Primes Get Together

Mathematics has long been an important tool in infectious disease epidemiology. I will provide a brief overview of compartmental models, the dominant framework for modeling disease transmission, and then contact network epidemiology, a more powerful approach that applies bond percolation on random graphs to model the spread of infectious disease through heterogeneous populations. I will derive important epidemiological quantities using this approach and provide examples of its application to issues of public health. Background Infectious diseases can have devastating impacts on human life and welfare. In the last three years, SARS, avian influenza, simian foamy virus, and monkeypox have jumped from animals into human populations. The uneven spread of SARS worldwide poignantly demonstrated that containment is possible, but depends critically on appropriate and aggressive management. With the growing threats of newly emerging diseases and bioterrorism, strategies to rapidly and effectively control outbreaks are vital to public health. Mathematics is an invaluable epidemiological tool. It allows public health officials to conduct virtual experiments that would be practically unfeasible or unethical. Controlled experiments to evaluate the efficacy of control strategies are impossible in practice as we cannot intentionally introduce disease into populations or withhold potentially lifesaving interventions for the sake of scientific study. Mathematical models of disease transmission dynamics enable systematic evaluation of strategies such as vaccination and quarantine, and thereby provide a way around this difficulty. In the 18 century, Daniel Bernoulli – the son, nephew and brother of mathematicians Johann, Jacob and Nicolaus II Bernoulli, respectively – made one of the first great mathematical contributions to infectious disease control [1]. While formally trained in medicine, Bernoulli is known for his research in biomechanics, hydrodynamics, economics, and astronomy. He also played an important role in the eradication of smallpox from Europe, which was likely introduced there in the early 16 century, and was endemic (maintained constantly) by the 18 century. Variolation is an inoculation technique whereby a scab or pus from an individual with a mild smallpox infection is introduced into the nose or mouth of healthy individuals. This practice began as early as 1000 AD in China and India and was introduced into England in 1717, where it was initially controversial. While variolation reduced the mortality probability of infected individuals from 30% to 1% [2], there was a small chance that the procedure would lead to death from a full-blown case of smallpox. Bernoulli developed a mathematical model with which he argued that the gain from variolation in life expectancy through the eradication of smallpox far outweighed associated risks [1, 3]. Assuming that all individuals had a one in n chance of catching smallpox, and a one in m chance of dying from an infection, he derived the following