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SIR Models and Lyme Disease

Modelling can be extremely helpful when trying to understand how a disease system works. The SIR model is a basic model that breaks up a population into three groups: susceptible, infected, and recovered/removed. There are several assumptions associated with this model, including a closed population, constant rates (transmission, removal rates), and a well-mixed population. This model is often too simplistic (and unrealistic) for wildlife diseases, such as Lyme disease (Lloyd-Smith et al. 2005). Another tool often used in epidemiology is the basic reproductive number (R0). This value is defined as the expected number of secondary cases produced by a single infection in a completely susceptible population. It is useful for evaluating the risk of the disease spreading (epidemic). This blog will discuss some SIR-variations and R0 calculations for Lyme disease.

Hosts of Borrelia burgdorferi vary in their competence as reservoir for the spirochete. While some are very competent (white-footed mouse), others are not (white-tailed deer). When developing models for a disease system, it is vital that you consider how long the infection lasts in the host and whether or not the host establishes immunity after it has been infected. Kurtenbach et al (2006) discussed this when considering SIR model alternatives for Lyme disease. The white-tailed mouse endures long-lived infections and maintains high infectivity. It is therefore pretty well-explained by the SI model, which only consists of two groups: susceptible and infected. Once you become infected, you pretty much stay infected. Other, less competent hosts of B. burgdorferi may be explained by SIR if they establish immunity to the bacterium after infection, or by SIS is they don’t build up an immunity to the bacterium. Kurtenbach et al (2006) argues that neither of these models truly fit the less competent wildlife hosts as these individuals never fully recover. Instead, they argue that these individuals become persistent carriers, but with low efficiency (making them less competent hosts). Potential causes for this persistence include how the bacterium can evade the host’s immune system, as discussed in previous blogs. A more appropriate model for these hosts would therefore be the susceptible-infected-carrier (SIC) model.

So what about us humans? Most physicians currently believe that an antibiotic treatment should remove B. burgdorferi from your system (Feder et al. 2007), although this is heavily debated (Miklossy 2012; Berndtson 2013; Stricker and Johnson, 2013). If doctors are correct in assuming this, then humans, or any individual treated for the infection, would probably follow the SIS model as re-infection of the bacterium often occurs (Nadelman and Wormser, 2016). If antibiotic treatments are not effective, however, the SIC model would better fit the human host.


Randolph (1998) proposed a mathematical model for the basic reproductive number (R0) of tick-borne diseases. She based this equation off of the equation used for insect-borne infections (Dietz 1980). Alterations were made to this equation due to the differences between Ixodid ticks (the vector involved with Lyme disease) and other insects (see table).


The equation for tick-borne parasites goes as follows:

R0 =
N ƒ β v - t β t - t β t - v pn F / H ( r + h )

The original insect-borne equation was:

R0 =
N a2 β v - i β i - v pn / H ( r + h ) ( -ln p )

Please see the box for specific parameter definitions.


This equation is performed for every life stage of the tick (larva, nymph, adult), which are then summed up to calculate the total R0 value for that tick. Changes were made to the original insect-borne pathogen equation for the following reasons. First, ticks only have one blood-meal per life stage instead of multiple as with other biting insects. The daily biting rate (a2) was therefore replaced by the probability of the tick feeding on an individual of a particular host species (f). Next, the transmission coefficients (β) includes β t - t for transmission between ticks (trans-stadially or transovarially). This term is a bit outdated, however, as researchers are now finding that this type of B. burgdorferi transmission between ticks does not occur (Rollend et al. 2013). The vector’s reproductive rate (F) is also added in the numerator for the tick-borne parasite equation. This term is highly dependent on the life stage and sex of the tick. The term “- ln p,” which stands for the vector’s daily mortality rate, was removed from the denominator. This is because the term confounds survival periods with development periods. Ticks can live for a long time (~ 2 years) at the same life stage. Longer survival, however, translates into a slower pace of transmission.

I think the mathematical equation for R0 proposed by Randolph (1998) is a pretty good method of calculating R0 for Lyme disease. Some alterations include removing the β t - t term, as we now know that this type of transmission for B. burgdorferi does not occur between ticks. I would also want to add in a few abiotic factors that can strongly affect a tick’s survival and host-seeking activity (Kurtenbach et al. 2006). For example, photoperiod, temperature, and humidity can greatly influence a tick’s behavior and distribution (Ogden et al. 2005; Randolph 2004). Including these abiotic factors will be especially important to assess the potential spread of both Ixodid ticks and B. burgdorferi as a result of climate change (Ogden et al. 2014).


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References

Berndtson, K. 2013. Review of evidence for immune evasion and persistent infection in Lyme disease. International Journal of General Medicine 6: 291-306.

Feder Jr, H.M., Johnson, B.J., O’Connell, S., Shapiro, E.D., Steere, A.C., Wormser, G.P., Agger, W.A., Artsob, H., Auwaerter, P., Drumler, J.S., Bakken, J.S., Bockenstedt, L.K., Green, J., Dattwyler, R.J., Munoz, J., Nadelman, R.B., Schwartz, I., Drapter, T., McSweegan, E., Halperin, J.J., Klempner, M.S., Krause, P.J., Mead, P., Morshed, M., Porwancher, R., Radolf, J.D., Smith Jr., R.P., Sood, S., Weinstein, A., Wong, S.J., Zemel, L. 2007. A critical appraisal of “chronic Lyme disease.” The New England Journal of Medicine 357: 1422-1430.

Kurtenbach, K., Hanincova, K., Tsao, J.I., Margos, G., Fish, D., Ogden, N.H. 2006. Fundamental processes in the evolutionary ecology of Lyme borreliosis. Nature Reviews Microbiology 4: 660-669.

Lloyd-Smith, J.O., Cross, P.C., Briggs, C.J., Daugherty, M., Getz, W.M., Latto, J., Sanchez, M.S., Smith, A.B., Swei, A. 2005. Should we expect population thresholds for wildlife diseases? TRENDS in Ecology and Evolution 20: 511-519.

Miklossy, J. 2012. Chronic or late Lyme Neuroborreliosis: analysis of evidence compared to chronic or late neurosyphilis. The Open Neurology Journal 6: 146-157.

Nadelman, R.B., Wormser, G.P. 2016. Reinfection in patients with Lyme disease. Clinical Infectious Diseases 45: 1032-1038.

Ogden, N.H., Bigras-Poulin, M., O’Callaghan, C.J., Barker, I.K., Lindsay, L.R., Maarouf, A., Smoyer-Tomic, K.E., Waltner-Toews, D., Charron, D. 2005. A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis. International Journal of Parasitology 35: 375-389.

Ogden, N.H., Radojevic, M., Wu, W., Duvvuri, V.R., Leighton, P.A., Wu, J. 2014. Estimated effects of projected climate change on the basic reproductive number of the Lyme disease vector Ixodes scapularis. Environmental Health Perspectives 122: 631-638.

Randolph, S.E. 1998. Ticks are not insects: consequences of contrasting vector biology for transmission potential. Parasitology Today 14: 186-192.

Randolph, S.E. 2004. Tick ecology: processes and patterns behind the epidemiological risk posed by Ixodid ticks as vectors. Parasitology 129: S37-S65.

Rollend, L., Fish, D., Childs, J.E. 2013. Transovarial transmission of Borrelia spirochetes by Ixodes scapularis: a summary of the literature and recent observations. Ticks and Tick-Borne Diseases 4: 46-51.

Stricker, R.B., Johnson, L. 2013. Persistent infection in chronic Lyme disease: does form matter? Research Journal of Infectious Diseases 1: 1-6.