side of system (1) to zero, we obtain

(4)

From the second, fourth and sixth equations of (4), we write and in terms of. From the sixth equation of (4) we have,

defining

.

Equation (2) reduces to (3).

3.1. Disease Free Equilibrium

This solution of 4 leads to the disease-free equilibrium point is given by

(5)

3.2. The Effective Reproductive Number, Reff

In this section, the threshold parameter that governs the spread of a disease referred to as the effective reproduction number is determined. Mathematically, it is the spectral radius of the next generation matrix [10] . The equations of the system (1) are re-written starting with infective classes, to obtain

(6)

From the system (6), and are defined as

substituting and, becomes,

The partial derivatives of and with respect to and and evaluating at the disease free point gives

is computed and obtained as

.

The eigenvalues of are and

The effective reproduction number measures the average number of new infections generated by a typical infectious individual in a community when treatment and trapping strategies are in place. As we increase trapping and treatment rates, have the effect of increasing because of linearity of in terms of taking into account that, treatment and trapping are effective.

Local Stability of the Disease Free-Equilibrium

The disease-free equilibrium point is Thus, the Jacobian matrix of the

system (1) is computed by differentiating each equation in the system with respect to the state variables. Hence, at the steady states the Jacobian matrix for system (1) is given by

(7)

where, , and The characteristic polynomial is given as

Using Birkhoff and Rota's theorem on the differential inequality (3) we obtain

.

From the matrix (7) we note that the first, third, fourth, fifth and sixth have diagonal entries. Therefore their corresponding eigenvalues are;

(8)

. (9)

With the help of mathematical software, the following characteristic equation is obtained

(10)

(11)

and

.

If, then and are all negative. These results are summarised with the following theorem

Theorem

The disease-free equilibrium point is locally asymptotically stable if and unstable if.

4. The Endemic Equilibrium, E3

In the presence s of infection, that is, , the model system (1) has a non-trivial equilibrium point, called the endemic equilibrium point which is given by, where from the system (4.3). In this case, the following solution is considered

(12)

where is derived above. Then from the equations of system (4.3) we obtain

(13)

(14)

(15)

(16)

(17)

. (18)

We let and then

(19)

Adding the last two equations of the system and making some simplifications we obtain

(20)

where

.

The equation, corresponds to a situation when the disease persists (endemic). In case of backward bifurcation, multiple endemic equilibrium must exist. This implies that while considering the equation (4.18) there are three cases we have to consider depending on the signs of and A since B is always positive. That is;

1) If and or, then Equation (4.18) has a unique endemic equilibrium point (one positive root) and there is no possibility of backward bifurcation.

2) If, and, then Equation (4.18) has two endemic equilibria (two positive roots), and thus there is the possibility of backward bifurcation to occur.

3) Otherwise, there is none.

However it is important to note that A is always positive if and negative if. Hence the above explanation leads to the following theorem.

Theorem 5 The rift valley fever basic model has,

1) Precisely one unique endemic equilibrium if

2) Precisely one unique endemic equilibrium if and or

3) Precisely two endemic equilibrium if, and

4) None, otherwise.

From (iii) it is observed that backward bifurcation is possible if the discriminant is set and solve for the critical value of. Thus, we get

where backward bifurcation occurs for values of lying in the range. The theorem below gives the condition of existence of the endemic equilibrium point,.

Theorem 5 The endemic equilibrium point, exists if.

5. Sensitivity Analysis

Sensitivity analysis determines parameters that have a high impact on and should be targeted by intervention strategies. We will use the approach done in [11] and Blower and Dowlatabadi, 1994 to calculate the sensitivity indices of the effective reproduction number,.

The indices are crucial and will help us determine the importance of each individual parameter in transmission dynamics and prevalence of the Rift Valley Fever Virus.

Definition 1 The normalized forward sensitivity index of a variable, u, that depends differentiably on index

on a parameter, p is defined as;

The analytical expression for the sensitivity of is for each of the parameter p in-

volved in. We used the following parameter values to determine the sensitivity indices;

Interpretation of Sensitivity Analysis

From Table 1, it shows that when the parameters and, are increased keeping other parameters constant they increase the value of implying that they increase the the burden of the disease among the human, animals and vector populations as they have positive indices. While the parameters, and decrease the value of when they are increased while keeping the other parameters constant, implying that they decrease the burden of the disease among the human, livestock and vector populations. The specific interpretation of each parameter shows that, the most sensitive parameter is the transmission rates for susceptible cattle individuals with infection followed by transmission rates for susceptible human individuals with infection and so on as the Table 1 indicates.

6. Numerical Simulation

We carry out numerical simulations for mathematical model of rift valley fever for the set of parameters from literature as shown in Table 1. The parameter values that changed the value of are: and.

We have the following simulation results (Figures 1-6). Figure 1 shows variation of the different populations for specified parameter values. As treatment rates and increase, both infected human population and cattle population rises quickly to reach maximum and then drops to a steady state. Corresponding to the rise of both human population and livestock population infective there is a drop in the susceptible human and livestock population until reaches the minimum values and then rises to a steady state. The reduction of mosquitoes and through trapping of both infected and susceptible lead to reduction in infected human and animal population because the two are infected by infected mosquitoes and they do not infect each other. The simulation results depicted in Figure 2 illustrating the the endemic state with the value of. The results show the introduction of trapping the mosquitoes, treating human and livestock populations reduce the reproduction number from 8.60276 to 0.4782, this implies the clearance of the disease.

Table 1. Parameter values and the calculated sensitivity indices.

Figure 1. Schematic diagram for Rift Valley Fever Model with interventions.

Figure 2. Population Dynamics of the rift valley fever without intervention model.

Figure 3. Effects of treatment of livestock on mosquitoes population.

Figure 4. Effects of treatment of human on mosquitoes population.

Figure 5. Effects of trapping of mosquitoes on human population.

Figure 6. Effects of trapping of mosquitoes on human population.

6.1. Variation of Different Parameters on the Dynamics of Rift Valley Fever Model with Treatment and Trapping

In this section parameters, , and representing the treatment rate for the infected human population, treatment rate for the infected livestock population and the trapping rate for the mosquitoes population respectively were varied to determine their effect on the different model populations. When the treatment rates of livestock and human increase the infected human, livestock and mosquitoes decrease as the Figure depicts. When the trapping rate and of the infected and susceptible mosquitoes respectively increase, the infected human and livestock decrease. This implies that endemicity of the disease among human and livestock decreases.

6.2. Discussion

The Rift Valley Model formulated in this study is well posed and exists in a feasible region where disease free and endemic equilibrium points are obtained and their stability investigated. The model has two interventions; treatment for human and livestock and trapping for mosquitoes. We have shown that disease free equilibrium exist and is locally asymptotically stable whenever its associated effective reproduction number is less than unity, and it has a unique endemic equilibrium u when exceeds unity. These results have important public health implications, since they determine the severity and outcome of the epidemic (i.e. clearance or persistence of infection) and provide a framework for the design of control strategies. Analysis of the model show that in the absence of treatment of livestock and human and trapping of mosquitoes (ie.) and if,the epidemic will develop,but if it will die out. At (all infected human and cattle have access to treatment for human and all mosquitoes are trapped), then, and the epidemic will be fully controlled. The main epidemiological findings of this study include:

Ÿ In the absence of treatment of human or livestock and trapping for mosquitoes: implying that treatment failure leads the epidemic persistence. Hence the combination of treatment for livestock, humans and trapping for mosquitoes can eradicate the rift valley fever infection if can be reduced to below unity.

Ÿ With human or livestock and trapping for mosquitoes, so; (a): human or livestock treatment and trapping for mosquitoes is effective, hence elimination of infection.

7. Conclusion

In this paper, the rift valley fever model with interventions was formulated and analysed. Using the theory of differential equations, the invariant set in which the solutions of the model are biologically meaningful was derived. Boundedness of solutions was also proved. Analysis of the model showed that there exist two possible solutions, namely the disease-free point and the endemic equilibrium point. Further analysis showed that the disease-free point is locally stable implying that small perturbations and fluctuations on the disease state will always result in the clearance disease if. In the final analysis treatment and trapping interventions program will effectively control the spread of rift valley fever.

Cite this paper
Lugoye, J. , Wairimu, J. , Alphonce, C. and Ronoh, M. (2016) Modeling Rift Valley Fever with Treatment and Trapping Control Strategies. Applied Mathematics, 7, 556-568. doi: 10.4236/am.2016.76051.
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