Understanding how individuals come into contact with each other is important in many fields from biology and ecology to robotics and physics. Interaction dynamics are central in understanding how information is spread between agents, how disease spreads through a population, and how group movement or behaviour occurs. However, in many applications, the underlying mode of movement is not considered, and instead, contacts are considered a fraction of all possible contacts amongst a population. This gives rise to the mass-action law which in turn implies a negative quadratic relationship between contacts and individuals. Here we consider how a simple but often used movement model, the correlated random walk, affects the contact rate in a standard Susceptible-Infection (SI) epidemiological model. Via extensive simulation, we show that the contact rate is not always well described by the assumed negative quadratic relationship, \(I(N-I)\) (where \(I\) is the number of infected at a given time and \(N\) the total number of individuals). Instead, we find that a contact rate proportional to \({\left[I(N-I)\right]}^{\alpha }\) with \(0<\alpha \le 1\) is a better qualitative fit, where \(\alpha\) depends upon parameters such as the straightness of the movement and the density of individuals. We highlight that the expected contacts at low densities increase with straight line movement, whereas, at high densities, they increase with more random movement.

了解个体如何相互接触对于从生物学和生态学到机器人学和物理学的许多领域都很重要。交互动力学对于理解信息如何在主体之间传播、疾病如何在人群中传播以及群体运动或行为如何发生至关重要。然而，在许多应用中，不考虑潜在的移动模式，而是将接触视为群体中所有可能接触的一小部分。这就产生了群体行动法则，而群体行动法则又意味着联系人和个人之间存在负二次关系。在这里，我们考虑一个简单但常用的运动模型（相关随机游走）如何影响标准易感感染（SI）流行病学模型中的接触率。通过大量的模拟，\(I(NI)\)（其中\(I\)是给定时间的感染人数，\(N\)是总人数）。相反，我们发现与\({\left[I(NI)\right]}^{\alpha }\)和\(0<\alpha \le 1\)成正比的接触率是更好的定性拟合，其中\(\alpha\)取决于运动直线度和个体密度等参数。我们强调，低密度下的预期接触会随着直线移动而增加，而在高密度下，预期接触会随着更多的随机移动而增加。