Exploring optimal control strategies in seasonally varying flu-like epidemics
Introduction
Mathematical models of disease transmission and control have become established tools for gaining an improved understanding of the transmission dynamics of infectious diseases and the potential impact of control interventions. In particular, the availability of computational power has allowed the numerical simulation of complex transmission models that involve multiple sets of rates of change equations or even models that keep track of detailed individual-level interactions and epidemiological transitions (e.g., Ferguson et al., 2005, Ferguson et al., 2006, Colizza et al., 2007). More recently, researchers have started to embark on the challenging task of using epidemic models for forecasting morbidity and mortality impact during epidemics as a function of various control interventions or changes in population behavior (Chretien et al., 2014, Chretien et al., 2015). This underscores the need to better understand the potential predictive ability of epidemic models that incorporate the complex interplay of time-dependent transmission mechanisms and control interventions.
In the context of respiratory disease transmission, multiple applications of epidemic models have been developed to explore the transmission dynamics and the effects of control interventions (Anderson and May, 1991, Arino et al., 2009, Qiu and Feng, 2010). For instance, some epidemic models have been designed to assess the relative impact of intervention strategies including social distancing and antiviral treatment on containing an influenza pandemic (Ferguson et al., 2005, Ferguson et al., 2006, Gani et al., 2005, Longini et al., 2004, Longini et al., 2005). Overall, these type of studies highlight the role of interventions in the context of realistic delays in the identification of an unfolding epidemic and initiation of control measures such as the implementation of targeted antiviral-based interventions. Apart from control interventions, for the particular case of influenza, there are a number of key mechanistic ingredients that need to be incorporated into mathematical models in order to capture realistic transmission dynamics at the population level. One important feature of influenza-like infections is the role of seasonal drivers that tend to limit the spread of the virus to particular periods of time of the year (Mummert et al., 2013, Shaman and Kohn, 2009, Tamerius et al., 2015, Towers and Feng, 2009). This mechanism could then allow a fraction of the population to remain susceptible and available for potential infection the next time window when the virus can enjoy the appropriate environmental conditions for relatively easy human-to-human transmission (Omori and Sasaki, 2013, Uziel and Stone, 2012). As a result of these seasonal drivers, respiratory infections are able to generate recurrent epidemics year after year or even multiple “waves” of pandemic influenza (Chowell et al., 2006, Chowell et al., 2008, Dowell, 2001) during relatively short time periods as a pandemic virus can initially enjoy a large fraction of susceptible individuals together with changing environmental conditions. In addition, confined settings such as schools have the potential to amplify the transmission potential by significantly increasing the transmission rate among school-aged children during school activity periods. For instance, the 2009 H1N1 pandemic influenza in Mexico displayed three waves of disease in 2009 with an initial wave in April–May, a second wave in June–July, and a widespread third wave in August–December associated with the return of student to school activities after summer vacations (Cauchemez et al., 2008, Chowell et al., 2011, Tamerius et al., 2015). The overall transmission dynamics are further complicated by additional factors including the role of control interventions such as annual immunization strategies, the role of antiviral medications for treatment of prophylaxis, and the role of temporary school closings, which could delay transmission and give time for the development of effective vaccines (Jackson et al., 2014, Washington Times, 2009).
Even relatively simple SIR-type compartmental models have been shown to provide surprising epidemic outcomes, e.g., when the transmission rate is time dependent (Feng et al., 2011). For instance, some periodic epidemic systems exhibit large epidemic cycles even when in the absence of any interventions (Bacaer and Gomes, 2009, Bacaer and Ait Dads, 2011). Moreover, using a relatively simple model with seasonality that incorporates the role of vaccination and treatment strategies proposed in Feng et al. (2011), researchers found counterintuitive results whereby vaccination or treatment could increase the peak size and the final epidemic size. Another study has shown that treatment interventions can increase the final epidemic size in a two-strain influenza model (Xiao et al., 2015). However, in those models vaccination and treatment rates have been kept constant for all time. Here, we build on the mathematical model in Feng et al. (2011) in order to assess the role of time variations in vaccination and treatment rates using an optimal control framework. We investigate how optimal control interventions affect epidemic outcomes for different transmission and control scenarios. For this purpose, we parameterize a simple model of influenza transmission and control to explore the effectiveness of intervention strategies in various scenarios. Our work aims to illustrate in a systematic and comparative analysis, the role of optimal control strategies on the sensitivity of epidemic outcomes by directly contrasting results based on seasonal transmission rates with those using a constant transmission rate.
Section snippets
Modeling optimal vaccination and treatment
Optimal control theory has provided useful insights in various fields including basic science, engineering, and economics (Fleming and Rishel, 1975, Pontryagin et al., 1962). The number of applications of optimal control theory in epidemiological and biological models has been on the rise (Lenhart and Workman, 2007). Recently, epidemiological models have incorporated vaccination, treatment and isolation controls to study the impact of optimal control on the spread of influenza (Lee et al., 2010
Numerical results
We examine epidemic outcomes in terms of the temporal variation in the proportion of infected individuals and final attack ratio assuming constant and time-dependent transmission rates. We first analyze the epidemic dynamics using simulations of the model in the absence of interventions. Next, we explore the impact of interventions in terms of vaccination, treatment or a combination of both interventions using our previously defined optimal control framework. In our simulation scenarios, we
Discussion
We have investigated the sensitivity of epidemic outcomes in terms of the epidemic trajectory and the final attack ratio in the context of optimal control interventions applied to seasonally varying flu-like transmission dynamics as a function of various key parameters including , the timing of epidemic onset t0, the strength of the seasonal forcing ϵ, and the start of the control interventions tc. We contrast our findings using a simple model of flu-like transmission dynamics assuming a
Conclusion
In summary, we have investigated dynamics of seasonally varying flu-like epidemics with and without control strategies. Using numerical simulations, we explored the impact of optimal control strategies as a function of the basic reproduction number ( ), the timing of introduction of the initial infectious individuals (t0), the timing of the start of control interventions (tc), and the strength of the seasonal forcing (ϵ). Our results underscore the sensitivity of the epidemic outcomes in
Acknowledgment
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 20152194). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve our manuscript.
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