by Dirk Brockmann

This explorable illustrates the mechanism of herd immunity. When an infectious disease spreads in a population, an individual can be protected by a vaccine that delivers immunity. But there's a greater good. Immunization not only projects the individual directly. The immunized person will also never transmit the disease to others, effectively reducing the likelihood that the disease can proliferate in the population. Because of this, a disease can be eradicated even if not the entire population is immunized. This population wide effect is known as herd immunity.

Press Play and keep on reading....

This is how it works

This explorable is actually a set of four similar explorables, all of which model the spread of a disease in a population with \(N\) individuals. An individual can be Susceptible, which means the person can acquire the disease and become Infected. Once infected, the person can transmit the disease to other susceptibles. An infected individual remains infectious for some time, recover subsequently, and become susceptible again.

This model is known as the SIS-model, one of the simplest dynamical models for infectious disease dynamics. If, on average, an infected person transmits to more than one other person during the infectious period, the disease will reach an endemic state in the population in which new infections and recoveries balance. You may also want to check the explorables Critical HEXersize and Epidemonic for more information on epidemic models.

Vaccination is modelled this way: All individuals can spontaneously decide to vaccinate at a certain rate such that in equilibrium a fraction \(P\) of the population is vaccinated.

Both, vaccine uptake and transmissibility of the disease can be controlled with a slider.

The system is initially fully susceptible with a few infected individuals randomly scattered into the population.

Model 1: The mixed population

In this version of the SIS-model, individuals move around randomly and interact with only with other individuals in their proximity. Transmissions occur by face-to-face contacts.

When you press play, the number of infected people will increase until a dynamic equilibrium is reached. Now turn the vaccine uptake up until you find the point at which the disease will be eradicated. The higher the transmissibility, the higher the critical threshold for the vaccine uptake.

Model 2: The static network model

In this model, individuals are stationary and linked forming a heterogeneous network in which some individuals possess more links than others. The disease is only transmitted across links in the network. Some nodes have more connections than others and are more likely to become infected and spread the disease.

As you increase the vaccine uptake, you should see that near the critical point pockets of infections exist, whereas other areas in the network remain disease free.

Model 3: The dynamic network model

In this model, the population is also connected by links network. Transmissions only occur between linked individuals. However, these links change over time. Individuals may rearrange their connections, cut old ones and establish new ones.

This particular "rewiring" generates little groups that are densely connected and individuals move between them.

Model 4: The spatial lattice model

This model is a bit more abstract and has been used to investigate spatial aspects of disease dynamics. Here we have a \(40\times 40\) square lattice. Every lattice site can be in one of the three states S, I and V. Transmissions only occur between neighboring lattice sites.

Further information


Related Explorables:


The SIRS epidemic model

Thrilling Milling Schelling Herings

Swarming behavior of animals that seek their kin

T. Schelling plays Go

The Schelling model

Berlin 8:00 a.m.

The emergence of phantom traffic jams

Facebooked Flu Shots

Network vaccination

Echo Chambers

A model for opinion dynamics