MODELING AND STABILITY ANALYSIS OF CRYPTOSPORIDIOSIS TRANSMISSION DYNAMICS WITH BEDDINGTON-DEANGELIS INCIDENCE

.


INTRODUCTION
Cryptosporidiosis is an infection caused by an apicomplexan protozoan known as Cryptosporidium.Cryptosporidium common parasites of vertebrates have recently attracted increasing interest due to several serious waterborne outbreaks, and the life-threatening nature of infection in immunocompromised patients, children, the elderly, and patients on chemotherapy, pregnant women; and also the realization of economic losses caused by these pathogens in livestock.It is a common enteric pathogen in humans and domestic animals worldwide with a very low infective dose of one to ten ooysts (Pereira [1]).The sporulated ooysts are immediately infectious when excreted in faeces as there is no intermediate host.Cattle are reared throughout Cameroon but the major production areas are in the West and North West Regions and from the Adamawa Province [2].The cattle are transported on foot to the cattle market and the dung they pass along the road is likely to contaminate the environment and the oocysts possibly end up in streams after torrential rains.In time past, following the description of Cryptosporidium in mice by Ernest Edward Tyzzer [3], the genus Cryptosporidium has been studied, and now discovered to contain numerous species and genotypes adapted to parasitic life in almost all classes of vertebrates.Over the years, our knowledge has expanded from microscopic observations of infection and environmental contamination to the knowledge obtained from large application spread of molecular techniques to taxonomy and epidemiology.Although, the medical and veterinary significance of this protozoan was not fully appreciated for an-other 70 years.The interest in Cryptosporidium escalated tremendously over the last two and half decades [4,5].It was later recognized as a cause of disease in 1976.As several methods were developed to analyze stool samples, the protozoa was increasingly reported as the cause of human disease [6].At first, Crypto was categorized as a veterinary problem because, majority of the early cases were diagnosed due to individuals rearing farm animals such as cows.Furthermore, 155 species of animals specifically mammals have been reported to be infected with Cryptosporidium parvum which is also known as C. parvum [7].Among the 15 named species of Cryptosporidium infectious to non-human vertebrate hosts C. Baileyi, C. canis, C. felis, C. hominis, C meleagridis, C. muris, and C. parvum have been reported to also infect humans.The primary hosts for C. hominis are Humans, except for C. parvum, which is widespread in non-human hosts and is the most frequently reported zoonotic species, the remaining species left have been reported primarily in immunocompromised or immunosuppressed humans [7].The first Cryptosporidiosis outbreak that was widely known occurred in 1987 [8] in Carrollton, Georgia.About 13,000 persons became sick as a result of the outbreak the disease.The main cause was traced to a large contaminated water system.In 1993, in Milwaukee area, Wisconsin, a massive outbreak of the disease occurred, causing approximately 400,000 people to fell sick as a result of contaminated drinking water in one of the two treatment plants serving the Milwaukee area [6].Therefore, motivated by the above discussion into account, in this paper, we propose and analyze a mathematical model of Cryptosporidiosis disease dynamics in humans and animals population with the Beddington-DeAngelis incidence .We construct the compartmental model by considering three different classes of individuals in the humans population and three different classes of individuals in the animals population.We believe that the findings of our work will be helpful in indicating appropriate measures to control the spread of the disease.The rest of the paper is organized as follows: The model description and formulation are discussed in Section 2.
The basic properties of the model including non-negativity and boundedness of solutions, the mathematical analysis of the model including the introduction of the threshold parameter R ha obtained using the Next-Generation method, the stability of the disease-free and endemic-equilibrium points as well as the bifurcation and sensitivity analysis are investigated in Section 3. In Section 4, we perform numerical simulations to support some of the analytical results.A brief discussion and conclusions are presented in the last section.

DESCRIPTION AND FORMULATION OF THE MODEL
2.1.Assumptions.The following assumptions will be used to simplify the model : • In the presence of the disease we divide the model into two parts, the total human and total animal (vector).These populations at any time, are also divided into six sub populations (compartments).The total human population also represented by H divided into sub-populations of susceptible humans H S , infected humans H I and recovered humans H R .The total human population is given by : H(t) = H S (t) + H I (t) + H R (t).The total animal population, represented by A, is divided into sub-populations of susceptible animals A S , infected animals A I and recovered animals A R .The total animal population becomes A(t) = A S (t) + A I (t) + A R (t).
• A susceptible human can be infected only by an infected animal.
• A susceptible animal can be infected only by an infected animal.
• The force of infection from infected animals to susceptible humans is modeling using Beddington-DeAngelis incidence form as   (1)

CRYPTOSPORIDIOSIS MODEL ANALYSIS
3.1.Positivity and boundedness of solutions.In this subsection, we must prove that at t ≥ 0 all solutions of the model system (1) are positive and bounded for the Cryptosporidiosis model to be meaningful and well posed.Theorem 3.1 A non-negative solution (H S (t), H I (t), H R (t), A S (t), A I (t), A R (t)) for model (1) exists for all states with positive initial conditions (H S (0) ≥ 0, Proof.According to the first equation of the system of differential equation (1) we have: that is: The integration of the inequality gives According to the second equation of the system of differential equation ( 1) we have: Let us take the third equation we have: Similarly, using the same argument, it can be shown that this completes the proof of the Theorem 3.1.

Theorem 3.2
The solution of the model system (1) with positive initial conditions are ultimately bounded in Proof.Human population at any time, t is given by: In the absence of mortality due to Cryptosporidiosis infection we obtain Animal population at any time t is given by: A(t) = A S (t) + A I (t) + A R (t).Similarly we get 0 ≤ A(t) ≤ Λ A µ A and we define Therefore the feasible solution set of Cryptosporidiosis model (1) remain in the following region Thus, the Cryptosporidiosis model ( 1) is well posed epidemiologically and mathematically.Hence, it is sufficient to study the dynamics of the Cryptosporidiosis model in Ω.
3.2.Disease-Free Equilibrium point.The disease-free equilibrium point (DFE) is the point at which no disease is present in the population of human and animal.However, DFE is obtain by setting H I (t) = H R (t) = 0 and A I (t) = A R (t) = 0.The DFE of the Cryptosporidiosis model ( 1) is given by: (2) Using the "Next Generation Matrix" approach, we determine R ha and its linear stability.Basic reproduction number refers to the number of secondary cases produced on average by one infected animal or person in completely susceptible population.This combines the biology of infections with the social and behavioural factors influencing contact rate [9,10,11].It is the threshold parameter that determines or governs the spread of disease.Considering only the infection classes in the system (1) Let F be the number of new infection coming into the system and V be the number of infections that are leaving the system either by death or birth, then The jacobian matrix of F and V at disease-free equilibrium is obtained by f and v as follows: The inverse of v is found to be The next generation matrix f v −1 is given by : (10) By finding the eigenvalues of matrix f v −1 we get ( 11) 3.4.Local stability of the disease-free equilibrium.Local stability of the disease-free equilibrium is given by Theorem 3.3 Theorem 3.3 The disease-free equilibrium is locally asymptotically stable if R ha < 1 and un- Proof.The disease-free equilibrium point E 0 is locally asymptotically stable if the real parts of the eigenvalues of the jacobian matrix corresponding to the system (1) around the DFE are all negatives.The Jacobian matrix corresponding to the system (1) around E 0 is given by: The characteristic equation of the Jacobian matrix ( 13) is given by: ( 14) Therefore the eigenvalues are a, b, c, d, e and h.Clearly, a, b, c, d and h are negative.
3.5.Global stability of the disease-free equilibrium point.In this section we investigate global asymptotic stability of the disease-free equilibrium point using the theorem by Castillo-Chavez and Song [12] as done in [13].To do so, we write system equation (1) as: Where X = (H S (t), H R (t), A S (t), A R (t)) ∈ R 4 denotes uninfected population and Y = (H I (t), A I (t)) ∈ R 2 represents the infected population.Let X * be the disease-free equilibrium of the system (16) dX dt = F(X, 0) Furthermore, we list two conditions and if met will guarantee the global asymptotic stability of Metzler and the Jacobian matrix of G(X,Y ) taken in (H I , A I ) and evaluated at E 0 = (X * , 0).

Theorem 3.4
The disease-free equilibrium point E 0 = (X * , 0) is globally asymptotically stable for the model (1) provided that R ha < 1 and that the conditions (i) and (ii) are satisfied.
Proof.We only need to show that the conditions (i) and (ii) hold when R ha < 1.From the model system (1) we obtain F(X,Y ) and G(X,Y ) as : From condition (i), we consider the reduced system dX dt = F(X, 0) and we get: is globally asymptotically stable equilibrium point for the reduced system dX dt = F(X, 0).We proved this by finding the solution of the equations in the system (19).
From the second and fourth equations of (19), we have H R (t) = H R , 0 e −µ H t , which approaches From the first and third equations of ( 19), we have Therefore, from the formula in condition (ii), we get the following expression and we obtain after some calculation From the invariant region Ω, we have Λ H µ H ≥ H S and Λ A µ A ≥ A S , therefore Ĝ(X,Y ) ≥ 0 for all (X,Y ) ∈ Ω, and we conclude that the DFE is globally asymptotically stable whenever R ha < 1.
3.6.Existence of Endemic equilibrium point.In this section, we explore the existence of the endemic equilibrium point (EE).In the presence of Cryptosporidiosis, H I (t) = 0, H R (t) = 0, A I (t) = 0 and A R (t) = 0, our model has an equilibrium point called endemic equilibrium point ). E 1 is the steady state solution where Cryptosporidiosis persist in the population of human and animal.For the existence of E 1 , the elements must We find the endemic equilibrium point by setting the right side of the model system equations (1) equal to zero, that is: If R ha > 1, the system (1) has a unique endemic equilibrium point given by: Local stability of the Endemic equilibrium point.This section explores the local stability of the endemic equilibrium point E 1 .We can see from (29),(30),(32), (33) that the endemic equilibrium point has long expressions and the standard linearization method which consist of finding the eigenvalues of the Jacobian matrix around the endemic equilibrium point can be mathematically complicated.Hence, in oder to investigate the local asymptotic stability of the endemic equilibrium point E 1 , we use the result based on the center manifold theory described in [12,13,14] to investigate if the model system (1) exhibits a forward or backward bifurcation when R ha = 1.When bifurcution is forward, it implies that the endemic equilibrium point is locally asymptotically stable for R ha > 1.This result is reproduced here for convenience.To use that method we make the following simplification and change of variables in the system (1).Let x 1 = H S , x 2 = H I , x 3 = H R , x 4 = A S , x 5 = A I and x 6 = A R .Futher by introducing the vector notation x = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) T , system (1) has the form dx dt = F(x), where We set the transmission rate δ 2 as the bifurcation parameter.Solving for δ 2 the equation Linearisation of the system (35) at the disease-free equilibrium point The above matrix J(E 0 ) has a simple zero eigenvalues.Moreover, be the right eigenvector of (37) associated with the simple zero eigenvalue.Then v is obtained by solving J(E 0 )v = 0.By direct calculation, we get: Let z = (z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) T be the left eigenvector of (37) associated with the simple zero eigenvalue.It satisfies zv = 1 and the matrix J(E 0 ) should be transposed so that J T (E 0 )z = 0.
By direct calculation, we get: z = (0, 0, 0, 0, 0, z 5 , 0) T .Now zv = 1 gives z 5 v 5 = 1.Assume that v 5 = σ > 0, we obtain z 5 = 1 σ .Then, the right and left eigenvectors turn out to be: Now we have to compute a and b give by the formulae We will only consider k = 5 because By direct calculation, we get: Therefore, a < 0 and b > 0 at bifurcation parameter δ 2 = δ * 2 .This scenario indicates that the Cryptosporidiosis model exhibits a forward bifurcation at R ha = 1.Its biological meaning is that as long as R ha < 1, the Cryptosporidiosis can be eliminated from the human and animal population.Hence the unique endemic equilibrium point ) is locally asymptotically stable whenever R ha > 1.  3.8.Global stability of the Endemic equilibrium point.In this section, we perform the global stability analysis of system (1) around the positive endemic equilibrium point using the method of Lyapunov functions with LaSalle'S invariant principle.Existing techniques for constructing Lyapunov functions have been improved by [15] because of the difficulties in constructing appropriate Lyapunov functions with nonlinear incidence.Theorem 3.5 The Endemic equilibrium point E 1 of the model ( 1) is globally stable if R ha > 1, and condition (46) is hold.
Proof.We define the lyapunov function U as (43) Hence U is C 1 on the interior of Ω, E 1 is the global maximum of U on Ω, and we then have The time derivative of U alongside the solutions trajectories of system (1) is: Separating positive and negative terms as U 1 and U 2 , we have is a singleton of E 1 with E as the endemic equilibrium.Therefore by the LaSalle'S invariant principle, E 1 is globally asymptotically stable in 3.9.Sensitivity analysis of the model parameters.In this section, we investigated the sensitivity of the parameters for the basic reproduction number of the model using the idea presented in [16,17].It is important to carry out the sensitivity of the basic reproduction number R ha for its parameters.This will give parameters with a high impact on the Cryptosporidiosis model (1) and therefore allow to target on control measures to reduce the transmission of the disease.To measure the sensitivity index of R ha to a given parameter p, we use the following relation: An analytical expression for the sensitivity index of each parameter involved in R ha is derived as follows: In the following table of the parameters most are assumed (due to the lack of data) while few are taken from the literature.
From Table 3, we can observe that only the parameter δ 2 , has the most positive influence on R ha .This means that the increase of this parameter while keeping other parameters constant will increase the value of R ha leading to an increase of the spread of Cryptosporidiosis in the human and animal population.We likewise observe that the parameters m 3 , α A , µ A and β A respectively have the most negative impact on R ha .This implies that the increase of these parameters while keeping the other constant will decrease the value of R ha , meaning that they will decrease the endemicity of Cryptosporidiosis in the human and animal population.

NUMERICAL SIMULATIONS
We performed numerical simulations of our proposed model (1) to support some of the analytical results.We use the set of parameters values given in Table 2 and the initial values of the model are set as: We set the final time as t f = 120 days.This was chosen on the basis of the assumption that a period of four month is enough for the disease spread.All simulations are done using Matlab with the ode45 function.

Simulation of the population dynamics of the Cryptosporidiosis showing the exis-
tence of a unique endemic equilibrium point (EE) when R ha > 1.We observe from Figure 3 that whenever R ha > 1, the susceptible humans population drop exponentially and converges to a steady state to acquire endemic equilibrium level while the infected humans increase exponentially to a certain maximum point before exponential drop to a certain endemic level.This is an indicator of Cryptosporidiosis outbreak.We also observe that the susceptible animals population decrease exponentially due to natural death and acquisition of Cryptosporidiosis infection and finally acquire the endemic equilibrium level.Hence, without intervention the populations approach the endemic equilibrium levels in the long run implying the existence and stability of endemic equilibrium point.In this fact, the saturation levels ultimately affect the dynamics of the model system and then, minimizing contacts between infected and susceptible populations are highly recommended.

1+m 3 A
S (t)+m 4 A I (t) from infected animal to susceptible animal where δ 1 and δ 2 are infection rate, m 1 , m 2 , m 3 and m 4 are parameters that measure the inhibitory effect.

FIGURE 1 .
FIGURE 1. Flow diagram for the Cryptosporidiosis disease transmission dynamics

FIGURE 3 .
FIGURE 3. Graph showing the population dynamics.

4. 4 . 6 (A)) and m 2 (
Simulation of the effects of saturation level m 1 and m 2 on the infected humans population.From the analytical results, we observed that the saturation levels m 1 and m 2 do not contribute on the basic reproduction number R ha , but they have the effects on infected population.Figure6is simulation results of the model showing the effects of the saturation levels m 1 and m 2 on the infected humans population.We observe that an increase in m 1 (Figure Figure6 (B)), respectively produces a decrease in the number of infected humans.

FIGURE 6 .FIGURE 7 .
FIGURE 6. Graph showing the effect of saturation levels m 1 and m 2 on the infected humans population.

4. 6 .
Simulation of the effects of natural death of animals µ A , and disease induced mortality α A on the infected animals population.

Figure 8 FIGURE 8 .
FIGURE 8. Graph showing the effects of µ A and α A on infected animals.

+m 4 A I (t) and die naturally at a rate µ A . Infected animals A I (t) are recruited at a rate δ 2 A S (t)A I (t) 1+m 3 H S (t)+m 4 A I (t) , die naturally at a rate µ A , die from infection at a rate α A , recover and has permanent immunity at
2.2.The model derivation.Our proposed model divides the total human population H(t) into three subclasses of susceptible H S (t), infected H I (t) and recovered H R (t).The total animal population A(t) is divided into three subclasses of susceptible A S (t), infected A I (t) and recovered A R (t) .For the model susceptible humans H S (t) are recruited at a rate Λ H , infected at a rate δ 1 H S (t)A I (t) 1+m 1 H S (t)+m 2 A I (t) and die naturally at a rate µ H . Infected humans H I (t) are recruited at a rate H S (t)+m 2 A I (t) , die naturally at a rate µ H , die from infection at a rate α H , recover and has permanent immunity at a rate β H . Recovered humans are recruited at a rate β H and die naturally at a rate µ H . Susceptible animals A S (t) are recruited at a rate Λ A , infected at a rate a rate β A .Recovered animals are recruited at a rate β A and die naturally at a rate µ A .The parameters of the model is summarized in Table 1 and the flow diagram of the model is shown in Figure 1.

Table 1 .
Notations and Description of model (1) parameters

Table 3 .
Sensitivity indices of R ha with respect to the model parameters