A Model for Coupled Outbreaks Contained by Behavior Change
Large epidemics such as the recent Ebola crisis in West Africa occur when local efforts to contain outbreaks fail to overcome the probabilistic onward transmission to new locations. As a result, there may be large differences in total epidemic size from similar initial conditions. This work seeks to determine the extent to which the effects of behavior changes and metapopulation coupling on epidemic size can be characterized. While mathematical models have been developed to study local containment by social distancing, intervention and other behavior changes, their connection to larger-scale transmission is relatively underdeveloped. We make use of the assumption that behavior changes limit local transmission before susceptible depletion to develop a time-varying birth-death process capturing the dynamic decrease of the transmission rate associated with behavior changes. We derive an expression for the mean outbreak size of this model and show that the distribution of outbreak sizes is approximately geometric. This allows a probabilistic extension whereby infected individuals may initiate new outbreaks. From this model we characterize the overall epidemic size as a function of the behavior change rate and the probability that an infected individual starts a new outbreak. We find good agreement between the analytical results and stochastic simulations leading to novel findings including critical learning rates that demarcate large and small epidemic sizes.
KeywordsEbola Epidemic model Behavior change Transmission rate Birth-death process Metapopulation model
Research reported here was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number U01GM110744 and the National Science Foundation under Rapid Award Number 1515194. The content is solely the responsibility of the authors and does not necessarily reflect the official views of the National Institutes of Health or the National Science Foundation.
- 2.Ball, F., Britton, T., House, T., Isham, V., Mollison, D., Pellis, L., Scalia Tomba, G.: Seven challenges for metapopulation models of epidemics, including households models. Epidemics 10, 63–67 (2015)Google Scholar
- 3.Brauer, F.: A simple model for behaviour change in epidemics. BMC Public Health 11 Suppl 1, S3 (2011)Google Scholar
- 4.Drake, J.M., Bakach, I., Just, M.R., ORegan, S.M., Gambhir, M., Fung, I.C.H.: Transmission models of historical Ebola outbreaks. Emerging Infect. Dis. 21, 1447–1450 (2015)Google Scholar
- 6.Drake, J.M., Kaul, R.B., Alexander, L.W., Regan, S.M.O., Kramer, M., Pulliam, J.T., Ferrari, M.J., Park, A.W.: Ebola cases and health system demand in Liberia. PLOS Biol. 13(1), e1002,056 (2015)Google Scholar
- 8.Johnson, N.L., Kotz, S., Kemp, A.W.: Univariate Discrete Distributions. Wiley, New York (1992)Google Scholar
- 12.Lofgren, E.T., Halloran, M.E., Rivers, C.M., Drake, J.M., Porco, T.C., Lewis, B., Yang, W., Vespignani, A., Shaman, J., Eisenberg, J.N.S., Eisenberg, M.C., Marathe, M., Scarpino, S.V., Alexander, K.A., Meza, R., Ferrari, M.J., Hyman, J.M., Meyers, L.A., Eubank, S.: Mathematical models: a key tool for outbreak response. Proc. Natl. Acad. Sci. USA 111(51), 18,095–18,096 (2014)Google Scholar
- 14.Poletto, C., Pelat, C., Levy-Bruhl, D., Yazdanpanah, Y., Boelle, P.Y., Colizza, V.: Assessment of the Middle East respiratory syndrome coronavirus (MERS-CoV) epidemic in the Middle East and risk of international spread using a novel maximum likelihood analysis approach. Eurosurveillance 19(23), 20824 (2014)Google Scholar
- 16.UN: World Population Prospects: The 2015 Revision, Key Findings and Advance Tables. Technical Report, United Nations, Department of Economic and Social Affairs, Population Division (2015)Google Scholar
- 17.WHO Ebola Response Team: Ebola Virus Disease in West Africa - The First 9 Months of the Epidemic and Forward Projections. New Engl. J. Med. 371(16), 1481–1495 (2014)Google Scholar