In this gallery, we consider models in epidemiology, capturing the dynamics of various infectious diseases including influenza, measles, and COVID-19. Once simulated, we visualize the infection dynamics across different choices of network.

Diffusion across networks is a widespread phenomena across many areas including the spread of viruses in computer networks, the dissemination of rumors across social networks, and the spread of traffic in transportation networks. Here, we explore the spread of infectious disease across social contact networks. First, we consider the simple case in which individuals can only be either susceptible or infected without any long-lasting immunity. This type of infectious disease is similar to the common-cold and influenza . Later, we generalize the infection to more complex settings, considering for example recovered individuals as seen in measles or asymptomatic spreaders as seen in COVID-19.

## Susceptible-Infected-Susceptible

Most models of epidemiology are the compartmental models of infectious disease. In these models, individuals lie in compartments or states, representing a "status" in the population. Consider for example the Susceptible-Infected-Susceptible model (SIS), where individuals are either susceptible or infected. SIS is the simplest setting since there are only two states for each individual and there are no other complex interactions in the population (for example, SIS does not take into account asymptomatic spreaders).

We simulate SIS with the following procedure. At graph initialization, each individual is represented as a node and is assigned unique parameters governing their infection dynamics (ie. infection, susceptibility, and healing rates). Next, at each timestep for each individual we compute an infection pressure as a linear combination of the individual's susceptibility and the infection pressure of its infected neighbors. Once an individual exceeds the threshold for infection, they become infected and start spreading the disease to their neighbors. An infected individual recovers as time increases according to their assigned healing rate.

Below, we visualize SIS across time in power law graphs and Watts-Strogatz small-world graphs. In each graph, we set the number of nodes to 200, initial rate of infection to 10%, and simulate across 50 timesteps. Nodes are colored according to their infection pressure at the current timestep, and increase in size once infected.

SIS Power Law

SIS Watts-Strogatz

We also explore infection dynamics across well-studied social networks, including a ring-of-cliques network and a relaxed caveman network. The former arranges nodes in cliques and connects them in the shape of a ring, while the latter arranges nodes in cliques and randomly rewires edges to connect the cliques.

SIS Ring Of Cliques

SIS Relaxed Caveman

## Asymptomatic Spreaders

We can add layers of complexity to the base SIS model with the addition of asymptomatic spreaders (SAIS). The addition of the asymptomatic state allows nodes to spread the disease without being labelled as infected. Thus, in the resulting networks, a node may have enough infection pressure to become infected, yet still remains small indicating that it is an asymptomatic carrier and not a symptomatic known infected. Here, we visualize SAIS processes on similar networks where 50% of the nodes are asymptomatic, while the other 50% are symptomatic.

SAIS Power Law

SAIS Watts-Strogatz

SAIS Ring Of Cliques

SAIS Relaxed Caveman