Let Therefore where F X denotes the rate of
Therefore,where F(X) denotes the rate of production of new people who get infected and V(x) represents the motion of individuals in compartments, which gives us
Now, the derivative of F and V calculated at an equilibrium point E0 gives matrices f and v of order 6 × 6 defined aswhere
Thus, the basic reproduction number R0 is
On solving the set of Eq. (1), we get an endemic equilibrium point as where
Stability analysis The disease-free equilibrium is stable if all the eigenvalues of the Jacobian matrix of the system (1) have negative real parts.
Optimal control For HCV-Alcohol, an optimal control model is analyzed to study the impact of rehabilitation for prevention of Hepatitis C. The objective of the model is to minimize the number of infected individuals in the time interval [0, T]. Considering the objective functions aswhere and A1, A2, A3, A4, A5, A6 denotes non-negative weight constants for the compartments S, L, H, A, C, R, respectively, and w1 is the weight constant for the control variable u1. The weight W1 is a constant parameter for u1 will standardize the optimal control condition. The relation between state variables and adjoint variables is To determine the minimal value of the Lagrangian, defining the Hamiltonian H for the control problem as,
The necessary conditions for Lagrangian function to be optimal for control is The property of control space u gives:
Sensitivity analysis Shah et al. (2017b) has described that PPDA sensitivity is the degree to which the change in input parameters affects the spread of disease. The normalized sensitivity index of the parameters is computed by using the formula: where θ denotes the model parameter (Shah et al., 2017b). Now, the sensitivity analysis for all model parameters is discussed in Table 2.
Numerical simulation According to all compartment in Fig. 2, within 30 days, 8 susceptible individual start liquoring. After 140 days, 3 liquor individual who have started liquoring will go back to the susceptible. 6 susceptible individuals will get habits of drinking high amount of liquor in 48 days. After 147 days, 3 high liquoring individuals will go back to the susceptible. During 56 days, 4 susceptible individuals get recovered. Within 86 days, 5 lower liquoring individuals becomes high liquoring and after 164 days high liquoring individuals get back to low liquoring. 8 lower liquoring will become chronic in 35 days. During 76 days, 5 lower liquoring individuals get recovered. Fig. 3 shows that initially 15% rehabilitation control is advocated to alcoholics. Thereafter decreases eventually individuals becomes chronic free. Fig. 4 suggests that 15% rehabilitation as a control reduces low liquored individuals by 20.7%. It can be seen from Fig. 5 that approximately 5 high liquored individuals are controlled through 15% rehabilitation. Fig. 6 shows that out of the total population 12% are low liquored whereas 9% of individuals are high liquored. Due to which 16% of individuals suffers from acute and 32% suffers from chronic diseases. Only 19% of individuals gets cured from this harmful attack.
Conclusion In this research, we developed a SIRS model for transmission of the hepatitis C virus (HCV) under effect of liquoring. The system of non-linear ordinary differential equations is formulated. Basic reproduction number R0 is computed using the next generation matrix approach. The stability of the model is worked out at the equilibrium points. Model analysis shows that the disease-free equilibrium is both locally and globally asymptotically stable. Sensitivity analysis with respect to key parameters of R0 indicates that control strategies should target reduction of usage of alcohol among individuals with HCV is to prevent or delay HCV disease progression. The control in our model is rehabilitation center which help people divert from high liquoring to low liquoring. Sensitivity analysis in our model showed about 4.15% of high liquoring individual opt for low liquoring due to the effect of rehabilitation. Numerical simulation has been carried out to show the impact of control on different compartment. With the impact of rehabilitation, 20.7% individual reduces their alcohol intake in low liquoring compartment while 5.38% individual were able to reduce their alcohol intake in high liquor compartment. This shows that earlier we start the rehabilitation, better and greater are the chances for individual to decrease their alcohol intake and subsequent HCV transmission.