# Forecasting Epidemics Through Nonparametric Estimation of Time-Dependent Transmission Rates Using the SEIR Model

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## Abstract

Deterministic and stochastic methods relying on early case incidence data for forecasting epidemic outbreaks have received increasing attention during the last few years. In mathematical terms, epidemic forecasting is an ill-posed problem due to instability of parameter identification and limited available data. While previous studies have largely estimated the time-dependent transmission rate by assuming specific functional forms (e.g., exponential decay) that depend on a few parameters, here we introduce a novel approach for the reconstruction of nonparametric time-dependent transmission rates by projecting onto a finite subspace spanned by Legendre polynomials. This approach enables us to effectively forecast future incidence cases, the clear advantage over recovering the transmission rate at finitely many grid points within the interval where the data are currently available. In our approach, we compare three regularization algorithms: variational (Tikhonov’s) regularization, truncated singular value decomposition (TSVD), and modified TSVD in order to determine the stabilizing strategy that is most effective in terms of reliability of forecasting from limited data. We illustrate our methodology using simulated data as well as case incidence data for various epidemics including the 1918 influenza pandemic in San Francisco and the 2014–2015 Ebola epidemic in West Africa.

## Keywords

Parameter estimation Regularization Volterra equation## References

- Bakushinsky AB, Smirnova AB, Liu H (2015) A nonstandard approximation of pseudoinverse and a new stopping criterion for iterative regularization. Inverse Ill Posed Probl 23(3):195–210MathSciNetMATHGoogle Scholar
- Biggerstaff M, Alper D, Dredze M, Fox S, Fung IC, Hickmann KS, Lewis B, Rosenfeld R, Shaman J, Tsou MH et al (2016) Results from the centers for disease control and prevention’s predict the 2013–2014 Influenza Season Challenge. BMC Infect Dis 16:357CrossRefGoogle Scholar
- Bjørnstad ON, Finkenstädt BF, Grenfell BT (2002) Dynamics of measles epidemics: estimating scaling of transmission rates using a time series sir model. Ecol Monogr 72(2):169–184CrossRefGoogle Scholar
- Camacho A, Kucharski A, Aki-Sawyerr Y, White MA, Flasche S, Baguelin M, Pollington T, Carney JR, Glover R, Smout E, Tiffany A, Edmunds WJ, Funk S (2015) Temporal changes in Ebola Transmission in Sierra Leone and implications for control requirements: a real-time modelling Study. PLoS Curr. doi: 10.1371/currents.outbreaks.406ae55e83ec0b5193e30856b9235ed2
- Capistrán MA, Moreles MA, Lara B (2009) Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus. Bull Math Biol 71(8):1890–1901MathSciNetCrossRefMATHGoogle Scholar
- Cauchemez S, Valleron A-J, Boelle P-Y, Flahault A, Ferguson NM (2008) Estimating the impact of schoolclosure on influenza transmission from sentinel data. Nature 452(7188):750–754CrossRefGoogle Scholar
- Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM (2004) The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J Theor Biol 229(1):119–126MathSciNetCrossRefGoogle Scholar
- Chowell G, Nishiura H, Bettencourt LM (2007) Comparative estimation of the reproduction number for pandemic influenza from daily case notification data. J R Soc Interface R Soc 4(12):155–166CrossRefGoogle Scholar
- Chowell G, Sattenspiel L, Bansal S, Viboud C (2016) Mathematical models to characterize early epidemic growth: a review. Phys Life Rev 18:66–97CrossRefGoogle Scholar
- Chowell G, Viboud C, Hyman JM, Simonsen L (2015) The Western Africa ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLoS Curr. doi: 10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261
- Chowell G, Viboud C, Simonsen L, Moghadas S (2016) Characterizing the reproduction number of epidemics with early sub-exponential growth dynamics. ArXiv preprint arXiv:1603.01216
- Chretien JP, Riley S, George DB (2015) Mathematical modeling of the West Africa Ebola epidemic. eLife 4:e09186Google Scholar
- Colgate SA, Stanley EA, Hyman JM, Layne SP, Qualls C (1989) Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States. Proc Nat Acad Sci USA 86(12):4793–4797CrossRefGoogle Scholar
- Dureau J, Kalogeropoulos K, Baguelin M (2013) Capturing the time-varying drivers of an epidemic using stochastic dynamical systems. Biostatistics 14(3):541–555CrossRefGoogle Scholar
- Engl H, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer Academic, DordechtCrossRefMATHGoogle Scholar
- Finkenstädt BF, Grenfell BT (2000) Time series modelling of childhood diseases: a dynamical systems approach. J R Stat Soc Ser C (Appl Stat) 49(2):187–205MathSciNetCrossRefMATHGoogle Scholar
- Hadeler KP (2011) Parameter identification in epidemic models. Math Biosci 229(2):185–189MathSciNetCrossRefMATHGoogle Scholar
- Kiskowski MA (2014) A three-scale network model for the early growth dynamics of 2014 West Africa Ebola epidemic. PLOS Curr. doi: 10.1371/currents.outbreaks.c6efe8274dc55274f05cbcb62bbe6070
- Lange A (2016) Reconstruction of disease transmission rates: applications to measles, dengue, and influenza. J Theor Biol 400:138–153MathSciNetCrossRefMATHGoogle Scholar
- Lekone PE, Finkenstädt BF (2006) Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study. Biometrics 62(4):1170–1177MathSciNetCrossRefMATHGoogle Scholar
- Lewnard JA, Ndeffo Mbah ML, Alfaro-Murillo JA, Altice FL, Bawo L, Nyenswah TG, Galvani AP (2014) Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis. Lancet Infect Dis 14(12):1189–1195CrossRefGoogle Scholar
- Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S, James L, Gopalakrishna G, Chew SK, Tan CC, Samore MH et al (2003) Transmission dynamics and control of severe acute respiratory syndrome. Science 300(5627):1966–1970CrossRefGoogle Scholar
- Meltzer MI, Santibanez S, Fischer LS, Merlin TL, Adhikari BB, Atkins CY, Campbell C, Fung IC, Gambhir M, Gift T et al (2016) Modeling in real time during the Ebola response. MMWR Suppl 65(3):85–89CrossRefGoogle Scholar
- Merler S, Ajelli M, Fumanelli L, Gomes MF, Piontti AP, Rossi L, Chao DL, Longini IM Jr, Halloran ME, Vespignani A (2015) Spatiotemporal spread of the 2014 outbreak of Ebola virus disease in Liberia and the effectiveness of non-pharmaceutical interventions: a computational modelling analysis. Lancet Infect Dis 15(2):204–211CrossRefGoogle Scholar
- Morozov VA (1984) Methods for solving incorrectly posed problems. Springer, BerlinCrossRefGoogle Scholar
- Newman MEJ (2002) Spread of epidemic disease on networks. Phys Rev E 66(1):016128MathSciNetCrossRefGoogle Scholar
- Pollicott M, Wang H, Weiss H (2012) Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem. J Biol Dyn 6(2):509–523MathSciNetCrossRefGoogle Scholar
- Ponciano JM, Capistrán MA (2011) First principles modeling of nonlinear incidence rates in seasonal epidemics. PLoS Comput Biol 7(2):e1001079MathSciNetCrossRefGoogle Scholar
- Rivers CM, Lofgren ET, Marathe M, Eubank S, Lewis BL (2014) Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. ArXiv preprint arXiv:1409.4607
- Szendroi B, Csnyi G (2004) Polynomial epidemics and clustering in contact networks. Proc R Soc Lond B Biol Sci 271(Suppl 5):S364–S366CrossRefGoogle Scholar
- Szusz EK, Garrison LP, Bauch CT (2010) A review of data needed to parameterize a dynamic model of measles in developing countries. BMC Res Notes 3(1):1CrossRefGoogle Scholar
- Taylor BP, Dushoff J, Weitz JS (2016) Stochasticity and the limits to confidence when estimating r_0 of Ebola and other emerging infectious diseases. ArXiv preprint arXiv:1601.06829
- Viboud C, Simonsen L, Chowell G (2016) A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15:27–37CrossRefGoogle Scholar
- World Health Organization, Statement on the 1st meeting of the IHR Emergency Committee on the 2014 Ebola outbreak in West Africa, retrieved 6/10/15. http://www.who.int/mediacentre/news/statements/2014/ebola-20140808/en/
- World Health Organization, Ebola Situation Report—30 March 2016, retrieved 6/01/16. http://apps.who.int/ebola/current-situation/ebola-situation-report-30-march-2016
- Xia Y, Bjørnstad ON, Grenfell BT (2004) Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. Am Nat 164(2):267–281CrossRefGoogle Scholar