Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation
Introduction
The innate ability of the influenza virus to generate secondary cases of infection over short windows in time mean that the timely implementation of pandemic containment strategies is critical (Colizza et al., 2007, Ferguson et al., 2005, Ferguson et al., 2006, Germann et al., 2006, Longini et al., 2004, Longini et al., 2005). The identification, evaluation, and implementation of effective regional, national or global pandemic mitigation plans can be enhanced with the assistance of mathematical frameworks and the extensive simulations and/or analysis of appropriate submodels. Preparedness plans dealing with the allocation of antiviral medications must account for a multitude of factors including the pathogens’ virulence (defined in terms of case fatality rates in the population) which plays a central role in the assessment of the size of the antiviral medication stockpile for a community (Lipsitch et al., 2009). These levels of assessment take on a new meaning when the antiviral drug supplies are insufficient (Bar et al., 2009, Merler et al., 2009) and consequently, the use of non-pharmaceutical interventions or treatment measures that include the distribution of face masks and/or the availability of ventilators must also be factored in any preparedness plan (Ferguson et al., 2006, Merler et al., 2009, Tracht et al., 2010).
The world's insufficient capacity to produce antiviral drugs and vaccines (specially the new 2009 H1N1pdm influenza vaccine) during an emergency such as the one posed by the 2009 H1N1pdm influenza virus raises concerns at multiple levels (Fedson, 2003, Gostin and Berkman, 2007, Kotalik, 2005, Ulmer and Liu, 2002). The stockpiles of antiviral drugs (and H1N1pdm influenza vaccine) during a pandemic event are expected to be in the hands of the industrialized nations. Countries with high population densities and limited access to quality health care like Mexico and/or India do not have the infrastructure to produce antiviral drugs to meet their needs during this type of emergencies. Poor nations did not get timely access to what one would consider minimally adequate drug stockpiles, equipment or vaccine supplies. In fact, without the efforts of the World Health Organization (WHO) a large number of nations would have had no access to the most basic pandemic medical supplies at all. The unusual levels of morbidity and mortality among young adults have raised additional concerns (e.g., Chowell et al., 2009a, Nishiura et al., 2009, Reichert et al., 2010). Who should be vaccinated first? The young, the elderly, expecting women, or emergency personnel? The usefulness of the World Health Organization's (WHO) definition of pandemic is now being questioned since it appears that the severity of this pandemic appears to be lower than that associated with past influenza pandemics.
The task of identifying “optimal” control strategies that minimize the impact of influenza pandemics through the judicious use of a limited antiviral drug supplies in combination with measures like isolation, is the focus of this manuscript. The parameters used in our dynamical systems model are initially calibrated using influenza pandemic data of the 1918 influenza pandemic (Chowell et al., 2006), our severe pandemic baseline scenario. The usefulness of intervention alternatives that involve the distribution of antiviral drugs and/or the isolation of hospitalized patients are explored. Optimal control theory (Fleming and Rishel, 1975, Lenhart and Workman, 2007, Pontryagin et al., 1962), with a history of successful applications in biological, medical and industrial problems (Behncke, 2000, Blayneh et al., 2009, Jung et al., 2002, Rowthorn et al., 2009), is the primary tool used in our analysis.
This paper is organized as follows: Section 2 describes the model including several control functions and defines the objective functionals used in the optimal control framework. We present and compare the results of numerical simulations for five scenarios in Section 3, and our thoughts and conclusions are summarized in Section 4.
Section snippets
Influenza pandemic model with controls
Optimal control theory is used to explore the impact of antiviral treatment and isolation strategies in situations that mimic 1918-like influenza pandemic scenarios (Chowell et al., 2006). We calibrate our model using parameter estimates that correspond to the worst influenza pandemic in record (Andreasen et al., 2008, Chowell et al., 2006, Mills et al., 2004). Intervention strategies (policies) are modeled by the functions ui(t) (i=1,2,3) that externally control the number of clinical cases
Numerical results
In this section we present the results of selected simulations generated by the numerical implementation of the intervention strategies described in Section 2 under unlimited and finite resource scenarios. Results of the sensitivity analyses on some of the model parameters in System (1) are given in Section 3.2. In this section we only report the sensitivity analyses associated with the unlimited antiviral resources scenarios.
Discussion
Mitigating the impact and spread of the ongoing influenza A (H1N1) pandemic was on the minds of every newscaster, government official, aspiring politician, public health officials and literally billions of individuals around the world just a couple of months ago. The rate of growth/spread of influenza A(H1N1) news reports even surpassed the rate of growth of this new strain of influenza A around the world (Ginsberg et al., 2009). The rather unusual age-dependent patterns of spread, morbidity
Acknowledgements
This project have been partially supported by grants from the National Science Foundation (NSF-Grant DMS-0502349), the National Security Agency (NSA-Grant H98230-06-1-0097), the Alfred T. Sloan Foundation and the Office of the Provost of Arizona State University. Finally, we would like to thank the anonymous referees for their valuable comments.
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