One of the interesting realizations of this morning came when I visited my professor, Dr Silal, for the infectious disease modelling course I am currently doing.
We have a static – fixed – sexual network that is “straw-man” example of how HIV can spread within a sexual network. In this network individuals are represented by nodes and the linkages between them represent epidemiological meaningful routes of transmission, in our case, sexual intercourse.
Compartmental models of infectious disease that are underpinned by differential equations, assume movement of individuals from susceptible (unifected) to infected states is proportional to products of the prevalence of infection and the fraction of the population that is susceptible. Here people mix randomly and each person has the possibility of contacting any other person. The network models, however, recognize that each person may not randomly mix with all other people, but has finite number of contacts. There is empirical evidence for this, given that sexual networks are well defined in many cases, as was with the emergence of HIV in South Africa – sexual networks of white gay and bisexual men in the 1980 likely had little overlap with those of black men who have sex with men (MSM). When one recognizes the importance of sexual networks, the network model can serve as tool to understand disease dynamics.
An important aspect of the dynamics is variability that is inherent in the number of sexual contacts. On average, we could say that out 100 days, a person will have 5 days of sexual activity with another person in the network, assuming 0.05 sexual contacts per day as the rate for this network. An approach to then building a model would be to say that out 100 days of the model, each person in the network will be sexually active for five of those days and abstain from sex for the remaining 95. Even if we chose those 5 days at random from the sequence of 100 days, we would still omit one important aspect of this type of dynamic model – stochasticity. The overall – average – behavior of the simulation over 100 days will include 5 days of sexual activity for each person, but there is nothing to guarrantee that any one run of 100 days will not include more or fewer days of sexual activity. In fact, if we take sexual contact to be a Bernoulli distributed random variable and then assume that a run of 100 days represents 100 trials, we have a binomial distribution for a new variable – the number of days of sexual activity. This variable must then have a variability surrounding the average given by the formula you have probably seen before:
The variance gives a measure of the variability and through mathematical modelling we can attempt to see what this variance has on the overall outcome of the system. In our case we are interested in knowing how long does it take for all individuals in the network to become sexually active.