Stochastic Population and Epidemic Models pp 21-27 | Cite as
Applications of Multi-Type Branching Processes
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Abstract
Two applications of multi-type branching processes to epidemic models are presented. The first application is to an SEIR epidemic model and the second application is to the same epidemic model but with dispersal. The SEIR epidemic is modeled as a two-type branching process. Occurrence of an outbreak depends on the number of exposed and infectious individuals. It is shown that the offspring pgfs for the exposed and infectious populations lead to an explicit formula for the probability of an outbreak.
Keywords
Epidemic Model Basic Reproduction Number Severe Acute Respiratory Syndrome Extinction Probability Infectious Individual3.1 Introduction
Two applications of multi-type branching processes to epidemic models are presented. The first application is to an SEIR epidemic model and the second application is to the same epidemic model but with dispersal. The SEIR epidemic is modeled as a two-type branching process. Occurrence of an outbreak depends on the number of exposed and infectious individuals. It is shown that the offspring pgfs for the exposed and infectious populations lead to an explicit formula for the probability of an outbreak. In the SEIR model with dispersal, the case of two regions with different healthcare situations are considered. One region has poor healthcare versus another region with excellent healthcare. It is shown that the rate and the direction of movement have a large impact on the occurrence of an outbreak. Branching process theory is used to investigate the probability of an outbreak when the movement rates differ between the two regions.
Although the SIR and SEIR epidemic models are simple, they are often used as a first approximation during or after disease outbreaks to provide estimates of the potential spread of the disease or to understand the pattern of spread. For example, SIR and SEIR epidemic models in conjunction with data provided useful information about the spread of the 2002–2003 SARS (Severe Acute Respiratory Syndrome) pandemic which began in China, the 2009–2010 H1N1 influenza pandemic which began in Mexico, and the 2014 Ebola outbreak in Africa [11, 22, 33].
3.2 SEIR Epidemic
Transition rates for the CTMC SEIR epidemic model (MC Rates) and for the corresponding branching process approximation for exposed and infectious individuals (BP Rates).
Event | \(\varDelta \mathbf{X}(t)\) | MC Rates | BP Rates | |
---|---|---|---|---|
1 | (−1, 1, 0, 0) | \(\beta \dfrac{X_{1}(t)} {N(t)} I(t)\) | β I(t) | |
2 | (0, −1, 1, 0) | δ X_{2}(t) | δ X_{2}(t) | |
3 | (0, 0, −1, 1) | γ X_{3}(t) | γ X_{3}(t) | |
4 | (0, 0, −1, 0) | α X_{3}(t) | α X_{3}(t) |
3.3 Epidemic Dispersal
Suppose disease is spread between two populations each occupying different regions or patches and modeled by the SEIR epidemic equations within each patch. In population 1, poor healthcare facilities result in frequent disease outbreaks. In population 2, better healthcare facilities and reduced mortality and recovery rates result in no major outbreaks. For population 1, the basic reproduction number is greater than one but for population 2, the basic reproduction number is less than one. With dispersal between these two populations, the outcome changes depending on the direction and the rate of dispersal.
A branching process approximation for the corresponding CTMC SEIR model for two patches can be applied if the population size is large but the exposed and infectious population sizes are small. For the branching process, we are only interested in the exposed and infectious stages. The direction and the rate of movement of individuals in these disease stages have a large impact on the probability of an outbreak.
3.4 Summary
The multi-type branching process application to an SEIR epidemic with dispersal illustrates the importance of controlling movement into and out of particular regions to prevent an outbreak. Although prevention and control measures are more complex in real epidemic or pandemic situations, the basic SIR and SEIR models are often used in conjunction with data to help estimate the potential spread of the disease, e.g., SARS, influenza, and Ebola [11, 22, 33]. The control measures in pandemic situations often include travel restrictions, quarantine, isolation, and drugs such as antiviral medication to prevent infection. Other specific applications of branching processes to infectious disease models include vector-transmitted diseases [4, 6, 9, 18], HIV infection, [12] and bovine respiratory syncytial virus [19].
References
- 4.L. J. S. Allen and G. E. Lahodny Jr. 2012. Extinction thresholds in deterministic and stochastic epidemic models. J. Biol. Dyn. 6: 590–611.CrossRefGoogle Scholar
- 6.L. J. S. Allen and P. van den Driessche. 2013. Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models. Math. Biosci. 243: 99–108.CrossRefMathSciNetzbMATHGoogle Scholar
- 9.M. S. Bartlett. 1964. The relevance of stochastic models for large-scale epidemiological phenomena. J. Roy. Stat. Soc., Series C 13: 2–8.Google Scholar
- 11.G. Chowell, C. Castillo-Chavez, P. W. Fenimore, C. M. Kribs-Zaleta, L. Arriola and J. M. Hyman. 2004. Model parameters and outbreak control for SARS. Emerging Infectious Diseases 10: 1258–1263.CrossRefGoogle Scholar
- 12.J. M. Conway, B. P. Konrad, and D. Coombs. 2013. Stochastic analysis of pre- and postexposure prophylaxis against HIV infection. SIAM J. Appl. Math. 73: 904–928.CrossRefMathSciNetzbMATHGoogle Scholar
- 18.D. A. Griffiths. 1972. A bivariate birth-death process which approximates to the spread of a disease involving a vector. J. Applied Prob. 9: 65–75.CrossRefzbMATHGoogle Scholar
- 19.M. Griffiths and D. Greenhalgh. 2011. The probability of extinction in a bovine respiratory syncytial virus epidemic model. Math. Biosci. 231: 144–158.CrossRefMathSciNetGoogle Scholar
- 22.Y.-H. Hsieh Y-H, S. Ma, J. X. Velasco Hernandez, V. J. Lee, W. Y. Lim. 2011. Early outbreak of 2009 influenza A (H1N1) in Mexico prior to identification of pH1N1 virus. PLoS ONE 6(8): e23853. doi: 10.1371/journal.pone.0023853Google Scholar
- 33.C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank, and B. L. Lewis. 2014. Modeling the Impact of Interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLOS Currents Outbreaks. 2014 Nov 6. Edition 2. doi: 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c.Google Scholar
- 36.P. van den Driessche and J. Watmough. 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180: 29–48.CrossRefMathSciNetzbMATHGoogle Scholar