A comprehensive spatial-temporal infection model


COVID-19 displays peculiar epidemiological traits when compared with previous coronavirus outbreaks of SARS-CoV and MERS-CoV. These include a large number of transmissions, both in nosocomial and community settings, that have occurred through human-to-human contact with individuals showing no or mild symptoms. Predictive mathematical models for epidemics are fundamental to understand the course of the epidemic and to plan effective control strategies. One commonly used model is the SIR model for human-to-human transmission, which describes the flow of individuals through three mutually exclusive stages of infection: susceptible, infected and recovered. More complex models can accurately portray the dynamic spread of specific epidemics. For the COVID-19 pandemic, several models have been developed. Generally, these models are based on three key elements: identifying the relevant populations such as the infected and recovered and their associated health and demographic histories, establishing the methods by which these populations interact with each other, and determining the rates by which each of the members of the identified populations convert into another.

Despite its popularity, the SIR model has several disadvantages that limit its practical applications. Its representations are based on the total populations rather than the areal density characteristics that are, in our opinioin, the crucial contagion variables. Additionally, it assumes large mobility and diffusion rates and ignores the spatial transfer effects. In contrast, agent-based models such as those that involve computing the trajectories of individual population members account for the effects of spatial density through collision, mobility and propagation rules. Recently, chemical reaction engineering-based models have been identified as a promising approach for modeling the problem. This continuum approach maps populations into chemical species: areal densities into molecular concentrations, the rate of infections into chemical reaction rates and spatial transport into diffusive and advective fluxes. As such, it allows for homogenization of the population distributions by expressing the continuum variables as functions of time and position.

Inspired by the analogies between the chemical processes and the spread of infectious diseases, Mr. Harisankar Ramaswamy, Professor Assad Oberai and Professor Yannis Yortsos from the Viterbi School of Engineering at the University of Southern California have developed a comprehensive spatial-temporal infection model to account for the effects of spatial density. In this model, the susceptible, infected and recovered populations corresponded to chemical species, and their areal densities were considered the key variables. Thereafter, expressions were derived for the kinetics of the infection rates, and for the important parameter R0. Importantly, these expressions directly included areal density of populations and their spatial distribution. This model, which is more representative of the spread of the contagion, was then used to examine several important practical scenarios. The original research article is now published in the journal, Chemical Engineering Science.

The model was first applied to a batch reactor, the chemical process equivalent of the SIR model, where it was used to examine how R0 varied with process extent, and how the initial density of infected individuals and fluctuations in population densities effected the progression of the disease. It was found that the key variable R0 did not remain constant throughout the spread but decreased with the extent of the contagion. It was also found that the effect of the lowering the initial density of infected individuals was to simply delay the active phased of the spread. Thereafter the model was applied to spatially heterogeneous systems. It was found that intermingling, which was modeled as diffusion, generated traveling waves that propagated at a constant speed, proportional to the square root of the diffusivity and R0. The wave-like propagation of the disease has been observed in earlier pandemics like the pneumonic plague in Europe (1347–1350) and in some of the phases of the current COVID-19 pandemic. Preliminary analysis revealed that when intermingling was modeled as stochastic advection it too produced a propagating wave with an enhanced effective diffusivity. It lea]d to microdispersion and subsequently to enhanced mixing that increased the spread of the infection.

In summary, the growth and spread of infectious diseases were modeled via chemical reaction engineering processes analogy, assuming the essential population elements that were mapped onto their corresponding chemical species. The applicability of this approach in obtaining the effective R0 was successfully demonstrated. The study results provided a number of new insights, including the representation of the kinetic parameters, density dependence and the role of advection and diffusion in the spatial spread of infections. In a statement to Advances in Engineering, the authors said their study will advance development of robust framework for modeling infections that spread from human to human and by this allow for better management of epidemics by considering key attributes like mobility and chemical reactions.


Ramaswamy, H., Oberai, A., & Yortsos, Y. (2021). A comprehensive spatial-temporal infection modelChemical Engineering Science, 233, 116347.

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