Coronavirus and Covid-19: A mathematical perspective – MediNick

By CoperNick

Alessandro Santoni

April 23, 2020

In these days the entire world is involved in the biggest pandemic in the last hundred years: the “coronavirus” outbreak! While you are reading this, thousands and thousands of people are becoming infected and unfortunately dying in some cases.

Let’s take a few steps back to firstly understand what we are talking about: the virus we are fighting against is the so called Sars-Cov-2 and is only one of the many viruses that belongs to the family of the “coronaviruses”. This name is due to their following characteristic structures.

Figure 1 – Appearance of a generic coronavirus

These things that you can see in the image all around the virus are called “spikes” and, being recognized by some receptors on the human cells’ membranes, allow the virus to fool our cells, enter in them and start replicating. In the case of Sars-Cov-2, they are made of a molecule called “protein S”, the receptor of which is the ACE-2, that is mainly present in cells of the respiratory apparatus.

Figure 2 – The key-receptor mechanism of Cov-Sars-2

That’s why Covid-19, namely the disease caused by this specific coronavirus, regard above all the respiratory tracts, lungs and their structures.

Regarding the history of the Sars-Cov-2’s origins, I hope it will be sufficient for you to know that the scientific community agree that it probably comes from a bat’s coronavirus, that in the poor hygiene conditions of a Chinese food market in Wuhan, was able to do the “species jump”, infecting humans for the first time. [1]

Since this is not my field of study I will not enter in more “biological details”, but I will focus myself on the mathematic side of this event. Don’t be afraid if in the next lines you will find too much math for your taste, I will try to pair it with rational explanations as much as I can!

SIR model

The simplest model in epidemiology that describes the spreading of a new disease is the SIR Model. Substantially, it consists into dividing the entire population in three categories, from which the model itself take the name

(S) Susceptible: these are the people that can be potentially infected by the pathogen
(I) Infected: clearly they represent the part of the population with the disease
(R) Removed: in this group are included all the people that got infected and now they are not, both because they have healed or because unfortunately they have died.

The dynamics of these three variables is guided by a set of differential equations

with the obvious constraint that, if we call the total population $N=S+I+R$, then $dN/dt = 0, which means this system is a closed one. The meaning of these equations is straightforward to understand:

The change of $S(t)$ in time is both proportional to the number of people already infected $I(t)$ and the new people to be Infected $S(t)$, multiplicated by the constant $\beta$ that represent in some way the probability per unit time of a healthy person to get the disease. Clearly its value will depend on different things, like the infectivity of the illness but also social factors, like the rate at which people meet each other.
The change of $I(t)$ in time clearly contains two terms, the first one that account for the new people that become infected, passing from $S(t)$ to $I(t)$, and a second one negative, that represents the people that overcome the disease or die. Obviously more people are infected and more people will be removed by the $I$ group for the latter reasons, that’s why the proportionality to $I(t)$. Furthermore, the constant $\gamma$, that has the physical dimension of the inverse of a time, represent the inverse of the mean time period needed for a person to get out from the disease.
Finally, the change of Removed people $R(t)$ in time is clearly the opposite of the second term present in the previous differential equation.

Figure 3 – Schematic representation of the groups and flows of people

Notice we are supposing that the ones that recover from the disease get an immunity such that they cannot become susceptible again. That is the case with lots of diseases, but it is not completely clear if it’s also the case for Covid-19.

Epidemic’s Beginning

Now that we know a little bit of the background, we can go further in the analysis: imagine the beginning of a new epidemic in a population with $N$ people and $I=1$ infected.

Since we suppose $N$ to be very large ($N>>1$), at this stage, the approximation $S \approx N$ is valid and then the second differential equation become

\[ \frac{dI}{dt} \approx \beta N – \gamma \]

Which imply, for the epidemic to start, that

\[ \frac{dI}{dt}>0 \,\,\, \rightarrow R_0 = \frac{\beta N}{\gamma} >1 \]

The $R_0$ value appeared in the last inequality, to which epidemiologists usually refer to as the “basic reproductive number”, is a constant of primary importance to determine the evolution of a disease outbreak. Rewriting the change of the infectious people in time we find

\[ \frac{dI}{dt} \approx \gamma (R_0 -1) \]

This essentially means that, during the $gamma^{-1}$ days for which the single sick person will be contagious, he will infect in average other $R_0$ people in the community, before passing in the $R$ group. Strictly speaking, cause to our approximations, this is valid only at the beginning of the disease spreading. While the number of infected increases, in fact, it becomes less and less probable to meet with a still susceptible person.

Let’s note that, as one would expect, if $R_0 < 1$, is to say that each contagious person generate less than one new infected, the epidemic dies out automatically. We should then postulate a $R_0 > 1$ in all our next considerations.

Furthermore, in this early stage, we can find another interesting characteristic of the spreading: as long as the number $S$ is much larger than , supposing to start with a few infected $I_0$, we have

\[ \frac{dI}{dt} \approx (\beta N – \gamma) I = \gamma (R_0 -1) I \]

So that integrating this differential equation we get

\[ I(t) \approx I_0 e^{\gamma (R_0-1) t } \]

Then, the number of infected will initially grow exponentially, with a doubling time $T_{x2}$ of

\[ T_{x2} = \frac{\ln{2}}{\gamma (R_0 -1)} \]

Epidemic’s Evolution

Let’s now focus on the complete evolution of the epidemic: it’s clear that at some point, while $S$ decrease, the rate of change of the infected in time will become negative. So we expect that, for a reasonable value of $R_0$, at the end a fair percentage of the population will not get infected.

Observe that dividing the first with the third differential equation

\[ \frac{dS}{dR}= -\frac{\beta}{\gamma}S = -\frac{R_0}{N}S \,\,\, S(R) \approx N e^{- \frac{R_0}{N}R} \]

And since the system is closed $I=N-S-R$, we write

\[ \frac{dR}{dt} = \gamma (N-R-S) = \gamma (N-R-N e^{- \frac{R_0}{N}R } ) \]

Letting time go to infinity, where $R(t \rightarrow \infty ) = R_\infty $ is a constant and so $dR/dt \rightarrow 0$

\[ R_\infty = N (1-e^{-R_0 \frac{R_\infty}{N}}) \,\,\, \rightarrow \,\,\, r_\infty = 1-e^{-R_0 r_\infty} \]

Where we have introduced the total portion of people that will get infected (and then removed) as $r_\infty = R_\infty /N$. Finally, finding numerically the solution of the equation for different values of $R_0$, we can plot the “$r_\infty \, vs \, R_0$”graphic

Figure 4 – Trend of the total portion of infected as a function of R₀

As you can see, the majority of the curve’s growth happens in the interval $R_0 \in (1,2)$, in which the percentage of total infected go from zero to the $80%$! This means for example that, if as a national government we really expect to contain the final number of infected people, then we should apply some very tough restrictive measures, to reduce $R_0$ near to $1$ or below, for a long enough period of time.

Another useful thing to study, is the maximum peak of infected: higher is this peak and harder will be for the different nations to manage the relative sanitary emergency. In fact, even if only a low portion of the infected need to be hospitalized, the hospital beds still are limited goods and the initial exponential increase can easily challenge the sanity structures, eventually bringing them to collapse.

To find the above mentioned peak, let’s divide the second differential equation for the first one

\[ \frac{dI}{dS} = -1 + \frac{\gamma}{\beta S} = \frac{N}{ R_0 S} – 1 \,\,\, \rightarrow \,\,\, I(S) -I_0 \approx \frac{N}{R_0} \ln{S/N} + N- S\]

Searching for the maximum of $I(S)$ we find

\[ S^* = \frac{N}{R_0} \, ; \,\,\, I_{MAX} =I(S^*) \approx N ( 1- \frac{\ln{R_0}+1}{R_0}) +I_0 \]

For $R_0=1$, we have $i_{MAX}= I_{MAX}/N \sim 0$ while $I_{MAX}$ approaches $N$ in the limit $R_0 \rightarrow +\infty $. The “$i_{MAX} \,vs \, R_0$”graphic then will look like this

Figure 5 – Trend of the peak of infected as a function of R₀

Observe also that, since $S^*=N/R_0$ is the number of susceptible at which the change of infected in time becomes zero and then negative, when the value of $R$ reach

\[ R_H = N-S^* = N(1-\frac{1}{R_0})\]

we have the so-called Herd immunity, since the disease can no longer spread effectively in the community, letting every new outbreak die out quickly.

The Covid-19 case

Now let’s see a quick application of what we learnt to the case of Covid-19: from the study done on the Diamond Princess cruise ship [2] we know that $R_0 \sim 2.3$ and from data all around the world that the time of duplication should be $T_{x2} \sim 2.5\, days$. Then, we can calculate $\gamma ^{-1} \sim 4.6 \, days$.

For such a basic reproductive number we find that, letting the virus free to propagate, approximately the $86%$ of the population could be infected, with a peak of sick people in the same day of $20%$, according to our simplistic model.

Here below you can see the online simulation [3] done with the Covid-19 parameters in a population of 1 million people

That’s obviously something that cannot be underestimated: in fact, even if only $10%$ of the sick people would need to be hospitalized, near the peak, we would need roughly $20000$ hospital beds for a population of only 1 million people, while for example the european average is $5000$ every 1 million people!

Furthermore, if the mortality is around 3%, like it seems from the data [4], it means that among the $800000$ infected people in our toy model, more than $24000$ of them will die… these are unsustainable numbers for sure! That’s why almost all the governments around the world, unfortunately often with bad timing, are doing what they can to control and lower the impact of the epidemy, trying at the same time to not shut completely down their economies.

We could now argue which strategy could be better for such a goal, and for this I refer you to this youtube video [5] in which the effects of different strategies are shown through a series of simulations. In particular, it’s shown that the effect of a “non-perfect” quarantine is not really to reduce the total number of infected but to slow the spreading down, avoiding the sanitary system collapse.

Keeping in mind that reality it’s always much more intricate than our simulations, it’s still possible to complicate the SIR model in many ways: introducing an incubation time for the virus, a rate of birth and death etc. however, as a wise man once said, “there are a time and place for everything, but not now”! In any case, I leave you with some links where you may satisfy your curiosity [6, 7].

I would like to end this article with a grating to sanitary workers all around the world, that are giving their hard work and lives, treating patients at their best in such a period of crisis. If we will win this fight, and I am sure we will, it will be above all thanks to them!

“Wherever the art of Medicine is loved, there is also a love for Humanity”

[Hippocrates]

Remember that all of us play an important role in this [8], stay at home and, to take advantage of this free time, don’t forget that you have plenty of articles to read over here!

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