# Bifurcation Analysis

Using the center manifold theory [27,86], the authors [16] have established the backward bifurcation phenomenon by taking into account the transmission rate *p _{H}* as a bifurcation parameter. = 1, if and only if

Denote the variables of the system as *S _{H} =х_{г},Е_{н} = х_{2},1н =x_{3},R_{H} =x_{if}S_{v} = x_{5},E_{v} = x_{6}, I_{v} = x_{7},* and the vector of variables as

*x = (х*

*1*

*,х*The model can then be reformulated in the form (

_{2},х_{3/}х_{л},х_{5},х_{6},х_{7})^{т}.*dx/dt*) =

*F(x),*with

*F =*(/i,/2,/3,/4,/5,

*ft, fi') ,*where

For the system (5.16), the Jacobian matrix evaluated at the DFE *E _{0}* is given by

The characteristic equation has a simple zero eigenvalue and the other eigenvalues are negative or have negative real parts (Problem 5.3, Exercise 5). Hence, the center manifold theorem [27] can be applied. For applying the theorem, we need to calculate the values of the parameters *a* and *b* as defined in the theorem. The right and left eigenvectors of *J(E _{0}) *denoted respectively by

*w = (w*

_{1/}w_{2},w_{3},W*4*

*,w*and v =

_{5/}w_{6},w_{7})^{T}*(v*are obtained as

_{u}v_{2},v_{3},v_{4},v_{5},v_{b},v_{7})The values of the non-zero derivatives are

The coefficients *a* and *b* are given by
Substituting the values of the derivatives, we obtain

We find *b>0.* It follows from the center manifold theorem [27] that the system (5.14) undergoes forward transcritical bifurcation if *a <* 0, that is, under the condition *к^руАнХн < кк1ц _{г}, + ptip_{v}A_{H}xh-* To test this conclusion, we performed the following computations. For example, for the values of the parameters Xh = 0.0022, = 0.0002,

*p _{v} =* 0.0009,

*p*0.01, <5i,=0.3,

_{H}=*Hv =*0.003, Л„ = 1.3, A

_{H}=0.4,

*T]*= 0.11, p = 0.029, and

*у*= 0.0614799, the values of

*a*and

*b*are obtained as

*a =*41.000124 < 0 and

*b =*23.104 > 0. The bifurcation diagram generated by the MATCONT package (in MATLAB) is presented in Figure 5.5. It shows that the DFE is locally asymptotically stable for 7?o < 1 and the unique EE is locally asymptotically stable for 7?o > 1. This confirms the existence of a transcritical bifurcation in the system. The branch point indicates the changes in the stability of the equilibrium points. (The authors [16] have shown that backward bifurcation occurs for д>0.)

# Optimal Control Analysis

To study the optimal control of the *SEIR* model of Zika virus, the authors [16] included time-dependent controls in the model and explored the appropriate optimal strategy for controlling the virus. We use the following three control variables: (i) *щ* (f) represents the efforts on preventing Zika infections through bednets; (ii) *u _{2}{t)* represents the efforts on the treatment of Zika-infected individuals; and (iii)

*u*represents the efforts to control/ eliminate Zika-spreading mosquitoes through insecticides spray. Consider the objective functional as [62,87,92],

_{3}(t)

where *B,C,D,E* are the balancing cost factors due to scales and *a _{u}a_{2},* and a

_{3}denote respectively the weighting constants for making uses of bednets, which have the potential

FIGURE 5.5

Bifurcation diagram showing transcritical bifurcation in the system (5.14).

to reduce the spread of the disease (prevention); effective treatment activities, which include the efficacy of the drugs and encouraging patients to take their drugs timely; and availability of insecticides for spraying against the mosquitoes. The costs associated with prevention, treatment, and insecticide are taken to be of the non-linear form. Optimal controls *u,u _{2},* and М3 are to be determined such that

The model system with control is taken as

The necessary conditions that an optimal solution must satisfy are obtained from the maximum principle of Pontryagin et al. [92]. This principle converts (5.18)-(5.19) into a problem of minimizing pointwise the following Hamiltonian *H,* with respect to *щ,и _{2},* and

*щ*

where As_{h},A_{£h} ,A_{1h} ,A_{Rh} ,A_{Sv}*,X _{Ev},* and A,

_{v}, constitute the adjoint variables or co-state variables. The solution of the system is obtained by appropriately taking partial derivatives of the Hamiltonian (5.20) with respect to the associated state variables. The result is summarized in the following theorem [16]:

Theorem 5.1

Given optimal controls *u,u _{2},u_{3}* and solutions

*S*of the corresponding state system (5.18) and (5.19) that minimize

_{H},E_{H/}I_{H},R_{H},S_{V},E_{V},I_{V}*](щ,и*over Г, adjoint variables As,, / A

_{2},щ)_{f:},,, A/,,, A

_{R},

*, A*

_{(}_{Sl}., A

_{f}, , A,

*exist satisfy ing*

_{v}

where *i = S _{H},E_{H},I_{H},R_{Hr}S_{V},E_{V},I*

_{v}; with the transversality conditions

. . ShPhIv(A.e„ -A_{s}„)+ Ih (pShPh (Ae„ ~ A-sh ) + S_{v}p_{v}(A,_{Ev} -X_{Sv})) |

and Mi = min <1, max 0,-------------— 1

Я] |

Simulation results of a study [16] suggest that the best strategy to minimize the spread of Zika virus is to optimize all the three controls. The reduction of the disease can only be attained when attention is given to all the three controls. The activation of all the controls has a greater effect on minimizing the number of infected and exposed humans in the communities. Application of all the three controls is the best strategy to minimize the number of infected mosquitoes *I _{v},* which eventually can lead to the reduction of the spread of Zika virus. The control profiles suggest that control

*щ*be kept at a maximum of 100% for about 40 days and gradually reduced to 25% and then kept the same during the entire 120 days period. The control

*u*is maintained at 8% and then gradually decreased and maintained during the entire 120 days. The control

_{2}*щ*is kept at a maximum of 100% for 20 days and then decreased to 25%, which is maintained throughout the entire 120 days.