MODELING THE TRANSMISSION DYNAMICS OF ANTHRAX DISEASE IN CATTLE AND HUMANS

Anthrax is a zoonotic disease caused by bacillus anthraces. In this study a deterministic mathematical model for transmission dynamics of anthrax in humans and animals is presented and quantitatively analyzed. To understand the dynamics of anthrax, the basic reproduction number R0 which measures average new infections is computed using next generation matrix operator and analyzed by normalized forward sensitivity index. Analysis shows that anthrax transmission rate to animals, animals’ natural death rate, anthrax natural death rate and animal’s birth rate are the most sensitive parameters to the disease transmission dynamics. When animal recruitment and anthrax transmission rates increase, the basic reproduction number R0 also increase proportionally. However, when animal natural and anthrax induced death rates increase, basic reproduction number R0 decreases. Numerical simulations using Runge-Kutta method show that animals drive the dynamics of anthrax. The study suggests control strategies such as vaccination, fumigation and decomposition of carcasses to eradicate the disease.


Introduction
Anthrax is an acute infectious disease caused by bacterium Bacillus anthraces [10].The disease is transmitted in four different forms that includes, inhalation, ingestion, contact and injection.Herbivorous are more vulnerable to infection when they eat spores in the soil or plants, [15].Omnivorous as well are vulnerable to the disease and they catch the infection through eating meat which is contaminated with anthrax.There are four types of anthrax which are inhalation, ingestion or gastrointestinal, injection and cutaneous anthrax [12].
Grasses serve as exploited habitats of bacillus anthracis and grazing animals such as sheep, goat and cattle are predominantly victims [13].The alkaline soil with a pH greater than 6.0, high nitrogen level caused by decaying vegetation in soil, balanced periods of rain, droughts and temperature higher than 15 degrees Celsius facilitate the occurrence of anthrax and influence the ecology and survival of the bacterium bacillus anthraces [9].The life span of bacillus anthraces is approximated to be 200 years [10].They die naturally and during rain season at a rate of 0.000014 day −1 [12].
Anthrax is a zoonotic disease which is transmitted from infected animal to human beings.
Humans are vulnerable to cutaneous type of anthrax [14].In the cycle of anthrax, the environment as the pathogen's reservoir serves as the central source of the infection.Cattle get the disease during grazing period especially in rain and dry season.An infected animal can cycle the spores or pathogens ( bacillus anthraces) again back to the environment during excretion [12].Humans catch the disease through contacting an infected animal's product during butchering and slaughtering and when eating raw meat from infected animal.
Anthrax has persisted in Africa where for example in Kruger National Park in South Africa, number of roan antelope declined from 450 to 45 animals [5].In Tanzania, 109 black wildbeest, 21 grant's gazelle,10 cattles and 26 goats died, [8].Also in Kilimanjaro region, three villages reported hospitalization of 36 who ate anthrax contaminate meat from a cow [11].In Arusha as well it has been reported that out of 134 people infected with anthrax, 8 died.Also, in Dar es Salaam out of 22 infected people, 6 died [11].
To provide better understanding of anthrax dynamics and suggest control measures to eradicate the disease mathematical models are of paramount importance.Few mathematical models have been developed.Keeling and Rohani [7] developed mathematical model for anthrax basing on environmental contamination and contact between uninfected animals and infected carcases.
The analysis shows that environmental contamination determines the threshold value for the model.[3] formulated the mathematical model basing on environmental contamination and direct contact between infected animals and non-infected carcasses.The model exhibit the threshold value determined by an environmental contamination parameter.Friedman and Yakubu [2] used a mathematical model developed by Hahn and Furnish and study anthrax is transmitted through carcass ingestion, spread of carcasses to the environment and migration rates on the persistence of animal population.The analysis shows that the spread of carcasses to the environment increases the transmission of the disease.Though the problem is addressed, infection between animals and human being in the transmission dynamics of anthrax is not considered.By modifying the models developed by , this work is intending to study the dynamics of anthrax when infection between animals and human being is considered.

Model Development
The model for anthrax is formulated by extending the models which were developed by Mushayabasa [10] and Sinkie [14] to include human beings.Dynamics of anthrax divides human and animal population each into two classes: susceptible S h and infected I h humans, and susceptible S a and infected I a animals respectively.Carcasses' class is represented by C.
Susceptible humans increase due to birth at a rate b h and decrease due to anthrax infection after eating meat from infected animal and carcasses at a rate σ .They also can acquire the disease when they eat or comes into contact with infected carcasses at a rate δ and when they come into contact with environment which is contaminated with pathogen at a rate φ .However, infected humans increase at a rate σ when susceptible human eat meat from infected animal and carcasses.Infected humans also increase when susceptible human eat or come into contact with infected carcasses at a rate δ and also when they come into contact with environment which is contaminated with pathogens at a rate φ .However, infected humans decrease due to anthrax and natural induced death at rates r and µ h respectively.Susceptible animals replenish due to birth at a rate b a , however, they decrease due to anthrax infection following contact with pathogens at a rate β .Infected animals increase following infection of susceptible animal at a rate β , they suffer disease induced death at a rate π.All animal classes suffer natural mortality at a rate µ a .
Infected animals die due to anthrax and become carcasses at a rate π.The carcasses increase at rate of π due to anthrax induced death rate and decrease due to decomposition at a rate of θ and shed pathogens to the environment at a rate θ .The pathogens increase at a rate of θ following carcass's decomposition and decrease at a rate of ω due to natural death and sometimes due to rain flushing off.The interaction between infected animals, environment and humans is well illustrated by the Figure 1 2

.2 Model assumptions
The model assumes that all animals and humans are susceptible to the disease.The recruitment rate for new individual is through birth.No incubation period for both animals and humans.Huamans and animals suffer natural mortality at rates µ h and µ a respectively.Infected animals and environment are the main source of infections.Carcasses shed pathogens to environment through excretion [12].

FIGURE 1. The model compartment-mental diagram
Placing together both formulation and assumptions, model which describes the interaction animals, humans and environment is given by:

Invariant region
Invariant region shows the boundedness of solutions.To determine the region, human, animal and pathogens' population are considered each separately.Humans, animal and pathogens' populations denoted by N h , N a and P respectively.Human population is given by: Solving the inequality (2) we get: but as t tends to infinity then:

For animal population:
Therefore the solution is given by: For N a (0) = S a (0) + I a (0) +C(0) Analysis of solution ( 6) considers two cases: Since: From the definition of N a , it follows that: For pathogen population we have: Solving the inequality (7) we get: It follows that, analysis of the solution (8) considers two cases, these are: P(0) > θ b a ω µ a and P(0) < θ b a ω µ a .When: P(0) > θ b a ω µ a we have: and when: , we have: Since: Therefore the model system (1) is positive invariant in the region: Solution for the model system (1) which begins on the boundary of the boundary region Γ converge to the region and remain bounded.Therefore the model ( 1) is mathematically and epidemiologically meaningful and we can consider flow generated for analysis.This result is summarized in the following theorem; Theorem: Solutions of the model system (1) enter the region:
Proof: Lets consider the equation number (1a) of the model system (1), we have: From the equation number (1f) of the model (1), we have; (17) The basic reproduction number R 0 The basic reproduction number is the average number of secondary infections generated by a single individual when introduced in an entirely susceptible population [1,6].It determines whether the disease persists or clears out.The disease clears out when R 0 < 1 and persists when To compute basic reproduction number, we adopt the next generation matrix method where new infections and transfer terms are considered.If the new infections are mathematically defined by f i and transfer terms by v i , then the matrices F and V are given by as defined by Van den Driessche and Watmough [16] .The basic reproduction number R 0 is therefore given by: From the model equations (1) the new infections and transfer terms are given by, (20) Derivatives of f i and v i with respect to infected classes at disease free equilibrium are: .
The inverse of the matrix V is: and the product of matrices F and V −1 is; From (19) the basic reproduction number R 0 is given by: The basic reproduction number R 0 depends on animal infection rate β , animal recruitment rate b a , animal anthrax induced death rate π, animal natural mortality µ a and pathogen's natural death rate ω.The basic reproduction number R 0 is directly proportion to β and b a therefore increasing β and b a will also increase the basic reproduction number R 0 .Parameters µ a and ω which are animal and pathogen's natural death rates are inversely proportional to the basic reproduction number R 0 .Increasing animal and pathogens' natural death rates will decrease the basic reproduction number R 0 .

Sensitivity Analysis
Sensitivity analysis assists to understand the potentials of each parameter to disease transmission [4].It identifies sensitive parameters which should be the target when designing disease interventions.We employ the Maximum Likelihood estimation, an inbuilt Matlab fminsearch function to estimate our parameters.In this section parameters' indices with respect to basic reproduction number R 0 are determined.The basic reproduction number R 0 depends on five parameters which are used to derive analytical expression for each parameters.If λ is a parameter in the basic reproduction number R 0 its sensitivity index with respect to R 0 is given by: Using equation ( 27), sensitivity index for each parameter is given in Table .

Parameter
Sensitivity Index

Numerical Simulation
In this section long and short terms behavior of anthrax dynamics in both humans and animal by considering sensitive parameter is analyzed.Parameter values from the data collected from the field are used.The general dynamics of anthrax with no controls is demonstrated in Figure 2 Susceptible humans and animals decrease due to anthrax infection rate as shown in Figure 2. Susceptible animals decrease exponentially as they acquire bacillus anthraces during grazing [3,12].Humans decrease as they acquire anthrax from environment or by eating meat from infected animals or by contacting infected carcasses or coming into contact with pathogen in the environment.The anthrax infection appear to be critical between five to fifteen years as demonstrated by Figure 2.However, pathogens are increasing due to shedding and decomposition of carcasses.

Conclusion, Discussion and Recommendation
A simple anthrax model in humans and animals is developed and analyzed to determine which parameters drive the transmission dynamics of the disease.The basic reproduction number R 0 is computed and sensitivity index for each parameter in the basic reproduction number R 0 is derived.The analysis shows that animal recruitment and infection rates are more sensitive to the disease transmission.Anthrax infection increases as more animals are recruited and it decreases as animals suffer natural mortality.These results are demonstrated by numerical simulation.To eradicate the disease, this study proposes control strategies such as animal vaccination, fumigation and decomposition of carcasses.
= b a − (β P + µ a )S a dS a dt ≥ −(β P + µ a )S a By separating variables we get: (10) dS a S a ≥ −(β P + µ a )dt By integrating both sides, we get; + µ a )dt ln S a ≥ − t 0 (β P + µ a )dt +C S a (t) ≥ e t 0 −(β P+µ a )t At initial condition, we get; (12) S a (t) ≥ S a (0)e t 0 (β P(s)+µ a )dt S a (t) ≥ S a (0)e t 0 (β P(s)+µ a )dtThen, S a (t) ≥ 0, ∀ t ≥ 0From the equation number (1b) of the model (1) we have;(13)dI a dt = β PS a − (µ a + π)I a dI a dt ≥ −(µ a + π) dI a I a ≥ −(µ a + π)dt dI a I a ≥ − t 0 (µ a + π)dt I a (t) ≥ I a (0)e −(µ a +π)t ≥ 0From the equation (1c) of the model (1), we have; ) ≥ C(0)e −θt ≥ 0.From the equation number (1d) of the model (1), we have; ) ≥ P(0)e −ωt ≥ 0 Again from the equation (1e) of the model (1), we have; From sensitivity indices we note that, positive indices imply proportional relationship with basic reproduction number R 0 .Any percentage increase in parameters with positive indices will make the same percentage increase in basic reproduction number R 0 .Parameters with positive indices are animal recruitment rate b a and animal infection rate β .On the other hand, negative indices imply inverse relationship with basic reproduction number R 0 .Any percentage increase in parameters with negative indices will make the same percentage reduction in basic reproduction number R 0 .Parameters with negative sensitivity indices are animal death rate µ a and pathogens' life span ω.

FIGURE 2 .
FIGURE 2. The general dynamics of anthrax in humans and animals FIGURE 3. Variation of anthrax transmission dynamics with animal infection rateFigure3shows the effect of varying animal transmission rate to the dynamics of anthrax in both susceptible and infected animals.It is found that as animal infection rate increases, 0016,R 01 = 1.7778 β = 0.0020,R 02 = 2.2222 β = 0.0030,R 03 = 3.3333 (B) Anthrax in carcasses population

FIGURE 4 .
FIGURE 4. Variation of animal infection rate in carcass and pathogen PopulationFigure4ashows the effect of varying animal transmission rate in both pathogens and carcasses population.The results show that as the animal transmission rate increases both carcasses and pathogens increase correspondingly.

FIGURE 5 .
FIGURE 5. Variation of animal infection rate in both susceptible and infected humans

FIGURE 6 .
FIGURE 6. Variation of animal natural death rate in both pathogens and carcasses Figure6shows the effect of varying the animal natural death rate in both pathogens and carcasses.The results shows pathogen and carcasses decrease as animal death rate increases.

FIGURE 7 .Figure 7
FIGURE 7. Variation of animal natural death rate in both susceptible and infected animal

FIGURE 8 .Figure 8
FIGURE 8. Variation of animal natural death rate in both susceptible and infected humans Figure 8 shows the effect of varying animals natural death rate to both susceptible and infected humans.The results show that when animal natural death rate increases susceptible humane increases and infected humans decreases.
Table shows the model parameters' values.Parameter values from the related literature were used as initial values to fit the model from the data collected from the field.The data were collected at Ngorongoro district in Tanzania from 2006 to 2016 [11].