In this model there are three functions S(t), I(t) and R(t). They represent the susceptible, infected and removed persons over the total population. Because they are in a closed system the sum adds up to the same number resulting in a stable total population, N = S(t) + I(t) + R(t). The model is initiated with a value for beta and gamma. The simulation is started with the amount of susceptibles, infected and removed at time zero with S(t=0), I(t=0) and R(t=0).

The change of susceptibles from day-0 to day-1 is calculated with:

dS/dt = - beta*S*I/N

Subsequently the addition in infected is calculated:

dI/dt = + beta*S*I/N - gamma*I

Then the addition in removed is calculated:

dR/dt = + gamma*I

This cycle from time-zero (t=0) outputs the values for time-one (t=1). All three functions have an added data-point for t=1. This cycle is repeated until values for S(t), I(t) and R(t) for all the time-points have been found.

However, in an actual situation of an epidemic, the beta value changes every day, due to lock-down measures or social distancing changes. The gamma values changes as well, due to hospitals overcrowding or financial troubles. Therefore, beta and gamma would have to be updated every cycle, resulting in the three functions changing every day. We can graph the 3 values for today and predict the functions, but beta and gamma would have changed within a few days..

The best hopes would be to act accordingly to optimize for beta and gamma, i.e. with social distancing and hospitals remaining accessible. In the SIR-model, the additions and subtractions of the flow diagram to construct the three functions are nearly arbitrary, however, with a constant population the changes are somewhat like during an epidemic. We see the susceptible people becoming ill, they then recover.

Hi Sebastian,

On Mar 27 I wrote a couple of notes in response to "Is spreading of Coronavirus like fractal branching and happens chaotic burst?" (I actually posted some before and several after Mar 27) and in them I outlined what I had done in my model for the US and what a global model should look like. We have more info now and some parameters may be better known but I will copy here what I wrote since it has some relevance to what you wrote. I agree with your points about delays in reporting, etc. and these are very important factors.

Here are my comments from Mar 27 but note that you don't see the questions/comments prior. One of the comments was about spatial diffusion.

" Transport of a cloud of viruses might be by diffusion in an ideal setting but in fact they move more on droplets through small currents of air after being breathed, coughed, or sneezed out of an infected person. An equally important means of transport is through {hand of infected} to surface to hand to face of susceptible person.

Those are details, however, that can be modeled at the population level as diffusion of infected people through a community of susceptible people. A good spatial model would have several islands (population centers) of differing sizes connected by lines of travel: A network of islands. Then one could simulate the spread throughout the world. Transportation along lines connecting islands could be modeled as one-dimensional diffusion, with differing diffusivities according to how busy each route was. E.g. Hong Kong to LA would have a high diffusivity coefficient while LA to Belfast, Ireland would have low diffusivity. That model could probably be derived from a stochastic system accounting for movement of individuals.

In the body of a person, the reproduction and transport is fairly well understood, I believe, and diffusion really doesn't play much of a role except for localized spread from infected tissue.

I think the system of PDE's modeling the world as a large collection of population centers (with diffusion within each of these) connected in a network of one dimensional diffusive channels is definitely worth studying and might have broad interest. Within each island (population center) the population should be age-structured (time scale of tens of years say) and the infected population should be (time from infection)-structured with a time scale of weeks, say. There should also be sub-populations getting quite ill and those having very slight symptoms but still being infectious. Testing would need to be included as a parameter to place people in another sub-population, those under care.

In any case, it is clearly quite complex but computationally just a bit messy with many parameters. It should be coded and run on a parallel computer system I think. But a toy model with about four centers could be run on a laptop, I think."

2nd post

"About my model for the viral epidemic in the US,

I have subdivided the population into 8 disjoint subsets according to susceptible, infected for less than a week (all undiagnosed and infectious), those infected for a time between one and two weeks and undiagnosed, those infected for more than two weeks and undiagnosed, those diagnosed and in their first week of care, those diagnosed and in their second week or more of care, those who have recovered, the dead. There are 20 or 21 parameters. I could include more to account for health care workers getting infected from those in quarantine, for instance, but I didn't. I also did not break down each of these sets by age, which really should be done.

Rather than a continuous dynamical system, I just used a discrete DS which I put into a spreadsheet.

Of course you are correct about saturation and my model includes this by having susceptibles move to infected according to mass-action kinetics (the probability that when two individuals meet, one infects the other is the fraction of infected who are loose in the population multiplied by the fraction of susceptibles in the non-quarentined (or dead) population). There is a prefactor parameter representing the average infectivity of an infected person. So, as the number of suscptibles gets below a certain threshold, the rate of increase of infected people decreases. Doing this in integer arithmetic causes the virus to die out before the entire population becomes infected. I have not used any curve fitting algorithm to tune my parameters as more data arrives but I have made adjustments according to intuition. One of my parameters is testing rate which in turn impacts the three rates in which undiagnosed infected people move into quarantine and care.

I can think of ways to improve the model, as I just pointed out, but I am doing this for my own information and not to publish or make claims to society. Even though this modeling is very important to society, there are specialists who I am sure have made more sophisticated and more accurate models than mine, so I feel I might do more harm than good by publicizing my amateur attempts."

Best,

Peter

A simple SIR model is general, and at best could only provide information on trend of disease progression through the population. Also, scientists are still debating about the recovery portion (R) and whether it provides immunity and methods of transmission, which could affect the dynamics when a susceptible person (S) interacts with an infected person (I). I would assume that at this point, there are still uncertainties in how to implement the model effectively.

Dear colleagues, if we consider the classic SIR model, obviously not, regardless of the fact that we do not have a closed system, there is population exchange and, most crucially, we are still not sure if all the infected remain immunized!

Also, the model generally make assumptions of homogeneous mixing i.e. everyone has the same chance of catching the virus from an infected person because the population is perfectly and evenly mixed. We know there are different contact rates and chances of being affected by /transmitting the disease.

Certainly there is a big difference in behavior (mixing) between the large population of "infected but asymptomatic/undiagnosed" and "infected and diagnosed => quarantined/treated." In my model I break the undiagnosed/untreated into 3 subgroups according to time since infection.

Dear Prof. Peter William Bates

Thank you very much for your answers!! Have you changed your mind about publishing/showing the equations of your model?? I'd love to see them!! Even though sometimes data wouldn't be enough for fitting parameters, I think that equations always have something to say.

I agree with Joanna Sooknanan

Hi,

The mathematical modelling can well predict a general trend of the epidemic disease like influenza, Covid-19 etc. by taking into account effective parameters (with regarding to the nature of the virus and its features of the spreading), especially at the beginning of the outbreak, because the nature of most of the epidemics is exponential (at the onset of outbreak), but after the progress of the virus, the problem (prediction of outbreak) will be more complex than its beginning, because of new effective parameters and control measures. However the mathematical modelling may lead to more accurate results by carrying statistical data and taking into account more involved variables during spread. For getting more details about mathematical modeling and spreading of Covid-19 I recommend following papers.

Thank you.

Well put Kamran.