The Spreading of Epidemics through Networks
The spread of epidemics through a network is like how information is spread through a social network in that it is done through nodes have connections to each other, edges. However, in a disease spreading network, edges are considered “contacts” instead of friendships, because it may not only be people who you are friends with who spread the disease to you. Epidemic networks specifically start with a single person who contracts the disease. This person is known as “patient zero.”
From patient zero the disease is spread through a branching process. The branching process can be simplified in the following manner. Patient zero comes in contact with x, an unknown variable, amount of people. Of these people, there is a probability, p, that the disease will be contracted. Then these people come in contact with x amount of people with p probability of spreading the disease. This process can either continue indefinitely or it can die out depending on the values of x and p. If x is zero, then the disease will die out. If p = 1 and x continues forever than the disease will never go away.
In the real world there are not many situations where diseases do not die out do to technology and vaccines. The general public as a whole can also reduce the spread of disease through networks by increasing sanitation, which will lower the probability of contracting the disease in the first place and by decrease the amount of people they come in contact with.
A specific example where it is difficult to completely stop the spread of disease in the real world is in sexually transmitted diseases (STD’s). The predominant epidemic model for the spread of STD’s is the “susceptible-infectious-susceptible (SIS) model (**see end of post).” This model is dulled down to neglect many details, which are too difficult to compute, but essentially the model “has been proved successful for a long time (**see end of post).” STD’s are currently a huge problem and through further research we may one day be able to understand how they spread through networks better and will be able to terminate the spread of the epidemic.
**Source: http://guava.physics.uiuc.edu/~nigel/courses/563/Essays_2017/PDF/XiangyuSong.pdf