Waning Sars-Cov2 Immunity, Yet Another SIR Model

July 21, 2020

Much concern has been raised about durability of immunity to Sars-Cov2 post-infection (or post-vaccination). Some studies suggest mildly symptomatic individuals mount a fairly weak antibody response, and other studies suggest that many recovered patients see their antibody levels decline rapidly, although there is apparently some room for disagreement on these conclusions. Similarly, studies that look at seasonal human coronaviruses suggests that immunity may last only up to 6 months to one year until reinfection occurs. Does this mean the epidemic curve can never get suppressed, doomed to resurface unsustainable number of cases every few months requiring masks and social distancing? I think the answer is probably not, because in an older seasonal human coronavirus study researchers actually attempted to challenge human beings at 1 year intervals to see if they could be reinfected, and while the re-exposed individuals did get infected a year after initial infection they exhibited no symptoms and their viral shedding period was significantly reduced compared to fully susceptible individuals. This suggests that partial immunity could still protect against serious disease, and furthermore reduce the chance of passing on the disease if it is re-acquired. I decided to modify a simple SIR model and explore what happens to epidemic dynamics under the following assumptions:

 1.  Recovered individuals are fully immune for a period of time.
 2.  After this period elapses, they become partially immune.
 3.  Partially immune individuals may get infected again, but are less likely to pass on disease.

Modified SIR Model

The basic “SIR” model uses three compartments: “Susceptible,” “Infected,” “Recovered” with a predefined rate of movement between these compartments. Once in “Recovered” state, that person is considered immune forever. To modify these equations I add two more compartments: “Primed-Susceptible” and “Primed-infected.” The “Primed” is meant to suggest that their immune systems can recognize Sars-Cov2 virus and readily react to it, preventing a full infection. The equations for this are as follows:

$$ \begin{aligned} \frac{dS}{dt} &= -\rho \beta \frac{I+P _ I}{N} S \\ \frac{dI}{dt} &= \rho \beta \frac{I+P _ I}{N} S - \gamma I \\ \frac{dR}{dt} &= \gamma I - \omega R + \delta P _ I \\ \frac{dP _ S}{dt} &= \omega R - \rho \psi \frac{I+P _ I}{N} P _ S \\ \frac{dP _ I}{dt} &= \rho \psi \frac{I+P _ I}{N} P _ S - \delta P _ I \end{aligned} $$

and the parameters may be described as follows

$$ \begin{aligned} S &: \text{ Susceptible compartment } \\ I &: \text{ Infected compartment } \\ R &: \text{ Recovered compartment } \\ P _ S &: \text{ Primed-susceptible compartment } \\ P _ I &: \text{ Primed-infected compartment } \\ \beta &: \text{ Number of contacts per day multiplied by probability of an infected contact spreading disease} \\ \rho &: \text{ Impact parameter of non-pharmaceutical-interventions. 0 means perfect impact, 1 means no impact} \\ \gamma &: \text{ Inverse of how many days it takes to become non-infectious (e.g. clear virus from system) } \\ \omega &: \text{ Inverse of how many days a recovered individual is fully immune } \\ \psi &: \text{ Number of contacts per day multiplied by probability of an infected contact spreading disease} \\ \delta &: \text{ Inverse of how many days a reinfected individual takes to become non-infectious} \\ N &: \text{ Population size} \end{aligned} $$

These kinds of models take longer to describe than to actually implement. I put a very simple implementation on GitHub. Note I used this study to get values for \( \gamma, \beta \).

Long-time view

So what does this model say to a completely uncontrolled outbreak where reinfections can occur, but are milder and less likely to be contagious? I use the following parameters to see:

$$ \begin{aligned} \beta &= 1.75 \\ \rho &= 1.0 \\ \gamma &= 0.5 \\ \omega &= 0.01 \\ \psi &= \beta \\ \delta &= 2 \gamma \\ N &= 100000 \end{aligned} $$

In words this means: the initial outbreak proceeds as before, but recovered individuals are fully immune for \( 100 \) days on average and partially immune thereafter. Reinfected individuals are half as contagious as they were during their initial infection. In this model I assume no “non-pharmaceutical-interventions” either, which gives \( \rho=1 \).

I plot the infection dynamics with these parameters below:


This plot shows that as expected there is a big sudden wave of infections followed by epidemic suppression that lasts for a while, but as people become partially immune a second smaller wave happens followed by what appears to be suppression, but that’s because the y-axis is very biased to the first wave it’s hard to see what’s going on with the second wave.

Active Infections Equilibrium

With these parameters an equilibrium occurs after the second wave where the number of active infections is constant, never going up or down, although this level of infection is much reduced compared to the first wave. The figure below shows the second wave zoomed in and simulated a little longer.


Can we still achieve suppression even with partial immunity?

Non-pharmaceutical Interventions

Suppose that in addition to acquired partial immunity from infection we implement things like widespread mask usage and testing and isolation. I will model this very simply by just taking \( \rho = 0.5 \), representing a lower probability of an infected individual spreading disease.


In this plot we see what appear to be suppression of disease. I show also a zoomed-in plot of the second wave.



It’s taking a long time for suppression when we have only partial immunity, although the disease levels do get to an arguably manageable level even though it never totally goes away. But I’m impatient, so suppose that on top of non-pharmaceutical-interventions we also vaccinated half of the population prior to the epidemic? I will model this simply by starting the simulation with half of the population already recovered - and so assuming that vaccines are equally as protective as a natural infection would be (though many believe they could be better).


The first wave in this case is very significantly reduced, but the long-term dynamics appear to essentially match what happens without the vaccinations


Obviously it is better to acquire immunity through a vaccine than through natural infection, however.

Caveat 1: Limitations of Simple Compartmental Models

I don’t think this model represents what a real epidemic looks like. This models an epidemic that happens in an enclosed room in which everyone is running around randomly for 2000 days. That said, it is still interesting and useful to use this model because you can test assumptions in a very controlled way, and the reactions of the model to change are likely of a similar magnitude as what you would see with a similar change in a real life epidemic.

Caveat 2: Don’t Use This For Real Life Conclusions

I am not an epidemiologist, virologist, immunologist, or any kind of -ologist. I basically don’t know what I’m doing. This was to satisfy my own curiosity about simple models and to explore what happens when I try to extend them as a form of “thought experiment” to provide insight on a problem that is relevant today.

Caveats Aside: Some Conclusions

A simple SIR model where recovery means you’re immune for life results in a curve that hits (almost) 0 infections and remains there forever. Waning immunity followed by partial protection on the other hand doesn’t quite hit 0, it has a small second wave (or a few small waves) that follows the first wave, and then hits an equilibrium that is not 0. We can make that equilibrium closer to 0 with Non-Pharmaceutical-Interventions. I think this is good news because reinfections (or infections following vaccination) are hopefully less severe, and we know that a certain low level of background infection is manageable without very significant changes in everyday life.


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