Mathematical modelling can help us understand the propagation of epidemic diseases.

We are a group of physicists from the Science School (Facultad de Ciencias) at UNED (Universidad Nacional de Educación a Distancia, Madrid) attempting to get a deeper understanding of the mathematical processes governing the expansion of epidemic diseases.

The confined-SIR model

We intend to characterize the effects of confinement on the expansion of an epidemic disease through a very simple geometric model, the so-called confined-SIR model. Agents move randomly on a lattice, although they can never wander too far away from their homes. In the simplest version of the model, the homes are randomly distributed and all agents are allowed the same wandering radius, which we may call R. Then, one (or a few) infected agents are introduced in the lattice. When an infected agent coincides with a healthy (susceptible) one (this means that their wandering circles overlap in some point), a contagion may take place. Also, infected agents may recover, with a certain probability (per unit time), becoming immune in the future. These are the basic tenets of a SIR model (susceptible-infected-recovered).

The next two figures show the locations of the homes (blue dots), while the blue circles show the wandering region (also called confined-region) of each agent. In black, we show the effective links between agents.

Mobility of the agents, percolation threshold and recovery probability

When the size of the wandering radius R is increased (so the mobility of the agent is larger), more effective links among agents take place, and consequently the spread of an epidemic burst will affect to a larger percentage of the population. There is a critical radius (the so-called percolation threshold), where a phase transition occurs and the pandemic become out of control. However, even at radios above of the percolation threshold, if the recovery probability is large enough a certain percentage of the population could not be infected. In the next videos we show these different situations.

Vaccination Program

But, even when the mobility of the agents is above the percolation threshold a vaccine could avoid the epidemic outburst. However, if we don’t have enough doses of vaccines for all population, which should be the vaccination programme to maximize the chances of halting the outburst. We have tests different schedules of vaccination in our confined-SIR model, which are only based on different properties. For example, the most simple vaccination schedule is merely to select randomly the individuals. Another simple scheme could be to select the agent with larger number of effective connections. Also, we can define a fragmentation degree associated to an agent, which intuitively inform of the reduction of the local connections in its neighbourhood when it is removed. However, the most promising observable is, nonetheless, the betweenness-centrality associated to each agent, defined as the fraction of the total number of geodesics which go through a particular agent.

According to our results, a vaccination program based on the betweenness-centrality property is the most effective to control the pandemic, but in practice, many other issues must be considered in this situation, such as the health conditions of the people or other social and politic subjects. However, we expect that some of our conclusions can be interesting for researchers in epidemic expansion and lead, in combination with insights from other specialists, to help to control the effects of present and future epidemic outbursts.

Our results have been published at Effects of confinement and vaccination on an epidemic outburst: A statistical mechanics approach, Physical Review E 104, 034310 (2021), also available at ArXiv.

Our code is freely available at this GitHub repository .