Predator-prey model of a virus

[etotheipi]

I was trying to put together a basic mathematical model for the Coronavirus and happened to stumble across the predator prey model; if $x$ is the number of humans and $y$ is the number of Covid-19 viruses, then
$$\frac{dx}{dt} = ax-bxy$$ and $$\frac{dy}{dt} = cxy - dy$$ I took the reciprocal of equation (1) and multiplied this with equation (2), which ultimately lead to the differential equation $$\int (\frac{a}{y} - b) dy = \int (c - \frac{d}{x}) dx$$which gives $$a\ln{y} - by = cx - d\ln{x} + C$$ I now want to try and find $x$ in terms of $t$; my instinct was to try and eliminate $y$ and substitute this into equation (1), however evidently $y$ exists in the $\ln$ and as a linear term so I can't isolate it. I was wondering if anyone could give me a hint of how to find $x(t)$? Or is this perhaps one I need to do numerically?

 

[fresh_42]

Lotka-Volterra is the wrong model. You must use a SIR model. The crucial factor is the infection rate: how many people get infected by one person. The curves are all very similar, but the amplitude of maximal infected people basically depends on the infection rate.

Here's a calculator for SIR models:
http://www.public.asu.edu/~hnesse/classes/sir.html

 

[DaveE]

This exercise can be as simple or complicated as you wish. The typical "simple" model is the SIR model described here: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease

In practice the quality of the data available is the limiting factor. We will know the real answers about 2020 in about 2022.

 

[etotheipi]

Lotka-Volterra is the wrong model. You must use a SIR model. The crucial factor is the infection rate: how many people get infected by one person. The curves are all very similar, but the amplitude of maximal infected people basically depends on the infection rate.

Here's a calculator for SIR models:
http://www.public.asu.edu/~hnesse/classes/sir.html

This exercise can be as simple or complicated as you wish. The typical "simple" model is the SIR model described here: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease

That's interesting, hadn't heard of this model! It appears to be$$\frac{dS}{dt} = -rSI$$$$\frac{dI}{dt} = rSI - \gamma I$$$$\frac{dR}{dt} = \gamma I$$I'll try and solve this on excel with Euler's method... since I don't think I'm going to get very far analytically!

 

[fresh_42]

A conservative estimation is 1 infects 3, and a death rate around 3%. The problem with the current outbreak is, that the situation is highly dynamic and not only numbers, but facts, too, change by the day. This means that those figures are not robust. They additionally depend on regions, the demographic situation etc. Containment in the middle of Montana is certainly easier than in the middle of NYC. Furthermore we have an unusual high number of people who aren't tested although they are carriers. And all of this is still a first world problem. I'm afraid things will turn ugly when Africa or India will be involved with high numbers. @DaveE is right when he says: we will know in 2022. I'd like to add: possibly later!

 

[etotheipi]

This is what Excel produced for the Euler solutions to the SIR model, and it seems to give a reasonable model. Orange is infected, grey is recovered, blue is susceptible.

I might try and make one for the predator prey model as well and see how similar they are!

 

[etotheipi]

On second thoughts, I've just found the Euler solutions to the predator-prey model and they don't really look anything like the SIR model, reaffirming that it's probably not a good model at all for viruses.

I did, however, manage to produce some cool looking graphs:

 

[fresh_42]

Here is a typical problem and image of predator-prey (problem 6)
https://www.physicsforums.com/threads/math-challenge-june-2019.972824/
https://www.physicsforums.com/threads/math-challenge-june-2019.972824/page-2#post-6203161

 

[fresh_42]

Here is a site with - more or less - current data:
https://blogs.sas.com/content/sasco...ualization-to-track-the-coronavirus-outbreak/

i think they are a bit behind, but it is better than guessing.

 

[FactChecker]

This is the type of thing where computer simulations (Monte Carlo if you want to inject random behavior) offer a great deal more modeling flexibility than exact equations will. You can have the simulation step through time and employ a variety of real-world policies and conditions that would be virtually impossible to get closed solutions to.

 

[FactChecker]

On second thoughts, I've just found the Euler solutions to the predator-prey model and they don't really look anything like the SIR model, reaffirming that it's probably not a good model at all for viruses.

I did, however, manage to produce some cool looking graphs:

View attachment 258655
View attachment 258656

This looks like the recovered cases are still able to be re-infected. That is currently an open question for the coronavirus, but I think the assumption is that a recovered person gains immunity. I think that would make these results look more like the SIR model.

 

[etotheipi]

This looks like the recovered cases are still able to be re-infected. That is currently an open question for the coronavirus, but I think the assumption is that a recovered person gains immunity. I think that would make these results look more like the SIR model.

Yeah, I think the Lotka-Volterra model just isn't cut out for this for precisely that reason.

 

[fresh_42]

Yeah, I think the Lotka-Volterra model just isn't cut out for this for precisely that reason.

Lotka-Volterra yields dynamic equilibriums like the two chasing sine curves you got, or the "cycle" in the example from the challenges: prey population grows, which increases predator populations (many mammals have more children if the food situation is fine), but more predators diminish first prey and then itself, so the prey population can recover and everything starts again.

SIR models are asymptotic models: You don't lose immunity again, at least not very quick, e.g. tetanus, so infected people die or become immune and then they are out of the relevant population, and healthy people stay healthy or get sick and leave again. Healthy people are limited from above, so neither pool can recover, only change status. It is not a cyclic but an asymptotic behavior. This is a fundamental difference, so you cannot model one with the other.

 

[FactChecker]

The time scales are also different. Most predator/prey models are on a time scale that allows new birth among the prey. An epidemic model for humans would not be on a time scale where births were significant.

 

[DaveE]

Listening to the latest "This Week In Virology" podcast interview with Ralph Baric, a Corona virus expert. He mentioned a few estimates that could illuminate some (more complicated) model parameters:

- Current estimates of SARS-CoV-2 Ro are in the 2.5 - 3.2 range (of course this depends on a lot of local things; it's higher on a cruise ship than in rural North Dakota).

- Assuming that there isn't an animal reservoir (like Bats, Camels, etc.), he estimates that a herd immunity of 70% of the population would probably result in extinction of the virus.

- Recent studies of MERS-CoV (a different infectious Corona virus) immunity via serum titers show that immunity may only last 1 - 2 years, and in rare cases just a few months. Other, more common, Corona viruses impart lasting immunity. This is early data and needs much more study. How this relates to SARS-CoV-2 is completely unknown.

- The panel (all virus experts) said that if we had a safe and effective vaccine in 18 months, that would be faster than any other similar effort.

"This Week In Virology" is the go to place for academic information about this virus (i.e. in depth, not sensationalized). They have been talking about it since the middle of January.

 

[fresh_42]

There is another big risk: evolution! Someone told me that we have already two human strains, but I haven't tried to confirm this. I personally think it is not true. Nevertheless, with high numbers of human infections, the chance raises, that we will have to deal with successful mutations.

 

[etotheipi]

Please excuse my ignorance (!), I’m still fairly new to differential equations: out of curiosity, can one get anywhere analytically with either of the two models? Though Runge-Kutta and Euler methods appear to work fine for numerical solutions...

If so, how would you go about it?

 

[fresh_42]

Wikipedia has some notes on the solutions of Lotka-Volterra. There is no one formula catches all solution for any IVP. We have constant equilibriums and a first integral which results in the image I quoted from the challenges. If we had a solution in closed form, we would use those equations instead of the differential ones. But deterministic numerical algorithms aren't such a big deal today.

 

[DaveE]

we have already two human strains

The problem here is defining what a strain is. My guess is that this depends on having enough genetic change to change the way the virus behaves in the infected individuals. This is tough to do since there is a ton of other variations in response not related to the virus's RNA, like the patients immune function or co-morbidities. Plus we only have a sparse data set so far.

This is all because viruses, especially RNA viruses, are constantly having minor mutations in their genome; mostly insignificant. So media reports of genetic change are hard to evaluate. They do, however, allow some really interesting possibilities in tracking the virus's history, essentially the same way 23 & me can tell you where your ancestors came from.

"Estimates of the timing of the most recent common ancestor (tMRCA) of SARS-CoV-2 using currently available genome sequence data point to virus emergence in late November to early December 2019."

This is from this recent paper that talks about SARS-CoV-2 genetic analysis:
http://virological.org/t/the-proximal-origin-of-sars-cov-2/398