Generalized differential equation compartmental models of infectious disease transmission

Abstract

For decades, mathematical models of disease transmission have provided researchers and public health officials with critical insights into the progression, control, and prevention of disease spread. Of these models, one of most fundamental is the SIR differential equation model. However, this ubiquitous model has one significant and rarely acknowledged shortcoming: it is unable to account for a disease’s true infectious period distribution. As the misspecification of such a biological characteristic is known to significantly affect model behavior, there is a need to develop new modeling approaches that capture such information. Therefore, we illustrate an innovative take on compartmental models, derived from their general formulation as systems of nonlinear Volterra integral equations, to capture a broader range of infectious period distributions, yet maintain the desirable formulation as systems of differential equations. Our work illustrates a compartmental model that captures any Erlang distributed duration of infection with only 3 differential equations, instead of the typical inflated model sizes required by differential equation compartmental models, and a compartmental model that capture any mean, standard deviation, skewness, and kurtosis of an infectious period distribution with merely 4 differential equations. The significance of our work is that it opens up a new class of easy-to-use compartmental models to predict disease outbreaks that does not require a complete overhaul of existing theory, and thus provides a starting point for multiple research avenues of investigation under the contexts of mathematics, public health, and evolutionary biology.

Modeling COVID-19 Pandemic by the \(\lambda\)ISR Volterra-Fredholm Integral Equation: A Case Study of South Africa

“…**Data used was from November 11, 2021 to March 6, 2022 S(x) denotes the increased susceptibility to severe infection with blood type A, and denotes the increased protection against severe infection when the rh factor is not present. The model accounts for the relationship between the number of initially removed, infected, and susceptible individuals, given by the equation below [14].…”

Modeling COVID-19 Nonpharmaceutical Interventions: Exploring periodic NPI strategies

“…This property can have profound consequences in the disease progression dynamics, ultimately affecting transmission. The boxcar method can alleviate the strong assumption of exponential dwelling time distributions substituting the exponential distribution with an Erlang distribution, which is more pathologically realistic [12]. This method requires breaking each disease state into a series of concatenated compartments of the same disease state and shorter internals.…”

The Health and Economic Impacts of Nonpharmaceutical Interventions to Address COVID-19: A Decision Support Tool for State and Local Policymakers

“…This is a known issue that can be corrected by concatenating multiple compartments that each describe the same disease state. This leads to a convolution of exponential dwelling time distributions and thus an Erlang distribution (Oguntunde, Odetunmibi, and Adejumo, 2014;Greenhalgh and Rozins, 2020).…”