Cervical Cancer and HIV Diseases Co-dynamics with Optimal Control and Cost Effectiveness

The deterministic model for co-infection of cervical cancer and HIV (Human Immunodeficiency Virus) diseases is formulated and rigorously analyzed. The optimal control theory is employed to the model to study the level of effort is needed to control the transmission of co-infection of cervical cancer and HIV diseases using three controls; prevention, screening and treatment control strategies. Numerical solutions show a remarkable decrease of infected individuals with HPV (Human Papilloma Virus) infection, cervical cancer, cervical cancer and HIV, cervical cancer and AIDS (Acquire Immunodeficiency Syndrome), HIV infection and AIDS after applying the combination of the optimal prevention, screening and treatment control strategies. However, Incremental Cost-Effective Ratio (ICER) shows that the best control strategy of minimizing cervical cancer among HIV-infected individuals with low cost is to use the combination of prevention and treatment control strategies.


Introduction
Cervical cancer is a major cause of morbidity and mortality among women in sub-Saharan countries and about 70% of cervical cancers are caused by Human Papillomavirus (HPV) types 16 and 18 which are transmitted sexually through body contact. Some studies have shown that HIV-infected women after being infected with HPV infection have a high risk to progress to HPV-related cervical diseases and invasive cervical cancer than women without having HIV infection [1,2,3].
The aim of this work is to study the effect of incorporating three optimal control strategies to the co-infection model of cervical cancer and HIV diseases. In [15] formulated coinfection model of cervical cancer and HIV diseases but the findings of this paper differ from the work presented in [15] because the co-infection model incorporates three optimal control strategies; prevention, screening, and treatment.

Optimal Control Analysis
Here, we introduce optimal control strategies to the coinfection model of cervical cancer and HIV disease as presented in [15]. The co-infection model in [15] is developed as follows: The total population of individuals at any time t , denoted as N is categorized into ten compartments according to the  The rates of transferring between different compartments are as described in Table 1.
The objective functional is defined as follows where i A and j B for The necessary conditions that an optimal must satisfy come from the Pontryagin's maximum principle [8]. This principle converts ( ) 2 and ( )   hpu  pu  hps  ps  hc  c  hl   S  I  I  I  I  I  I  I Proof. Corollary 4.1 of Freming and Riches [9] gives the existence of an optimal control due to the convexity of the integrand J with respect to 1 u , 2 u and 3 u a priori boundedness of the state solutions and the Lipchitz property of the state system with respect to the state variables. Differentiating Hamiltonian functions with respect to state variables gives differential equations governing the adjoint variables as follows;

Numerical Solutions
In order to obtain optimal control solutions, the optimality system which consists of two systems namely; the state system and the adjoint system is solved. In solving state equations using forward fourth order Runge-Kutta, an initial guess of all controls over time are made and the initial value of state variables are introduced. Having the solution of state functions and the value of optimal controls, the adjoint equations are solved using backward fourth order Runge-Kutta by using transversality condition. In this simulation, the weights are chosen be 1 80 Other parameter descriptions and values used in getting numerical results of the co-infection model of cervical cancer and HIV diseases are presented in Table 1. Control strategies are formed and studied numerically as follows

Strategy I: Combination of Prevention and Treatment Control Strategies
The combination of prevention control strategy 1 u and treatment control 2 u are used to optimize objective functional while setting screening control 3 u equal to zero.
The results show that applying optimal prevention and treatment control strategies; the population of susceptible individuals increases whereas thepopulation of all infected compartments decreases as illustrated in Figure1

Strategy II: Combination of Prevention Control Strategy and Screening Strategy
The combination of prevention control strategy, 1 u , and screening control strategy, 3 u , are used to optimize objective functional while setting treatment control strategy, 2 u , equal to zero. Results illustrate that the population of susceptible individuals increases (see Figure 2: A) while the population of infected individuals decreases (see Figure 2: B-J).

Strategy III: Combination of Screening and Treatment Control Strategies
The combination of treatment control 2 u and screening

Strategy IV: Combination of Prevention, Treatment and Screening Control Strategies
The combination of prevention control strategy, treatment control strategy 2 u , and screening control strategy 3 u are used to optimize objective functional. Results illustrates that the population of susceptible individuals increases as shown in Figure 4: A while the population of infected individuals decrease compared with other combination of control strategies as shown in Figure 4: B -J.

Cost-effectiveness Analysis
Here, Incremental Cost Effectiveness Ratio (ICER) is used to quantify the cost-effectiveness of different strategies. This approach is useful to understand which strategy saves a lot of averted species while spending low cost. This technique is needed to compare more than one competing interventions strategies incrementally, one intervention should be compared with the next less effective alternative [10]. The ICER formula is given by Difference in intervention cost ICER= Difference in the total number of infection averted The total number of infection averted is obtained by calculating the difference between the total number of new cases of individuals having HPV infection without control and the total number of new cases of individuals having HPV infection with control.
Comparing strategy III and strategy II, ICER of strategy II is less than ICER of strategy III. Hence strategy III is more costly and less effective than strategy II. Thus, strategy III is omitted and ICER is recalculated. Comparing strategy II and strategy I, ICER of strategy I is less than ICER of strategy II. Thus, strategy II is omitted and the ICER is recalculated. By comparing strategy I and strategy IV, ICER of strategy I is less than ICER of strategy IV. Therefore, strategy IV is dropped and strategy I is considered.
Thus, according to Incremental Cost Effectiveness Ratio analysis, the combination of optimal prevention and treatment control strategies is the best way of minimizing cervical cancer among women with or without HIV infection in our community following the combination of all control strategies.

Conclusion
This paper designed and analyzed a deterministic model for co-infection of cervical cancer and HIV diseases. The optimal control theory was employed to the main model and analyzed using Potrayagin's Maximum Principle. The different combination of three optimal control strategies; prevention, screening, and treatment were studied numerically. Also, cost-effectiveness analysis was performed and the following results were obtained (a) The combination of optimal screening and treatment strategies is not much powerful way of controlling HPV infection and cervical cancer in the community compared with other combination of optimal control strategies as shown in Figure 3. Also, applying the combination of all control strategies (prevention, screening, and treatment) is the best way of minimizing co-infection of cervical cancer and HIV diseases in a community as shown in Figure 4. (b) In the case of Incremental Cost Effectiveness Ratio analysis, the combination of optimal prevention and treatment strategies is the most cost-effective control to minimize cervical cancer among women with or without HIV infection. The second is the combination of all optimal control strategies; prevention, screening, and treatment. The third is the combination of optimal prevention and screening control strategies.