6. Applications to Disease Prediction and Spread Prevention

And, we note that since the sum S+I+R is assumed to be a constant value of the total population, we have the fact of

[latex]\frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}=0[/latex]

And, due to the fact that [latex]\tau=\frac{1}{\gamma}[/latex] we note the basic reproduction number can be rewritten as

[latex]\frac{\beta}{\gamma}=R_0.[/latex]

[latex]\frac{dI}{dt}=\left(R_0\frac{S}{N}-1\right)\gamma\ I\left(t\right)[/latex]

we can observe that the value within the parentheses tells a lot of practical information. First, it is worthy to note that the value of [latex]\gamma\ I\left(t\right)[/latex] will always be positive in sign. Thus we can conclude that if

[latex]R_0>\frac{N}{S(0)}[/latex]

while it will not be expected to if

[latex]R_0<\frac{N}{S\left(0\right)}[/latex]

This information is one of the most powerful pieces of information that doctors and/or governmental officials can be provided with in real time when making policy decisions for a new disease. The only major issue with this is that by the time enough data has been collected to obtain this critical piece of information is obtained, in real time the disease has often spread so far that there is often not much that can be done to stop the spread of the disease, other than attempts to reduce the function [latex]S\left(t\right)[/latex] such as social distancing.

Now, to actually obtain a model solution for this system of equations, some algebraic manipulation is needed. To begin if the first is divided by the third it is obtained that

[latex]\frac{dS}{dR}=-R_0\frac{S\left(t\right)}{N}[/latex]

and by routine variable separation for the first order differential equation this becomes

[latex]\frac{1}{S}dS=\left(\frac{-R_0}{N}\right)dR.[/latex]

Then, by conducting a definite integration it is found that

[latex]Ln\left[S\left(t\right)-S\left(0\right)\right]=\frac{-R_0}{N}R\left(t\right)+\frac{R_0}{N}R\left(0\right)[/latex]

From which the solution

[latex]S\left(t\right)=S\left(0\right)e^{-\frac{R_0}{N}\left(R\left(t\right)-R\left(0\right)\right)}[/latex]

is obtained. While this is not exact solution to our 3×3 systems of equations, as the value of [latex]S\left(t\right)[/latex] obtained depends on the value of [latex]R\left(t\right),[/latex] it is an extremely useful piece of information which can be used in real time. If, in real time, data is collected to measure the value of [latex]R\left(t\right)[/latex] and possibly even [latex]I\left(t\right)[/latex] this solution, which can be rewritten as

[latex]N-I\left(t\right)=R\left(t\right)+S\left(0\right)e^{-\frac{R_0}{N}\left(R\left(t\right)-R\left(0\right)\right)}[/latex]

is an extremely powerful solution. From here further methods can be applied to either obtain actual individuals closed from solutions for [latex]\left(t\right)[/latex] and [latex]I\left(t\right),[/latex] or can yield an approximation for the basic reproduction number. Namely, if a nonlinear regression data fit (AKA using the command ~nls in R or python coding) a data fit solution can be created and then by comparing the two one can extract an approximation value for

While this modeling is an extremely interesting mathematical model we will not continue further development on the topic here, but rather we will quickly look in the next section at one real world data example from New York City of the COVID-19 disease spread in to illustrate how the statistical methods learned can be applied to actually model a regression solution. Then we will end this textbook in the proceeding section to discuss an extremely interesting research question, how to determine what factors drive citizens to actively participate in social distancing measures. This is a very important topic to study as within the scientific community it accepted that once a disease is activity spreading in an epidemic, or even worse a pandemic, spread then the most effective way to stop the disease is social distancing. This is due to the fact that while these mathematical solutions are beautiful to study, the downfall is that in real life once a disease starts spreading from person to person there is absolutely nothing that can be done to stop it. The only tools we have available to us such as humans in such a battle against a virus are common sense preventative measures to spread the disease in daily life ( e.g. wearing filtration masks and gloves or other personal protective equipment ), or taking societal measures such as social distancing. While this is accepted by most in the community, we can now validate it mathematical as from our solution of the second equation we noted the disease will not become a pandemic if

[latex]R_0<\frac{N}{S\left(0\right)}[/latex]

While we do not have any control over R naught, and the value of N is fixed, we can greatly reduce the value of [latex]S\left(0\right)[/latex] by implementing social distancing measures; in fact as that value approaches zero the right hand side of this bound will become infinite which ensures society will win the battle against the virus, or here mathematically a battle against the direction of an inequality symbol!

6.2 Data illustrations of seeking the exponential inflection point

Now, prior to conducting the data analysis for this data set it is useful to make a note about how, in real time, it is preferred to look at a very simple measure which is the rate of change of cases not the actual increase in raw numbers. For example, the day over day change from March 24^th to March 25^th was 31.7% which is easily computed as number of cases reported on March 25^th – cumulative number of cases up to March 24^th, then to make this a percentage ratio the result is divided by the cumulative number.

[latex]\%=\frac{\#new}{cumulative}=\frac{total\ cumulative}{cumulative}=\frac{688-503}{503}[/latex]

This very simple computation is one of the most important data values to watch in real time, as while the number of new cases may still be a large value it is the percentage change that really tells the story, really tells when the spread is slowing. Obviously, in the early days of the disease spread that value will be rapidly changing due to small numbers, but once the progression of the disease continues it is noted that this number becomes more stable. Namely, in the New York City data this percentage change was steadily growing up to March 26^th , when it reached its maximum value of just under 37%. Then, the value steadily declined over the next few days, 30.4% on March 27^th and 29.5% on March 28^th . The value continued a steady decline reaching 19.7% on April 1^st and then falling to 9.3% on April 8^th staying below 20% for the reminder of the month. Thus, this numerical value is a measure of when the spread of the disease starts to slow, and as one can see within this timeline the peak corresponds to the time shortly after the strict social distancing measures were put in for place for NYC residence. Furthermore, it is debatable as to what measure is the most accurate to use, the number of infected or the number of deaths, and while each is now without some doubt it is generally accepted that the number of deaths is more practical in real time as in order to actually know the total number of infected persons each member of the population would be needed to be tested and it is not practical to do so in real time. However, to study the models in the prior section, related to data analysis methods, a data fit for [latex]I\left(t\right)[/latex] is desired, thus we will conduct one now but not address the actual validity of the data obtained nor any corrections to the data values due to advanced sampling methods that could be applied.

If a simple regression model was run on the natural log of the data of the number of cases during the month of March, the following result is obtained.

[latex]\ln{\left(\hat{y}\right)}=0.37x+1.62[/latex]

Thus, one can back solve this to see the approximate exponential growth model for the number of infected cases as

[latex]I\left(t\right)\approx{I_0e}^{0.37x}[/latex]

And, from this one can compare to the either one of two things: the formal solution to the methods learning in the prior section, hence matching parameters, or a training/testing data set from current time data. Either way, the model should be effective at making short term predictions on the future spread of the data. It would be at interesting study to look at this same exercise, but at different times in the future; hence, one could conduct a post hoc type statistical analysis to look if measure taken by local authorities had any effect of the spread?