We introduce a two sex age-structured mathematical model to describe the dynamics of HPV disease with childhood and catch up vaccines. We find the basic reproduction number ($R_0$) for the model and show that the disease free equilibrium is locally asymptotically stable when $R_0\le 1$. We introduce an optimal control problem and prove that optimal vaccine solution exists and is unique. Using numerical simulation, we show that 77% childhood vaccination controls HPV disease in a 20 years period, but 77% catch up vaccine does not. In fact, catch up vaccine has a slight effect on HPV disease when applied alone or with childhood vaccine. We estimate the optimal vaccine needed to control HPV in a 25 year period. We show that reducing the partners between youths and adults is an effective way in reducing the number of HPV cases, the vaccine needed and the cost of HPV. In sum, we show that choosing partners within the same age group is more effective in controlling HPV disease than providing adult catch up vaccination.

我们引入了两性年龄结构的数学模型来描述儿童时期HPV疾病的动态并追赶疫苗。我们找到基本的复制编号（${\mathrm{[R}}_0$）的模型，并证明当以下情况时，无病平衡局部渐近稳定 ${\mathrm{[R}}_0\le \mathrm{1个}$。我们介绍了最佳控制问题，并证明最佳疫苗溶液存在并且是唯一的。通过数值模拟，我们显示77％的儿童接种疫苗可在20年内控制HPV疾病，但77％的追赶疫苗却不能。实际上，单独使用或与儿童疫苗一起使用时，追赶疫苗对HPV疾病影响不大。我们估计在25年内控制HPV所需的最佳疫苗。我们表明，减少年轻人和成年人之间的伴侣是减少HPV病例数，所需疫苗和HPV成本的有效方法。总而言之，我们表明，选择相同年龄组内的伴侣比控制成人疫苗更有效地控制HPV疾病。