Continuous and Discrete Compartmental Models for Infectious Disease

Author(s)

Sousa, Gustavo A. ; Souza, Diogo L. M. ; Gabrick, Enrique C. ; dos Reis, Patricio D. C. ; Bentivoglio, Lucas E. ; Batista, Antonio M. ; Szezech Jr, Jose D.

• Document type: Journal article
• Source: Brazilian Journal of Physics; v. 55, n. 3, p. 10-pg., 2025-06-01.

Abstract

The study of infectious disease propagation is essential for understanding and controlling epidemics. One of the most useful tools for gaining insights into the spread of infectious diseases is mathematical modelling. In terms of mathematical epidemiology, the main models are based on compartments, such as Susceptible-Infected (SI), Susceptible-Infected-Recovered (SIR), and Susceptible-Exposed-Infected-Recovered (SEIR). These models offer mathematical frameworks for representing the proliferation dynamics of various diseases, for instance flu and smallpox. In this work, we explore these models using two distinct mathematical approaches, cellular automata (CA) and ordinary differential equations (ODEs). They are able to reproduce the spread dynamics of diseases with their own individuality. CA models incorporate the local interaction among individuals with discrete time and space, while ODEs provide a continuous and simplified view of a disease propagation in large and homogeneous populations. By comparing these two approaches, we find that the shape of the curves of all models is similar for both representations. Although the growth rates differ between CA and ODE, one of our results is to show that the CA yields a power-law growth, while the ODE growth rate is well-represented by an exponential function. Furthermore, a substantial contribution of our work is using a hyperbolic tangent to fit the initial growth of infected individuals for all the considered models. Our results display a strong correlation between simulated data and adjusted function. We mainly address this successful result by the fact that the hyperbolic function captures both growing: the power law (when considered the first terms of infinite sums) and combinations of exponential (when the hyperbolic function is written via exponential). Therefore, our work shows that when modelling a disease the choice of mathematical representation is crucial, in particular to model the onset of an epidemic. (AU)

• FAPESP's process: 24/14478-4 - Machine Learning in Complex Systems
• Grantee: Iberê Luiz Caldas
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• FAPESP's process: 24/14825-6 - Characterization and Identification of Transport Barriers in Dynamic Systems
• Grantee: Iberê Luiz Caldas
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• FAPESP's process: 25/02318-5 - Dynamics of a host-vector model with nonlocal coupling
• Grantee: Enrique Chipicoski Gabrick
• Support Opportunities: Scholarships in Brazil - Post-Doctoral