# Can SARS-COV2 strain competition be modelled by inter-species lotka volterra equations?

Currently studying infectious diseases epidemiology and never studied ecology. But I was wondering if the interspecies Lotka Volterra equations could model and explain strain dominance.

r = could be the effective reproductive number of the virus;

x = prevalence of people currently infected with a certain strain;

alpha = formally is the inhibitive effect of species one on species two; Here I think they'd be equal to each other. if a strain infects someone, he or she cannot be infected by the other competing strain.

Finally K, which is in the ecological context, the carrying capacity defined as the maximum number of individuals of that species the ecosystem is able to accommodate, where the number of births is equal to that of deaths so for this case it may be defined as the prevalence of the strain where the number of infected individuals remain at an equilibrium i.e. every infected person infects, on average, one other person so the effective reproductive number = 1.

So for instance Omicron and Delta in the UK. where it took omicron only one month to supplant delta as the dominant strain.

• frontiersin.org/articles/10.3389/fmicb.2020.572487/full Commented Jan 6, 2022 at 19:22
• Welcome to Biology.SE. This is not simply an answer site, but instead a site that promotes self-learning with some expert help. Consequently, questions that show little or no prior research effort are off-topic on this site. Please edit your question and tell us where you've looked for answers, what you do know about the topic, and where exactly you still have questions. Please take the tour and see the help center starting with How to Ask for details. Commented Jan 6, 2022 at 22:25
• In addition, note that "if a strain infects someone, he or she cannot be infected by the other competing strain" is unlikely to be true — one of the ways that viruses change occurs when different viruses infect the same cell. If you have evidence that this can't happen for coronaviruses, please add that to your question. Commented Jan 6, 2022 at 22:27
• @tyersome its more parsimonious to assume that active infection with a particular strain will constrain another strain from establishing infection when exposed, given that the innate immune response is heightened making the respiratory tract inhospitable. And besides as far as we know human coronaviruses do not undergo antigenic drift. So what i really meant was once a person is infected and becomes symptomatic, they will not be able to be infected with the competing strain until the person becomes susceptible to the competing strain again (because of cross protection) Commented Jan 7, 2022 at 23:29
• From everything I’ve read about SARS-CoV-2, it absolutely has experienced antigenic drift throughout the pandemic. Of course, the sensitivity of an existing immune response (or priming due to vaccine or a prior infection) to that drift is still an area of research. That said, the degree of vaccine escape by omicron (along with the many mutations relative to past variants) implies some amount of antigenic drift. Commented Jan 8, 2022 at 1:01

Presumably by "interspecies Lotka-Volterra equations" you mean the L-V competition equations (the predator-prey version is much better known).

Let's start with a two-strain epidemiological model without coinfection, i.e. a host can only be infected by one strain at a time:

$$\begin{split} S' & = S (-\beta_1 I_1 + \beta_2 I_2) \\ I_1' & = \beta_1 S I_1 - \gamma_1 I_1 \\ I_2' & = \beta_2 S I_2 - \gamma_2 I_2 \\ R' & = \gamma_1 I_1 + \gamma_2 I_2 \end{split}$$

This doesn't look like the L-V competition equations, which reach some sort of competitive equilibrium, because there is no renewal of the "resource" (in this case, susceptible people); the epidemic runs its course and dies out. However, we could allow waning immunity and make an SIRS model so that the recovered people eventually become susceptible again:

$$\begin{split} S' & = S (-\beta_1 I_1 + \beta_2 I_2) + \phi R \\ R' & = \gamma_1 I_1 + \gamma_2 I_2 - \phi R \end{split}$$

(the $$I$$ equations stay the same). This is getting closer, but still doesn't quite match the LV equations. We have to take one more step (which is actually away from the direction of a realistic COVID-19 model) and allow recovering people to become susceptible immediately (i.e. no immunity, as in a disease like gonorrhea), so that we get:

$$\begin{split} S' & = S (-\beta_1 I_1 + \beta_2 I_2) + \gamma_1 I1 + \gamma_2 I_2 \\ I_1' & = \beta_1 S I_1 - \gamma_1 I_1 \\ I_2' & = \beta_2 S I_2 - \gamma_2 I_2 \\ \end{split}$$

This is an "SIS" model. In the SIS and SIRS models the population size is constant, so in particular for the SIS model we can write $$S = N - I_1 - I_2$$ and drop the redundant $$S$$ equation so we get

$$\begin{split} I_1' & = \beta_1 (N - I_1 - I_2) I_1 - \gamma_1 I_1 \\ I_2' & = \beta_2 (N - I_1 - I_2) I_2 - \gamma_2 I_2 \\ \end{split}$$

which we can rearrange to

$$I_1' = (\beta_1 N - \gamma_1) I_1 - \beta_1 I_1^2 - \beta_1 I_1 I_2$$

and a symmetric equation (exchanging $$_1$$ and $$_2$$ subscripts) for $$I_2$$. We can rewrite this as

$$I_1' = (\beta_1N-\gamma_1) I_1 \left( 1 - \frac{I_1 + I_2}{N-\gamma_1/\beta_1}\right)$$

Comparing this with (one particular parameterization of) the L-V competition equations,

$$N_1' = r_1 \left( 1 - \frac{N_1 + \alpha_{21} N_2}{K_1} \right)$$

We see that these are equivalent with $$r_1 \equiv \beta_1 N - \gamma_1$$, $$K_1 = N-\gamma_1/\beta_1$$, $$\alpha_{21} = 1$$.

• a lot of your guesses are right. For this case the two strains are equivalent (each occupies one unit of "resource", i.e. one susceptible host), so $$\alpha_{12} = \alpha_{21} = 1$$.
• the r in the L-V equations is a growth rate (i.e. rate of exponential increase when rare), not a growth multiplier per generation ($$R$$ or $${\cal R}$$), so it's $$\beta N - \gamma$$ (with units of 1/time) rather than $$\beta N/\gamma$$ (unitless)
• The carrying capacity is indeed the number of hosts infected by $$I_1$$ at a monoculture equilibrium, i.e. $$N ( 1- 1/{\cal R}_{0,1})$$