Modelling the Effects of Vertical Transmission in Mosquito and the Use of Imperfect Vaccine on Chikungunya Virus Transmission Dynamics

In this paper, a deterministic mathematical model for Chikungunya virus (Chikv) transmission and control is developed and analyzed to underscore the effect of vaccinating a proportion of the susceptible human, and vertical transmission in mosquito population. The disease free, and endemic equilibrium states were obtained and the conditions for the local and global stability or otherwise were given. Sensitivity analysis of the effective reproductive number, c R (the number of secondary infections resulting from the introduction of a single infected individual into a population where a proportion is fairly protected) shows that the recruitment rate of susceptible mosquito ( M Λ ) and the proportion of infectious new births from infected mosquito ( β ) are the most sensitive parameters. Bifurcation analysis of the model using center manifold theory reveals that the model undergoes backward bifurcation (coexistence of disease free and endemic equilibrium when 1 C R < ). Numerical simulation of the model shows that vaccination of susceptible human population with imperfect vaccine will have a positive impact and that vertical transmission in mosquito population has a negligible effect. To the best of our knowledge, our model is the first to incorporate vaccinated human compartment and vertical transmission in (Chikv) model.

gested alternative way for controlling the 8disease. Yakob and Clements [18], analysed a simple, deterministic mathematical model for the transmission of the virus between humans and mosquitoes. They fitted the model to the large Reunion epidemic data and estimated the type reproduction number for Chikungunya, their model provided a close approximation of both the peak incidence of the outbreak and the final epidemic size.
In this work, we proposed a deterministic mathematical model for the spread, and control of Chikv. Our model attempt to bridge identified gaps in the works cited above. Specifically, our model incorporated an imperfect vaccinated human compartment and vertical transmission in the mosquito population.

Model Formulation
The chic model is represented by nine non-linear ordinary differential equation consisting of human-sub population and mosquito sub-population. The human sub-population is divided into; susceptible human H S , vaccinated human H V , exposed human H E , infected symptomatic human 1 I , infected asymptomatic human 2 I , recovered Human R, such that the total human population, into; susceptible mosquito M S , exposed mosquito M E , and infected mosquito 3 I , such that the total mosquito population, The parameters of the model and their values are given in Table 1 ε < < ) is the efficacy of the imperfect vaccine. Members of the exposed population move to either symptomatic infectious population at the rate 1 σ or to asymptomatic infectious population at the rate ( ) 1 1 σ − . The recovered population is generated as both symptomatic and asymptomatic infected populations recover with lifelong immunity at the rate γ . All human population are decreased by natural death at the rate 1 µ , except the two infected populations that are decreased by disease induced death at the rate δ . The susceptible mosquito population is generated by M Λ , this population is decreased by birth from infected mosquito (vertical transmission) at the rate M βΛ ; and as its members take a blood meal from either symptomatic or asymptomatic infected human (horizontal transmission) at the rate 2 α . The σ . It is assumed that births from infected mosquito do not pass through the exposed class. All sub-populations of mosquito die naturally at the rate 2 µ .

The Model Equation
From the model formulation, and schematic diagram Figure 1, we hereby present the model equations.

Basic Properties
For the Chikungunya model (1) to (9) to be epidemiological meaningful, it is necessary to prove that all its state variables are non-negative for all time. This means that the solution of the model Equations (1) to (9) with non-negative initial data will remain non-negative for all time 0 t > .
respectively. Thus a standard comparison theorem as in Lakshmikantham and Martynyuk, [25] can be used to show that  Hence D is attracting, that is all solutions in 9 + ℜ eventually enters D. Thus in D, the basic model Equations (1) to (9) is well posed epidemiologically and mathematically according to [26]. Hence it is sufficient to study the dynamics of the basic model Equations (1) to (9).
, , , , , , , , Then the solution ( ) So that, Hence, Similarly, it can be shown that For the second part of the proof, note that, (10) and (11), as required.

Local Stability of Disease Free Equilibrium (DFE)
The basic model (1) to (9) has a DFE, 0 E obtained by setting the right-hand sides of the model equations to zero, which gives: The linear stability of 0 E can be established using the next generation Matrix operator method on the system (I) to (9). Using the notation in [23], the matric- and, where, Hence using theorem 2 of [23] the following results are established.

Proof
From Equations (1) to (9) and (20), we have that, the only non-zero compartments at disease free equilibrium are; Such that, Hence, Similarly, it follows from Equation (7)  We have that, Hence, Thus if In summary, we have shown that 1 D is positively invariant and attracting with respect to the solutions of our model Equations (1) to (9).
Theorem 2 The DFE of the basic model (1) to (9) is Global Asymptotical Stability (GAS) in 1 D , whenever 1 , G X Z , the right hand side of 1 2 , , , , Next we consider the reduced system: be an equilibrium of (37) we show that * X is a global stable equilibrium in 1 D .
To do this, we solve the Equations (37), which gives as t → ∞ . This asymptotic dynamics is independent of initial conditions in D. Hence the solution of xxx converges globally in 1 D .
Next we are required to show that ( ) , G X Z satisfies the following two conditions in [19] pp246 namely; where, Therefore, by the theorem 2 in [28], the disease-free equilibrium is globally asymptotically stable since in the absence of disease induced mortality the human population is constant.

Sensitivity Analysis
Here we present the sensitivity index of the parameters of the effective reproductive number ( ) C R . Sensitivity tells us how important each parameter is to disease transmission. Such information, is crucial not only to experimental design, but also to data assimilation and reduction of complex nonlinear model [29].
Sensitivity Analysis is commonly used to determine the robustness of model Sensitivity indexes allows us to measure the relative changes in a variable when a parameter changes. The normalized forward sensitivity index of a variable with respect to a parameter is the ratio of relative changes in the parameter when the variable is a differentiable function of the parameter. The sensitivity index may be alternatively defined using partial derivatives. The sensitivity index of our model is given in Table 2.  Table 2, the most sensitive parameter of C R is the recruitment rate of susceptible mosquito ( M Λ ) followed by the proportion of infectious new birth from infected mosquito ( β ) while the natural birth rate of mosquito ( 2 µ ) and the rate at which exposed mosquito become infectious ( 2 σ ) are equally sensitive to the C R according to the model. This means that any policy or practice capable of reducing these parameters will go a long way in reducing the menace of Chikungunya and at the long run, result to eradication.

Vaccine Impact Analysis
Vaccine was believed to confer life-long immunity until 1990s. This was the norm as it was approximately correct for most available vaccine for infectious children diseases. But most vaccines used for combating adult infectious diseases today are defective and thus immunity conferred on the recipients wane with time. It is expected that the future Chikv vaccine will also be defective and hence the need to assess its effectiveness in C R a community. In this paper, the vaccine impact analysis is done by differentiating effective reproductive number with respect to the proportion p of susceptible individuals vaccinated at equilibrium, according to [32], i.e., ( ) R is a decreasing function of p. This means that a vaccination program with 0 p > and 0 ε > at equilibrium, the future vaccine will have a positive impact. Besides, there exist a C p such that and for vaccination of proportion of susceptible C p p > the number of new-cases reduces to zero faster than when C p p < .

Numerical Simulation
To further verify the analytical results in the model, the ode 45 code embedded in matlab was used to simulate some parameters of the model. Table 1 Figure 4 is the simulation of some compartments Figure 2. Plot of the various populations with parameters as in Table 1. (A) is the simulation of susceptible human against time, the plot shows that the susceptible human decreases with time due to the proportion that gets infected but slows down after some days, perhaps due to the vaccination and other control measures. (B) is the simulation of the vaccinated compartment. The plot shows a steady increase initially, but began to slope down after few days, this could be due to the fact that a proportion of the class are infectious as the vaccine is imperfect. (C) is the simulation of the exposed compartment with time, the plot shows a steady decline as members become infectious and progress to either the symptomatic or asymptomatic compartment. Finally (D) is the simulation of the symptomatic compartment with time. The plot shows a steady decline and tends to zero after about 20 days. This could be attributed to recovery from the infection.      Table 1. Figure 6. Simulation of the chikv model displaying a contour graph of ( c R ) as a function of vaccinated human population and vaccine efficacy ( ε ); with parameter values as listed in Table 1.    number C R with respect to the vaccine efficacy ε and the proportion of susceptible vaccinated ( Figure 6) gave the rates at which the C R is above, below and equal to unity, this confirms that the use of imperfect vaccine will be effective. Figure 6 also reveals a linear relationship between the effective basic reproductive number and the two parameters in question unlike Figure 5. Also the graph of Chikungunya new case ( Figure 7) shows a decrease in new cases with high vaccine efficacy ε and proportion of vaccinated susceptible ν . Hence buttressing the point made in Figure 6.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper. φ  , 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0 1 φ <  , 0 is unstable and there exists a negative, locally asymptotically stable equilibrium; 2) 0, 0 a b < < , when 0 φ < with 1 φ  , 0 is unstable; when 0 1 φ <  , 0 is locally asymptotically stable equilibrium and there exists a positive unstable equilibrium; 3) 0, 0 a b < > , when φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.