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A discrete-time infectious disease model for global pandemics
Thursday, 2021/10/21 | 07:22:09

Abdul-Aziz Yakubu; PNAS October 19,

2021 118 (42) e2116845118


The ongoing global pandemic of coronavirus (COVID-19), an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has raised concerns about the effectiveness of current preventive pharmaceutical and nonpharmaceutical interventions (1). In addition, the upward global trends in the numbers of emerging and reemerging infectious diseases, as evidenced by the reported cases of Ebola, Zika, Chikungunya, SARS, West Nile virus, and other serious infections, have dramatically expanded the demand for mathematical models of infectious diseases across multiple entities that include the pharmaceutical industry, health and medical organizations, and local and international governments, and that span the public and private sectors (23). With this increased demand comes the opportunity to meaningfully reassess the variety of existing mathematical epidemic models. Such an assessment is an important step in capturing the ways in which these models contribute to the understanding of infectious disease surveillance data and the policies, programs, and practices that emerge from these data (1).


Mathematical models of infectious diseases are powerful tools that are used in extending societal understanding and forecasting of disease transmission dynamics and for evaluating the effects of different interventions and changing on-the-ground conditions for epidemiological outcomes. Thus, it is important that we make use of the full range of the available models and disease data to study disease dynamics. Mathematical models can be classified based on how they model variability, chance and uncertainty, time, space, and the structure of the population. On the data side, disease surveillance data are reported at discrete time intervals, for example, daily, weekly, monthly, or yearly disease incidence or number of disease-induced deaths (1). However, many of the existing infectious disease models are continuous-time models that implicitly assume the availability of a continuous stream of these data. While these models have produced useful information, insights, and interventions, it may be worthwhile to consider discrete-time infectious disease models and other models that are more closely aligned with the discrete nature of disease surveillance data.


The discrete-time version of the Kermack–McKendrick model, a system of difference equations introduced in ref. 1, is more compatible with the data that are available to the modeling community. As a result, parameters of the model can be related directly to disease surveillance data without additional model assumptions. Furthermore, the discrete-time model in ref. 1 is very easy to implement computationally. To investigate the factors that determine both magnitude of the “bell-shaped geometry” associated with most disease epidemics and their termination within a given population (see Fig. 2), in 1927, Kermack–McKendrick introduced an age-of-infection model (4), that is, a model in which the infectivity of an individual depends on the time since the individual became infective. To describe the classic Kermack–McKendrick model, we consider a population that is partitioned into the following three nonintersecting classes by an infectious disease: the class of susceptible individuals or susceptibles (S), infected individuals or infectious (I), and recovered individuals or removed (R) (45). Fig. 1 is the flowchart of the Kermack–McKendrick SIR epidemic model. Childhood diseases, such as chickenpox, smallpox, rubella, and mumps, are modeled by SIR disease models. To account for how the number of individuals in each of the three classes changes continuously, the classic Kermack–McKendrick model was formulated as an initial value problem for a system of continuous-time ordinary differential equations (Newtonian derivative). The deterministic continuous-time SIR model framework was a significant milestone in the development of subsequent infectious disease models.



Figure: Flowchart for the classic Kermack–McKendrick SIR model.

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