An assessment of the contact rates between individuals when movement is modelled by a correlated random walk

摘要

Abstract 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, $$I(N-I)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>I</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> (where $$I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi></mml:math> is the number of infected at a given time and $$N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> the total number of individuals). Instead, we find that a contact rate proportional to $${\left[I(N-I)\right]}^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mfenced><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>I</mml:mi><mml:mo>)</mml:mo></mml:mfenced></mml:mrow><mml:mi>α</mml:mi></mml:msup></mml:math> with $$0&lt;\alpha \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> is a better qualitative fit, where $$\alpha$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> 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.