Mathematical modeling and infectious diseases
Mathematical modeling is a powerful tool used to predict the spread and impact of infectious diseases. It employs complex algorithms and statistical methods to assess how diseases might transmit through populations, enabling health officials and epidemiologists to hypothesize about potential outbreaks and the effectiveness of public health interventions. The origins of this approach trace back to the early 20th century, notably with Sir Ronald Ross, who emphasized the importance of a mathematical framework in epidemiology.
Key factors in these models include reproduction rates of the disease, susceptibility within the population, the typical age of infection, and life expectancy, along with various demographic and environmental considerations. While mathematical models provide valuable insights, they also come with limitations; the accuracy of predictions relies heavily on the assumptions made and the dynamic nature of diseases and human behavior. Nonetheless, these models play a critical role in informing vaccination strategies and guiding public health policy, ultimately aiming to mitigate the spread of infections and protect communities.
Subject Terms
Mathematical modeling and infectious diseases
- ALSO KNOWN AS: Epidemic modeling, mathematical epidemiology
Definition
Mathematical modeling is the use of a complex mathematical formula or algorithm to predict the outcome of a disease, to hypothesize its likely spread, and to determine what public health actions could limit its transmission.
Epidemiologists, healthcare workers, public health officials, and the general public all have a vested interest in whether and how a particular infectious disease will spread, in what portions of the population it will affect, and in whether it will turn into an epidemic. Scientists have created sophisticated statistical, mathematical, and computer programming methods to develop hypotheses as to how a disease might spread through a population and whether any factors are likely to inhibit its growth. Scientists also run mathematical calculations to determine if and how vaccination against a disease will affect its spread.
History
In 1911, British doctor Sir Ronald Ross was one of the first to determine that mathematics could be used to study disease transmission. He believed that epidemiology, which is the study of the variation of disease from time to time and place to place, to be considered scientifically, it must be considered mathematically. Doing so, he argued, is the only way to apply careful reasoning to epidemiology’s methods. Accordingly, Ross attempted to apply the law of mass action, a mathematical model used in chemistry, to explain how an epidemic was transmitted. Another mathematical model of epidemics was constructed in the 1920s by Lowell Reed and Wade Hampton Frost of Johns Hopkins University. Since this time, scientists have applied many mathematical models from different disciplines and have created a few new models to explain disease transmission.
Basic Ideas
An infinite number of assumptions can be made and many variables can be factored into a model to predict the transmission of an infectious disease. However, the following four fundamental ideas are most often used in mathematical modeling:
Reproduction. The average number of people a person with a given disease would infect if no one had immunity to that disease
Susceptibility. The proportion of people in the population who do not have immunity and do not have the disease
Age. The average age at which a person in the general population is likely to get the disease
Life expectancy. The average life expectancy of the population in the transmission path of the disease
Modelers add many other factors to their calculations, such as the proportion of children to adults in the population, how close people live to one another in the area, and whether or not school is in session. They may also add real-world examples of how another, possibly related, epidemic occurred to ensure their model is as close to reality as possible. A person’s contact pattern may also need to be considered. For example, an infectious disease would have a very different pattern of spreading if two different people contracted the disease: one who lived alone and went immediately to the hospital to be isolated, and another who died at home, undiagnosed, after exposing his or her multigenerational family (as happened in Canada with the introduction in 2003 of the virus that causes SARS).
Limitations
In any type of modeling, the model created is only as good as the assumptions with which the creator populated the model. If a model is correctly constructed mathematically, yet the results are still out of line with the disease’s observed patterns, the assumptions may need to be revisited. Real-world issues also come into play; a mathematical model is, at its best, a highly likely scenario. Genetic mutations of a disease or a shift in human behavior can alter how the scenario actually evolves. These factors, for example, can lead to the development of a vaccine for influenza that seemed, at the beginning of flu season, to cover all likely influenza viruses but, in fact, did not cover all the actual viruses that were active that year, or can lead to the development of a vaccine that did not account for the particular virulence of a given strain.
Impact
By using well-reasoned and highly accurate mathematical models to predict the spread of infectious diseases, scientists using mathematical modeling inform public health policymakers and public health workers of the factors that affect the spread of disease. Mathematical models can determine what diseases are likely in a particular population and thus can help determine what types of vaccines are necessary to prevent the occurrence or recurrence of a given disease.
Mathematical models for infectious diseases were revised after the COVID-19 pandemic in the early 2020s. New mathematical models showed significant disparities associated with demographic variables. These variables included population density, household characteristics, and urbanicity.
Bibliography
Castillo-Chavez, Carlos, ed. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory. New York: Springer, 2002.
Daley, D. J., and J. Gani. Epidemic Modeling: An Introduction. Reprint. New York: Cambridge University Press, 2005.
Diekmann, O., and J. A. P. Heesterbeek. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation. New York: John Wiley & Sons, 2000.
Keeling, M. J., and L. Danon. “Mathematical Modelling of Infectious Diseases.” British Medical Bulletin 92 (2009): 33-42.
Keeling, Matt J., and Peiman Rohani. Modeling Infectious Diseases in Humans and Animals. Princeton, N.J.: Princeton University Press, 2007.
Ma, Stefan, and Yingcun Xia, eds. Mathematical Understanding of Infectious Disease Dynamics. Hackensack, N.J.: World Scientific, 2008.
Ma, Zhien, and Jia Li, eds. Dynamical Modeling and Analysis of Epidemics. Hackensack, N.J.: World Scientific, 2009.
Richard, Dianelle M. "What's Next: Using Infectious Disease Mathematical Modelling to Address Health Disparities." International Journal of Epidemiology, Feb. 2024, doi.org/10.1093/ije/dyad180. Accessed 4 Feb. 2025.
Sokolowski, John A., and Catherine M. Banks, eds. Principles of Modeling and Simulation: A Multidisciplinary Approach. Hoboken, N.J.: John Wiley & Sons,2009.