of γ H and grows at a frequency of λ H in the exposure class E H . Human infections I H rise at γ H and fall at α H due to disease-related death.
Natural death occurs in every human compartment at a rate of μ H . Due to the decreasing influence of vaccine effectiveness with 1 − d ∈ 0, 1 , vaccinated humans may transition to the exposure compartment at a rate of dλ H .At a rate of ϕ, humans may become sensitive and lose their immunity.
Supplicant animals S A are bred and transported into communities at an average of Λ A ; thereafter, they acquire a latent bovine TB infection through dairy consumption and interaction with ill humans and animals.

Journal of Applied Mathematics
Using the Caputo derivatives of order α, let us examine the fractional model where 0 < α < 1.
Using Diethelm's method [14], we will use the following system of fractional-order equations from Figure 1: Under initial conditions,   Proof.

Model Analysis
(i) By summing the system equations of human population from model 3, the fractional derivative of the entire human population is as follows: Note: compute without α on the right side in order to simplify the expressions.On both sides of Equation ( 12) using the Laplace transform [15], On the LHS, Infection rate of animals from I H , I A , and C e , 0.25, 0.70, 0.50 [5], given, given Regarding the RHS, Equation ( 13) now looks like the following: Assuming that s α−1 N H 0 = 0 at t = 0 [12], then The Mittag-Leffler function and the inverse Laplace transform of N H s are used; we obtain (ii) Using the same methodology, the animal population will yield 4

Journal of Applied Mathematics
The 8 th equation of model (3) yields the following: Using the equality case and performing the Laplace transform of Equation ( 24) on both sides, we have Using the identical calculus method as the human population example, we obtain the following: On the LHS, On the RHS, Now, Equation (28) becomes Taking s α−1 C e 0 = 0 at t = 0, then Using (29)'s inverse Laplace transform, we obtain and so Given a system of fractal-order equations in ( 5), the via-ble region is as follows: That comprises a set of positive invariants.
This exhibits the model solution's boundedness.Lemma 2 (see [16]).Assuming that k t ∈ C x, y and C 0 D , C e t are nonnegatives for all time t > 0.
Proof.Taking all of the model's equations in Equation ( 5) at t = 0, we obtain The solution escape from the hyperplanes of U t = 0, S A 0 , E A 0 , I A 0 , C e 0 are nonnegatives, as per Equations ( 35)-(42) and by using Remark (3).Consequently, for all t > 0, all of the model's solutions with initial conditions in the set Ψ stay in Ψ.This area is a positive invariant set as a result.

Disease-Free Equilibrium (DFE).
According to Φ 0 , this is reached when there exists no disease in both the animal and human populations as a whole.

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Following some math, we obtain 3.4.The Basic Reproduction Number.We use the nextgeneration matrix algorithm to generate the bTB fundamental reproduction number R 0 , as described in [17,18].
Let F i represent the total quantity of newly acquired infections entering the system and V i represent the number of infections leaving the system owing to births or deaths.
Denote the Jacobian matrices of F i and V i by F and V, respectively.
Journal of Applied Mathematics Compute the eigenvalues of FV −1 and select the dominant one.
X is the eigenvalue of the matrix.
Equation (49) can be translated as follows: Characteristic equation is given as Therefore, the highest eigenvalue is Evaluating and substituting A 1 , A 3 , B 1 , and B 3 at the DFE Φ 0 , we obtain The terms 1/ μ H + σ H + κ and 1/ μ A + σ A in Equation (53) represent the mean amount of each humans and animals in their respective exposed compartment, and 1/ μ H + ϕ denotes the mean duration of time that each individual is exposed to in their respective classes, 1/ μ H + γ H denotes the mean ratio of time spent by each human in the vaccinated class, and 1/ μ A + γ A denotes the mean ratio of time each infectious populace devotes time to their infectious compartment.
The proportion of diseased animals is represented by Equation (54) gives the sum of infected bTB through contact with diseased animals through ingestion of dairy products.

Analysis of Local Stability for DFE.
If the eigenvalues of the Jacobian matrix are negative or have a negative real portion, DFE is considered locally asymptotically stable [5].

Journal of Applied Mathematics Matrix (58) reduces now to
If det R > 0 and tr R = 0, thus DFE is locally stable.Equation (60) gives the trace of matrix R.
Assume det R = 0; thus, Eigenvalues have a negative real portion as a result of the trace and determinant criteria being proven.For every α ∈ 0, 1 , arg λ j > απ/2.When R 0 < 1 and condition (63) holds, the DFE ψ 0 of the model ( 5) is locally asymptotically stable; otherwise, it is unstable.

Global Stability of DFE.
Theorem by [19] is used to study the global asymptotic stability (GAS) of the model's DFE state.Based on (5), The uninfected population consists of V = S H , V H , S A , and infected population consists of T = E H , I H , E A , I A , C e .
For the system dV/dt = F V, 0 , assume that V * is DFE, with It is assured that the model's disease-free equilibrium (DFE) point Φ 0 is GAS if R 0 < 1, which is LAS (local asymptotical and stable).
(i) A1: for dV/dt = F V, 0 , V * is GAS for model ( 5) The model is biologically meaningful if Ψ 0 , and A = ∂G Φ 0 /∂T is an M-matrix with nonnegative nondiagonal elements.
If the two aforementioned assumptions are met by model ( 5), then the following theorem holds.Theorem 6.If R 0 < 1 is LAS and assumptions A1 and A2 are true, then DFE, Φ 0 , is GAS for model (5).
Proof.In order to demonstrate that V ⟶ V * , we must first demonstrate that assumptions A1 and A2 hold for R 0 < 1.
The α's order linear ODEs are represented by the second and third equations of (66), and their solutions look like this H S H ; using Laplace transform, we will get Journal of Applied Mathematics We can now obtain the following using the Mittag-Leffler function and the Laplace inverse transform of S A W : Using the same technique, we get Subtracting V H t from the initial equation of (66) produces We get the following from the Laplace transform of (70): From the Laplace inverse transform, we obtain So all points concerning these conditions converge at Thus, V * is asymptotically stable globally.Consider We then obtain

Journal of Applied Mathematics
where We have S H + dV H /Ω H ≪ 1 and μ α H κ α d − 1 ≪ 1 since all parameters are positive as well.Since G1 ≥ 0, G3 ≥ 0 must also follow.
denotes bTB endemic equilibrium point.Given that C * e be the global stability.
Proof.To define a function, we use the Volterra-type Lyapunov functional approach in [3,20]. where * , the equivalence may be demonstrated [3].
Let us demonstrate that D α t L ≤ 0 at the EE point.
We get from Equation (80) the following by simplification: It can be established that D α t L ≤ 0. Hence, if R 0 > 1, (81) is less than zero (0) and equivalent to zero (0

Numerical Simulations
This indicates that 86% of humans and animals should receive vaccination if R * 0 = 7 4298.

Sensitivity Analysis.
The sensitivity analysis of parameters in Table 2 of R 0 determines the contribution of each parameter [22].Using the normalized forward sensitivity analysis in [22,23],

Journal of Applied Mathematics
For every 10% increase in dairy products, R 0 rises by 0.018%.There is an increase in the human disease-induced death rate γ H , the human natural mortality rate μ H , and the animal natural mortality rate α A .

Numerical Simulation.
We demonstrate the model behaviour by varying fractional orders α ∈ 0, 1 and parameter values.We use values in Table 1 for our simulations.
As shown in Figure 2, the number of susceptible individuals and animals decreases.However, susceptible class reduces more than animals, due to human vaccination.
The effects of changing α on humans, animals, and vaccinated humans are depicted in Figures 3(a)-3(c), respectively.
As seen in Figure 4, the population of animal appears more infected than in human population.This can be attributed or explained through vaccinations.

Influence of Vaccination Rate on Animals and Humans
Who Are Infected.Figure 5 illustrates how the vaccination rate κ varies across the human population while maintaining constant values for other parameters.It can be observed that, as κ rises, the results have no impact on the animal population.The biological implication is that the number of infected individuals falls as the vaccination rate κ increases.

Impact of Decay Rate on the Environment That Is
Contaminated.But while the other parameters remain constant, changing the rate of decay ω for the contaminated environment (meat and dairy products) demonstrates in Figure 6 a direct correlation between an increase in infectious humans and animals.A decrease in the decay rate plays a major role in eliminating the illness in both human and animal populations.

Effects of the Rate at Which Animals Become Infected.
The outcome of adjusting the animal infection rate η 5 while holding other constant parameter is shown in Figure 7.A rise in η 5 from 0.5 to 0.8 results in a higher number of individuals and animals afflicted.
The percentage of ill individuals and animals is higher for η 5 = 0 8. Based on numerical findings, there is a direct correlation between the rate of animal infection and the quantity of humans and animals that become infected.
To reduce the animal infection rate η 5 from affected animals and halt the spread of the disease, policy makers and health authorities should think about strategies to isolate infectious animals.

Conclusion
We created a fractional-order mathematical framework that outlines the evolution of bovine tuberculosis, using environmental pollution and vaccination as compartments.In Section 3, we examined the model's qualitative behaviours by defining the area that is feasible, the solution's positivity, equilibrium points and their global and local stability, and the model's fundamental reproduction number.
The reproduction rate was subjected to a sensitivity analysis, which demonstrated a significant impact on the management of tuberculosis in cows.A parameter contribution analysis and numerical simulation were performed.The effect of parameters κ, ω, and eta 5 on the fractional-order model was explored.
Based on our findings, we can infer that a substantial reduction in the rate of bovine tuberculosis transmission in both human and animal populations can be achieved by raising the vaccination rate, κ, of both populations.Controlling the disease requires reducing the rate of animal infection from diseased animals eta 5 while raising the pace at which the contaminated habitat decays.
Caputo fractional derivative.Note: let us use the notation D α instead of C 0 D α in the rest of the discussion.

3. 2 .
Positivity.Since the initial values of each model equation are positive, then all of the model equations' solutions (5) remain positive for future times.

Figure 2 :
Figure 2: Bovine tuberculosis dynamics in the human and animal populations for α = 65.
Then, the model's system equation has a feasible solution set S H t , E H t , V H t , I H t , S A t , E A t , I A t , C e t , which is bounded in the region Ψ.

Table 1 :
Parameter values and descriptions.
Increasing vaccination rates in human and animal populations is most effective in reducing bovine tuberculosis.4.1.1.Threshold of Herd Immunity.Let H 1 denotes the Herd immunity; then,