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This paper examines the computational modelling of cholera bacteriophage with treatment. A nonlinear mathematical model for cholera bacteriophage and treatment is formulated and analysed. The effective reproduction number of the nonlinear model system is calculated by next generation operator method. By using the next generation matrix approach, the disease-free equilibrium is found to be locally stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Globally, the disease free equilibrium point is not stable due to existence of forward bifurcation at threshold parameter equal to unity. Stability analysis and numerical simulations suggest that the combination of bacteriophage and treatment may contribute to lessening the severity of cholera epidemics by reducing the number of Vibrio cholerae in the environment. Hence with the presence of bacteriophage virus and treatment, cholera is self-limiting in nature.

A highly pathogenic gram-negative bacterium Vibrio cholerae is the causative agent of the water-born diarrheal disease; cholera [

Treatment of cholera with massive doses of bacteriophage is not as effective as treatment with tetracycline. However, bacteriophage can selectively eliminate the majority of Vibrios without affecting the other intestinal flora toxic effect on the patient.

Bacteriophage might be useful as a research tool [

A nonlinear mathematical model is proposed and analysed to study the impact of bacteriophage and treatment in the environment while the cholera epidemic is in progress. The proposed mathematical model divides the human population,

A Susceptible, Infective, Treatment, Recovery and Susceptible (SITRS) model is developed to study the role of bacteriophage and treatment in the environment during the cholera outbreaks. The SITRS model indicates that the passage of individual is from the susceptible class

In formulating the model, the following assumptions are taken in consideration:

1) The population varies.

2) There is natural death of human, Vibrio cholerae and bacteriophage at the rates of

3) The disease is fatal.

4) The rate of transmission is directly proportional to the susceptible population and also to the ratio between the members of infected population to the environment.

5) The population is homogeneously mixed and each susceptible individual has equal chance of acquiring cholera.

However, the model also assumes that the infected individual can recover at the rate of

probability that an individual in contact with untreated water with pathogenic Vibrios is infected with Vibrio cholerae.

Taking into account the above considerations and assumptions then we have the following schematic flow diagram (

The model is thus governed by the following system of nonlinear ordinary differential equations:

where

The total population of human at time t is given by

Thus it follows that

This implies that

This reduces to

It follows that from equation (2), that in the absence of the disease i.e.

The model system of Equations (1) will be analysed qualitatively to get insight into its dynamical features which will give a better understanding of the effects of bacteriophage and treatment while cholera epidemic persist in a given population. The effective reproductive number

The disease free equilibrium of the model system of equations (1) is obtained by setting

At disease-free equilibrium, we have

Therefore the disease free equilibrium (DFE) denoted by

The effective reproduction number,

The disease-free equilibrium of the nonlinear model system (1) is given by

Theorem 1

The local stability of the disease-free equilibrium of the cholera Bacteriophage and treatment model system (1) is locally asymptotically stable if

This is shown by computing the Jacobian matrix of the model (1). The Jacobian matrix is computed by differentiating each equation in the system with respect to the state variables

It follows that

This gives

The characteristic equation corresponding to

It follows that

when

when

Other eigenvalues are

Since all Eigen values of the characteristic equation have negative real parts then the disease-free equilibrium

Theorem 2

The disease-free equilibrium point

Proof

From system (1) we have

where the matrices

Since

Using the fact that the eigenvalues of the matrix

and fourth equations of the model system (1), we obtain

The endemic equilibrium of the nonlinear model system (1) is given by

where

From Equation (9) it follows that

implying that

which corresponds to the disease free equilibrium.

This gives

It follows that

where

Corresponding to unique endemic equilibrium

Theorem 3

The unique endemic equilibrium

From Equation (9), it follows that there is no backward bifurcation since the value of

From

The local asymptotic stability of endemic equilibrium will be analysed by using the Centre Manifold theory [

Theorem 4

If

Proof:

To establish the global stability of the endemic equilibrium

Direct calculation of the derivative of L along the solutions of (1) gives

But

This implies that

Substitute

Therefore

which gives

Collecting positive and negative terms together in the system (13), we obtain

If we let

And

then

system (1). Then by LaSalle’s invariant principle it implies that

Now we consider the situation when there is no treatment of the acute infective i.e.

We note that

In this case, we consider the situation where there is no bacteriophage in the model system (1). Since there is no bacteriophage then,

In this situation, it is observed that

the infection in this case increases in the absence of bacteriophage which may contribute to lessen the severity of cholera epidemics by reducing the number of Vibrio cholerae in the environment. After analysing the two epidemiological situations discussed above, it may be concluded that in the presence of both bacteriophage and treatment in the model system, the disease tends to the disease free equilibrium points, otherwise the disease tends to endemic state. Therefore the presence of bacteriophage and treatment can reduce the number of Vibrio cholerae in the environment and the number of infectives within the society is also decreased, hence the disease tends to die out.

In order to illustrate analytical results of the study, numerical simulations of the nonlinear model system (1) are carried out using the set of estimated parameter values below

1.

2.

3.

1.

2.

3.

The equilibrium point of the endemic equilibrium

It is observed from

Figures 5(a)-(c) show the variation of proportion of total population in different classes, Treated population, Recovery population for different values of

From

From

From

From

From

A nonlinear mathematical model has been analysed to study the effect of bacteriophage and treatment in the environment while the cholera epidemic is in progress. This study is the extended work done by [

society is also decreased hence, the disease tends to die out.

Based on the results of this study, we conclude that the most effective way to control cholera epidemic is well achieved by involving both bacteriophage and treatment. However, it is important to note that phage can reduce the number of Vibrio cholerae in the environment. Consequently, number of infected population within the society is also decreased and severity of the disease is also checked. Hence by using phage as a biological control agent in the endemic areas, cholera is self-limiting in nature. Moreover, therapeutic treatment which includes hydration therapy, antibiotics and water sanitation should be administered during the cholera epidemic. Furthermore, people should be educated on the awareness of the effective prevention methods which includes provision and use of clean drinking water, hand washing, environmental hygiene and sanitation, and also avoidance of potentially contaminated foods.

Daniel S.Mgonja,Estomih S.Massawe,Oluwole DanielMakinde, (2015) Computational Modelling of Cholera Bacteriophage with Treatment. Open Journal of Epidemiology,05,172-186. doi: 10.4236/ojepi.2015.53022