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 R0 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 R0 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 R0 ≤ 1 are the catch-up vaccine rates of 0.004 and 0.005 for females and males, respectively; 100% is needed to reduce R0 to its minimum value. To reduce the expenses for HPV disease and its vaccines, we need 100% childhood vaccines (both genders) for the first 13–14 years and then gradually reduce the vaccine to reach 0% at year 20. For adults (both genders), we need maximum rates (one) for the first 9 years, then 0.2 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.

中文翻译：

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人乳头瘤病毒的两性逻辑模型和最佳疫苗

我们制定了一个两性易感-感染-易感 (SIS) 模型，以在 20 年的时间里寻找最佳的儿童疫苗和补种疫苗。最佳疫苗应最大限度地降低随机后勤增长人口中人乳头瘤病毒 (HPV) 疾病的成本。我们找到基本的再生数R0用于模型并用它来描述无病平衡（DFE）的局部渐近稳定性。我们估计模型的解以显示疫苗在减少R0和控制疾病。我们制定了一些最佳控制问题，以找到在有限资源下控制 HPV 所需的最佳疫苗。需要保持的最佳疫苗R0 ≤ 1分别是女性和男性的追赶疫苗接种率 0.004 和 0.005；100%需要减少R0到它的最小值。为了减少 HPV 疾病及其疫苗的费用，我们需要 100%前 13-14 岁儿童疫苗（男女），然后逐渐减少疫苗接种量以达到0%在第 20 年。对于成年人（男女），我们需要前 9 年的最高费率（一个），然后0.2在接下来的 3-4 年中，在第 20 年逐渐降至零。虽然儿童疫苗提供了非常早期的 HPV 保护策略，但其控制 HPV 的时间比成人疫苗长。因此，在足够长的时期内只接种成人疫苗是一种以最低成本和短时间控制 HPV 的可行选择。