Two-sex logistic model for human papillomavirus and optimal vaccine

Abstract

We formulate a two-gender susceptible–infectious–susceptible (SIS) model to search for optimal childhood and catch-up vaccines over a 20-year period. The optimal vaccines should minimize the cost of Human Papillomavirus (HPV) disease in random logistically growing population. We find the basic reproduction number [Formula: see text] for the model and use it to describe the local-asymptotic stability of the disease-free equilibrium (DFE). We estimate the solution of the model to show the role of vaccine in reducing [Formula: see text] and controlling the disease. We formulate some optimal control problems to find the optimal vaccines needed to control HPV under limited resources. The optimal vaccines needed to keep [Formula: see text] are the catch-up vaccine rates of 0.004 and 0.005 for females and males, respectively; 100[Formula: see text] is needed to reduce [Formula: see text] to its minimum value. To reduce the expenses for HPV disease and its vaccines, we need 100[Formula: see text] childhood vaccines (both genders) for the first 13–14 years and then gradually reduce the vaccine to reach [Formula: see text] at year 20. For adults (both genders), we need maximum rates (one) for the first 9 years, then [Formula: see text] for the next 3–4 years before reducing gradually to zero rate at year 20. Although the childhood vaccines provide very early protection strategy against HPV, its time to control HPV is longer than that for adult vaccines. Thus, full adults’ only vaccines for enough period is a viable choice to control HPV at minimal cost and short time.