Numerical simulation of agent-based model describing human migration

HTLV-I是导致ATL的逆转录病毒。HTLV-I的传播途径是（i）由受感染的母亲传给新生婴儿;（ii）由受感染的男性（丈夫）经长期性交传给女性（妻子）;及（iii）由受感染的女性（妻子）传给男性（丈夫）。在参考文献的第一篇论文中，我们考虑了HTLV-I感染的离散时间数学模型中的年龄结构的人口动态。从该模型出发，导出了HTLV-I灭绝的充分必要条件。为了定量地描述区域间迁移，在文献的第二篇论文中，我们构建了两个随机的基于主体的模型，该模型由大量的主体在一个离散的有界区域中重新定位以获得更高的效用。在一个模型中，我们假设效用被定义为代理密度的递增仿射函数。在另一个模型中，我们假设效用等于代理密度的凹二次函数。当主体数量和时间变量趋于无穷大时，我们获得了模型行为的估计。在参考文献的第三篇论文中，我们从数学上证明了，如果移动所需的努力很大，那么模型在以下意义上彼此接近：主方程具有周期边界条件的混合问题具有唯一的解，该解非常接近福克-普朗克方程的解，其中努力是一个社会动力学量，表示移动中产生的成本。

HTLV-I is a retrovirus that causes ATL. The transmission routes of HTLV-I are (i) from infected mothers to their new born babies, (ii) from infected males (husbands) to females (their wives) by long-term sexual intercourse, and (iii) from infected females (wives) to males (their husbands). In the first paper of the references we consider the population dynamics of HTLV-I infection in a discrete-time mathematical model incorporating an age structure. The necessary and sufficient condition for the extinction of HTLV-I is derived from the model. In order to quantitatively describe interregional migration, in the second paper of the references, we construct two stochastic agent-based models that consist of a large number of agents relocating to obtain higher utility in a discrete bounded domain. In one model we assume that the utility is defined as an increasing affine function of the density of agents. In the other model we assume that the utility is equal to a concave quadratic function of the density of agents. We obtain estimates for the behavior of the models when the number of agents and the time variable tend to infinity. In the third paper of the references we mathematically prove that if the effort required in moving is large, then the models are close to each other in the following sense : the mixed problem with the periodic boundary condition for the master equation has a unique solution that is very close to a solution of that for the Fokker-Planck equation, where the effort is a sociodynamic quantity that represents a cost incurred in moving.