Fig. 5. Boundary of the Hopf bifurcation of the endemic steady state … ^{1} |

I don’t pretend to be a mathematician or to understand the more complex disease models that are out there, but I do think modeling is an essential way of understanding how to effectively deal with diseases. A recent paper^{1} looks at epidemic diseases and seems to reach some interesting conclusions (though I will cheerfully admit that I don’t even remotely understand this paper, which is heavily mathematical).

The authors have built on models of infectious disease that incorporate immunity to the disease, and incorporated the assumption that immunity to the disease can wane over time, as opposed^{2} to the simpler, but less realistic, assumption that the immunity is either on or off. I don’t think they are the first to do this, and I don’t understand any of the details of how their techniques differ from other models,^{3} but what I think they’re saying is that temporary immunity can lead to disruption of a simple, constant level of infection, and can actually drive periodic epidemics:

[W]hen the temporary immunity period is within a certain range, there will be periodic outbreaks of epidemic, and the disease will not be eradicated from the population. … The main feature is that

temporary immunity leads to a possible destabilization of endemic steady state,and an interesting open question is what effects would vaccination have on the dynamics of an epidemic in such situation.^{1}

(My emphasis) If I’m understanding this correctly, it leads to the possibility that where vaccines lead to relatively short-term immunity compared to the natural infection,^{4} it’s conceivable that vaccination could actually shift the disease from a steady state to an epidemic mode. Obviously, if this can happen, it would be nice to be able to predict it.

Offhand, I can’t think of any examples where this might have happened in real life. The most notorious epidemics, like influenza and norovirus, both tend to have fairly short-term immunity to start with. Something like Marek’s Disease of chickens would be an interesting case study, but the logistics of the poultry agribusiness is going to have a bigger impact than the vaccine (I would think). The chicken-pox vaccine is the best example I can think of for a vaccine with relatively short-term immunity where the disease was endemic before the vaccine, and we’re not seeing any sign, that I know of, that chicken-pox is entering an epidemic situation.

The more relevant situation, I think, is for the natural epidemics. As I say, both influenza and norovirus are well known for short-term immunity from natural infection, so maybe this is a factor there. On the other hand, measles, which is spectacularly epidemic, has pretty long-term immunity from both the vaccine and the natural disease, and I don’t see any sign that the vaccine changed the personality of measles epidemics qualitatively (though of course, quantitatively the epidemics are much smaller now).

On the other other hand, of course it’s entirely possible that I completely misunderstand this paper, so if someone has a better grasp than I do please feel free to correct me.

- Blyuss, K., & Kyrychko, Y. (2009). Stability and Bifurcations in an Epidemic Model with Varying Immunity Period Bulletin of Mathematical Biology, 72 (2), 490-505 DOI: 10.1007/s11538-009-9458-y[↩][↩][↩]
- I think[↩]
- “For numerical bifurcation analysis of system with weak and strong kernels, we use a Matlab package traceDDE, which is based on pseudo-spectral differentiation and allows one to find characteristic roots and stability charts for linear autonomous systems of delay differential equations … “[↩]
- This is true for some vaccines, though not all[↩]