Abstract
The rabies virus infects a wide range of mammalian carnivores across the world with spillovers to non-competent hosts including humans. While not always classified as rabies, there are a wide range of related lyssaviruses of bats that can also spill over to humans. These viruses are transmitted from saliva during aggressive encounters involving biting.
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Notes
- 1.
In this context, non-competent means host species that has no onward transmission; Sect. 15.1 will discuss interspecific spillover in more detail.
- 2.
Recall the incubation period is the time to onset of symptoms, while the latent period is the time to new pathogen presentation such as to be ready for onward transmission.
- 3.
And a “center” that produces amplitude-neutral oscillations like that seen in the Lotka–Volterra predator–prey model if it has only imaginary parts.
- 4.
NB: Unstable equilibria will not be found with this protocol, and if there are more than one steady state (as discussed further in Chap. 11), only one will show up depending on chosen initial conditions.
- 5.
The notation has been changed from the original publication to conform to conventions in the current text.
- 6.
Note again how initial values are log-transformed in start, the first line in the function is x = exp(logx), and the last line returns dS/S, etc., in place of dS that comes from the chain-rule of differentiation and the fact that \(D(\log x) = 1/x\).
- 7.
This equation stems from the result that if \(\tilde {\mathbf {X}}(\varpi )\) denotes the Fourier transform of the vector of state variables, the transform of \(\frac {d\mathbf {X}}{dt}\) will be \(\tilde {\mathbf {X}}(\varpi ) \imath \varpi \); rearranging in matrix form yields \(\tilde {\mathbf {X}}(\varpi ) = \tilde {\mathbf {T}}(\varpi ) \tilde {\mathbf {A}}(\varpi )\). For discrete-time systems, the transfer function is \(\tilde {\mathbf {T}}(\varpi ) = (\tilde {\mathbf {I}}-e^{-\imath \varpi } \tilde {\mathbf {J}})^{-1} \tilde {\mathbf {A}(\varpi )}\) because the Fourier transform of X t−1 is \(\tilde {\mathbf {X}}(\varpi ) e^{-\imath \varpi }\) (see example 2 below).
- 8.
References
Anderson, R. M., Jackson, H. C., May, R. M., & Smith, A. M. (1981). Population dynamics of fox rabies in Europe. Nature, 289(5800), 765–771.
Axelsen, J. B., Yaari, R., Grenfell, B. T., & Stone, L. (2014). Multiannual forecasting of seasonal influenza dynamics reveals climatic and evolutionary drivers. Proceedings of the National Academy of Sciences, 111(26), 9538–9542.
Bjørnstad, O. N., Nisbet, R. M., & Fromentin, J.-M. (2004). Trends and cohort resonant effects in age-structured populations. Journal of Animal Ecology, 73(6), 1157–1167.
Bjørnstad, O. N., Sait, S. M., Stenseth, N. C., Thompson, D. J., & Begon, M. (2001). The impact of specialized enemies on the dimensionality of host dynamics. Nature, 409(6823), 1001.
Bjørnstad, O. N., Shea, K., Krzywinski, M., & Altman, N. (2020b). The SEIRS model for infectious disease dynamics. Nature Methods, 17(6), 557–559.
Bjørnstad, O. N., & Viboud, C. (2016). Timing and periodicity of influenza epidemics. Proceedings of the National Academy of Sciences, 201616052.
Carrat, F., Vergu, E., Ferguson, N. M., Lemaitre, M., Cauchemez, S., Leach, S., & Valleron, A. J. (2008). Time lines of infection and disease in human influenza: A review of volunteer challenge studies. American Journal of Epidemiology, 167(7), 775–785.
Childs, J. E., Curns, A. T., Dey, M. E., Real, L. A., Feinstein, L., Bjørnstad, O. N., & Krebs, J. W. (2000). Predicting the local dynamics of epizootic rabies among raccoons in the united states. Proceedings of the National Academy of Sciences, 97(25), 13666–13671.
Coyne, M. J., Smith, G., & McAllister, F. E. (1989). Mathematic model for the population biology of rabies in raccoons in the mid-Atlantic states. American Journal of Veterinary Research, 50(12), 2148–2154.
Dushoff, J., Plotkin, J. B., Levin, S. A., & Earn, D. J. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proceedings of the National Academy of Sciences, 101(48), 16915–16916.
Hope-Simpson, R. E. (1981). The role of season in the epidemiology of influenza. Journal of Hygiene, 86(01), 35–47.
Keeling, M. J., & Rohani, P. (2008). Modeling infectious diseases in humans and animals. Princeton University Press.
Koelle, K., Cobey, S., Grenfell, B., & Pascual, M. (2006). Epochal evolution shapes the phylodynamics of interpandemic influenza A (H3N2) in humans. Science, 314(5807), 1898–1903.
Kooperberg, C., Stone, C. J., & Truong, Y. K. (1995). Logspline estimation of a possibly mixed spectral distribution. Journal of Time Series Analysis, 16(4), 359–388.
Murray, J. D., Stanley, E. A., & Brown, D. L. (1986). On the spatial spread of rabies among foxes. Proceedings of the Royal Society of London. Series B, 111–150.
Nisbet, R. M., & Gurney, W. (1982). Modelling fluctuating populations. John Wiley and Sons Limited.
Priestley, M. B. (1981). Spectral analysis and time series. Academic press.
Rosatte, R., Sobey, K., Donovan, D., Allan, M., Bruce, L., Buchanan, T., & Davies, C. (2007). Raccoon density and movements after population reduction to control rabies. Journal of Wildlife Management, 71(7), 2373.
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Bjørnstad, O. (2023). Stability and Resonant Periodicity. In: Epidemics. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-031-12056-5_10
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