key: cord-0965186-k6u1l3si
authors: Thomas, Diana M.; Sturdivant, Rodney; Dhurandhar, Nikhil V; Debroy, Swati; Clark, Nicholas
title: A primer on COVID‐19 Mathematical Models
date: 2020-05-09
journal: Obesity (Silver Spring)
DOI: 10.1002/oby.22881
sha: a884e80656f05465ff920e89ab3955460bb960d3
doc_id: 965186
cord_uid: k6u1l3si

The emergence of severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2 or COVID‐19) disease has led to a wide‐spread global pandemic (1). COVID‐19 symptoms and mortality are disproportionately more severe in people with obesity and obesity related comorbidities (2, 3). This is of concern for the United States, where ~42% have obesity and of these, 85% have type 2 diabetes.

do COVID-19 models yield such variable predictions? How are the different COVID-19 models developed? How should models be applied? Here, we provide a brief tutorial addressing these questions.

The majority of models fall into two categories which are applied for different purposes;

projections versus statistical forecasts. Projections are deterministic and explain what could happen under a set of underlying hypotheses, while statistical forecasts use observed data to predict what will happen (6) .

In 1927, Kermack and McKendrick (7) , developed the first continuous variable projection model of epidemic population dynamics, often referred to as a "susceptible, infected and removed" or SIR model. They compartmentalized a constant population into three states. The first state represents individuals susceptible to the disease with the number of susceptible individuals on day of the epidemic denoted by

The second state is the number of infected individuals on day of the ( ).

Finally, individuals removed from the infectious disease dynamics ( ).

through death or recovery with immunity on day of the epidemic is denoted by ( ).

SIR models assume susceptible individuals contract the infection by interacting with infected individuals. The term that models this interaction is represented as a proportion of the product of the susceptible and infected:

. If the mortality/recovery rate is modelled as a direct proportion ( ) ( ) of infected individuals, we arrive at the final Kermack-McKendrick Model who's flow diagram is depicted in Figure 1 with mathematical formulation as a system of three ordinary differential equations:

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The key property of projection models like the Kermack-McKendrick system is that they are based on logical assumptions of the underlying mechanics of a process and can be developed without data to immediately address "what if" questions.

It is from SIR-like models we can see the effects of social distancing on flattening the curve. SIR models, however, are sensitive to underlying model assumptions. When these assumptions are modified, projections sometimes change dramatically. For example, we could assume there is a time lag to infection that accounts for the virus incubation period by modifying the term, to ( ) ( ) . The transmission rate, , could also be assumed dependent on the currently observed ( ) ( -) infected population. Additionally, instead of assuming is constant, we could model as a function of using the Hill function (8) ( )

Here, r declines as the number of infected individuals gets higher, reflecting increased social distancing during peak infectivity. On the other hand, increases when the number of infected individuals decrease. The values , K and n are parameters that could be fit to data once available. β max

The changes in assumptions alter the projections as depicted in Figure 2 .

Once data is fit to model parameters, SIR models can predict when infections peak or how high the peak may be, but as pointed out in Jewell et al. (5) and observed in Figure 2 these are dependent on the underlying model assumptions .

Forecasting models are more useful after data has been gathered. For instance, most SIR COVID-19 models use fixed parameters which lead to a constant referred to as or " naught". 

This article is protected by copyright. All rights reserved Within the SIR framework, where is the time to recover and is the population 0 = size. Absent of interventions, the estimated of COVID-19 is between 1.5 and 6.7 (9) . However, 0 this value is not constant but changing daily. Renewal equations allow us to estimate values (10) of that can be fit to a statistical model. Similar to (6) we fit an exponential decay model to New ( ) York's data yielding , though other statistical models such as one with an ( ) = 0.5 -0.02 asymptote could be considered. New York fit to data with an exponential curve.

represents the average number of secondary ( ) cases due to one infected person at the beginning of the epidemic and in order for the epidemic to decline, should be less than 1. The points are the calculated daily effective reproduction ( ) number calculated by number of observed cases divided by the number of expected cases on a given day while the gray curve forecasts future effective reproduction numbers that can be used in dynamic projection SIR models. Here, went below 1 (represented by the dashed line) around 28 days.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease-2019 (COVID-19): The epidemic and the challenges

COVID 19 and the Patient with Obesity -The Editors Speak Out. Obesity (Silver Spring)

Presenting Characteristics, Comorbidities, and Outcomes Among 5700 Patients Hospitalized With COVID-19 in the New York City Area

What 5 Coronavirus Models Say the Next Month Will Look Like

Predictive Mathematical Models of the COVID-19

Pandemic: Underlying Principles and Value of Projections

Forecasting versus projection models in epidemiology: the case of the SARS epidemics

A contribution to the mathematical theory of epidemics

The reproductive number of COVID-19 is higher compared to SARS coronavirus