If you are completely unfamiliar with modeling, this is a good place to start!

A basic exponential model is a simple mathematical model used to represent exponential growth in a population, or growth that starts off small but rapidly increases over time. Exponential growth is a specific way that a quantity, such as cases of COVID-19,  increases over time. Because exponential models keep growing until the maximum limit is reached, COVID-19 is a major concern when no preventative measures are taken. With exponential growth, the virus will keep spreading more and more rapidly as more people get infected until everyone is either infected or has passed on.  In other words, this model demonstrates what would happen if no preventive measures, such as social distancing or wearing masks, were put in place to prevent the spreading of COVID-19. For our purposes, we decided to calculate the exponential growth of an infected population because eventually we will be modeling the spread of COVID-19.

Now that the basic exponential model has been defined, we will discuss the difference between the discrete- and continuous-time exponential models.

• Discrete Exponential Model: In a discrete exponential model, changes occur at specific time intervals. For example, cases and deaths of COVID-19 typically modeled on a daily basis.
• Continuous Exponential Model: In a continuous exponential model, changes occur at every instant in time. Radioactive decay is an ongoing process and can be modeled using a continuous exponential model because the amount of substance remaining is constantly changing values.

Discrete exponential (Logistic) model Formula

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With the discrete model, the population of individuals that have contracted COVID-19, $I(t)$, is sampled at specific points in time, so $t = \Delta t \, i$, where $\Delta t$ is the time interval between measurements (e.g. daily reporting of incidence) and $i$ is an index for the time. The discrete infected population is then defined as $I(\Delta t \, i) \equiv I_{i}$ so that
\begin{gather}
I_{i+1} = I_{i} + \alpha \, I_{i}, {\quad \longrightarrow \quad} I_{i+1} = (1 + \alpha) \, I_{i}.
\end{gather}

As a result, the population of infected individuals can be solved to be $I_{i} = I(\Delta t \, i) = (1 + \alpha)^{i} \, I(0)$, where $I(0)$ is the initial number of infected individuals.

In this model $\alpha$ represents the rate at which infected individuals transmit the virus to others. The number this variable represents will be multiplied by the total number infected population. Considering COVID-19, since this virus is highly contagious, the rate of population becoming infected will be higher compared to a different virus, such as the flu.

Continuous exponential model FORMULA

For the continuous-time model, the population of infected individuals is described by a differential equation, defining the rate of change ($\frac{dI}{dt}$) of the population as
\begin{gather}
\frac{dI}{dt} = \beta I, \quad \longrightarrow \quad I(t) = I(0) \, \exp (\beta \, t).
\end{gather}

The infection rate $\beta$ in the continuous time description and the rate $\alpha$ in the discrete time formulation are related, as
\begin{gather}
1 + \alpha = \exp(\beta \, \Delta t).
\end{gather}

For simplicity we will focus on the discrete model.

Exponential Growth

Mathematically, exponential growth occurs because the instantaneous rate of change (infection rate) causes the quantity (infected population) to increase by a constant percentage at each measurement. The number of newly infected people depends on the number of individuals that are currently infected ($I_{i}$) and the rate at which they infect someone ($\alpha$). When increasing a quantity by a constant percentage of that quantity, a positive feedback loop develops in which the quantity and the amount being added to the quantity both increase over time.

For example, assume three populations are increasing by 10% each time. The first population has 10 people, the second has 100 people, and the third has 1000 people. After an initial increase of 10% the populations are now 11, 110, and 1100 respectively. Although each population increased by the same percentage the smaller population only increases by 1 person, the second by 10 people, and the third by 100 people.

In this equation, the population of infected people is added to the infection rate times the population to get the next number of infected people with COVID-19. Essentially, the number of people that will be infected tomorrow depends on how many people are infected today multiplied by the rate of infection, or how fast the virus is spreading. Eventually, everyone in the population would either become infected or previously infected. So, the main difference between an exponential and linear model is that a linear model just keeps increasing by a constant rate (and will look like a straight slope or line) whereas an exponential will grow by an increasing amount each time (and will look like a curve). This model is why experts and government officials are taking such drastic measures to stop the spreading of COVID-19. Luckily, we have taken preventative measures, so we can use other models to predict the spread of COVID-19.

Graph of exponential vs LInear Growth:

Interactive matlab exponential model:

1. Open the following link
2. Login with with MATLAB account
3. Open folder “COVID-19 Interactive Models”
4. Find the file “ExponentialModel.m”
5. Right click the file and hit “Run”

Next, check out SIR Models!

Check out the Current COVID-19 Cases tab for current data in the state of Ohio and internationally.