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Mathematical Biosciences & Engineering

2011 , Volume 8 , Issue 1

Special Issue on
Mathematical Models, Challenges, and Lessons Learned from the 2009 A/H1N1 Influenza Pandemic

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Carlos Castillo-Chavez and Gerardo Chowell
2011, 8(1): i-vi doi: 10.3934/mbe.2011.8.1i +[Abstract](107) +[PDF](172.7KB)
A new A/H1N1 influenza virus strain was officially reported/identified in Mexico City on April 13 2009 and over a period of two-weeks the WHO pandemic alert was moved from level 3 to level 5. By May 2, a total of 141 cases had been confirmed in 19 states across the USA. Additional cases were soon confirmed in fifteen countries in Europe, Canada, New Zealand, and Asia. The global reach of this novel strain became evident when a summer influenza incidence high was reached in Japan by May 16, 2009 [2, 4] just about a month after its identification in the New World.

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Pandemic influenza: Modelling and public health perspectives
Julien Arino, Chris Bauch, Fred Brauer, S. Michelle Driedger, Amy L. Greer, S.M. Moghadas, Nick J. Pizzi, Beate Sander, Ashleigh Tuite, P. van den Driessche, James Watmough, Jianhong Wu and Ping Yan
2011, 8(1): 1-20 doi: 10.3934/mbe.2011.8.1 +[Abstract](120) +[PDF](377.8KB)
We describe the application of mathematical models in the study of disease epidemics with particular focus on pandemic influenza. We outline the general mathematical approach and the complications arising from attempts to apply it for disease outbreak management in a real public health context.
Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different "waves" of A-H1N1pdm cases observed in México during 2009
Marco Arieli Herrera-Valdez, Maytee Cruz-Aponte and Carlos Castillo-Chavez
2011, 8(1): 21-48 doi: 10.3934/mbe.2011.8.21 +[Abstract](153) +[PDF](1853.3KB)
Influenza outbreaks have been of relatively limited historical interest in México. The 2009 influenza pandemic not only changed México's health priorities but also brought to the forefront some of the strengths and weaknesses of México's epidemiological surveillance and public health system. A year later, México's data show an epidemic pattern characterized by three "waves''. The reasons this three-wave patterns are theoretically investigated via models that incorporate México's general trends of land transportation, public health measures, and the regular opening and closing of schools during 2009. The role of vaccination is also studied taking into account delays in access and limitations in the total and daily numbers of vaccines available. The research in this article supports the view that the thee epidemic "waves" are the result of the synergistic interactions of three factors: regional movement patterns of Mexicans, the impact and effectiveness of dramatic social distancing measures imposed during the first outbreak, and the summer release of school children followed by their subsequent return to classes in the fall. The three "waves" cannot be explained by the transportation patterns alone but only through the combination of transport patterns and changes in contact rates due to the use of explicit or scheduled social distancing measures. The research identifies possible vaccination schemes that account for the school calendar and whose effectiveness are enhanced by social distancing measures. The limited impact of the late arrival of the vaccine is also analyzed.
Joint quantification of transmission dynamics and diagnostic accuracy applied to influenza
Hiroshi Nishiura
2011, 8(1): 49-64 doi: 10.3934/mbe.2011.8.49 +[Abstract](115) +[PDF](975.8KB)
The influenza A (H1N1) pandemic 2009 posed an epidemiological challenge in ascertaining all cases. Although the counting of all influenza cases in real time is often not feasible, empirical observations always involve diagnostic test procedures. This offers an opportunity to jointly quantify transmission dynamics and diagnostic accuracy. We have developed a joint estimation procedure that exploits parsimonious models to describe the epidemic dynamics and that parameterizes the number of test positives and test negatives as a function of time. Our analyses of simulated data and data from the empirical observation of interpandemic influenza A (H1N1) from 2007-08 in Japan indicate that the proposed approach permits a more precise quantification of the transmission dynamics compared to methods that rely on test positive cases alone. The analysis of entry screening data for the H1N1 pandemic 2009 at Tokyo-Narita airport helped us quantify the very limited specificity of influenza-like illness in detecting actual influenza cases in the passengers. The joint quantification does not require us to condition diagnostic accuracy on any pre-defined study population. Our study suggests that by consistently reporting both test positive and test negative cases, the usefulness of extractable information from routine surveillance record of infectious diseases would be maximized.
Pandemic mitigation: Bringing it home
Tom Reichert
2011, 8(1): 65-76 doi: 10.3934/mbe.2011.8.65 +[Abstract](68) +[PDF](415.0KB)
In the US, national, regional and even institutional plans for ameliorating the effects of pandemic influenza focus on stockpiling antiviral medications, early production and distribution of vaccine, mass and personal social distancing, and a number of personal hygiene activities. Essential personnel are the first scheduled to receive preventive and therapeutic pharmaceuticals, followed by high risk groups, the largest of which are the elderly. Specific recommendations for protection embody a bunker mentality with a time horizon of two weeks, emulating preparation for a natural disaster. The epidemiology of pandemic influenza is scarcely considered.
   We summarize here the envelope of mortality attributable to epidemic and pandemic influenza in the last 90 years of the last century as a lead in to a presentation of the multinational case age distribution of the novel H1N1 pandemic of 2009. We discuss the sparing of elderly subpopulations in pandemics and the subsequent abrupt resurgence of mortality in the spared age groups as drift variants emerge. The general decline in the baseline of age-specific excess mortality in economically developed countries is characterized and its importance assessed.
   Models of acute and chronic care facilities are discussed and an argument is advanced that society as a whole as well as acute care facilities cannot be protected against incursion and widespread infection in pandemics of severity above low moderate. The key findings of models of chronic care institutions and others that can control public access, such as corporations, are used to describe programs with a realistic chance of providing protection in even severe pandemics. These principles are further mapped onto individual residences. Materials directing institutional and home planning are cited.
The impact of vaccines and vaccinations: Challenges and opportunities for modelers
Roy Curtiss III
2011, 8(1): 77-93 doi: 10.3934/mbe.2011.8.77 +[Abstract](87) +[PDF](340.9KB)
This review focuses on how infectious diseases and their prevention and control by development of vaccines and widespread vaccination has shaped evolution of human civilization and of the animals and plants that humans depend on for food, labor and companionship. After describing major infectious diseases and the current status for control by vaccination, the barriers to infection and the attributes of innate and acquired immunity contributing to control are discussed. The evolution in types of vaccines is presented in the context of developing technologies and in improving adjuvants to engender enhanced vaccine efficacy. The special concerns and needs in vaccine design and development are discussed in dealing with epidemics/pandemics with special emphasis on influenza and current global problems in vaccine delivery.
Prioritization of delayed vaccination for pandemic influenza
Eunha Shim
2011, 8(1): 95-112 doi: 10.3934/mbe.2011.8.95 +[Abstract](85) +[PDF](451.5KB)
Limited production capacity and delays in vaccine development are major obstacles to vaccination programs that are designed to mitigate a pandemic influenza. In order to evaluate and compare the impact of various vaccination strategies during a pandemic influenza, we developed an age/risk-structured model of influenza transmission, and parameterized it with epidemiological data from the 2009 H1N1 influenza A pandemic. Our model predicts that the impact of vaccination would be considerably diminished by delays in vaccination and staggered vaccine supply. Nonetheless, prioritizing limited H1N1 vaccine to individuals with a high risk of complications, followed by school-age children, and then preschool-age children, would minimize an overall attack rate as well as hospitalizations and deaths. This vaccination scheme would maximize the benefits of vaccination by protecting the high-risk people directly, and generating indirect protection by vaccinating children who are most likely to transmit the disease.
Evaluation of vaccination strategies during pandemic outbreaks
Christopher S. Bowman, Julien Arino and S.M. Moghadas
2011, 8(1): 113-122 doi: 10.3934/mbe.2011.8.113 +[Abstract](75) +[PDF](503.9KB)
During pandemic influenza, several factors could significantly impact the outcome of vaccination campaigns, including the delay in pandemic vaccine availability, inadequate protective efficacy, and insufficient number of vaccines to cover the entire population. Here, we incorporate these factors into a vaccination model to investigate and compare the effectiveness of the single-dose and two-dose vaccine strategies. The results show that, if vaccination starts early enough after the onset of the outbreak, a two-dose strategy can lead to a greater reduction in the total number of infections. This, however, requires the second dose of vaccine to confer a substantially higher protection compared to that induced by the first dose. For a sufficiently long delay in start of vaccination, the single-dose strategy outperforms the two-dose vaccination program regardless of its protection efficacy. The findings suggest that the population-wide benefits of a single-dose strategy could in general be greater than the two-dose vaccination program, in particular when the second dose offers marginal increase in the protection induced by the first dose.
Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks
Diána H. Knipl and Gergely Röst
2011, 8(1): 123-139 doi: 10.3934/mbe.2011.8.123 +[Abstract](98) +[PDF](432.0KB)
Finding optimal policies to reduce the morbidity and mortality of the ongoing pandemic is a top public health priority. Using a compartmental model with age structure and vaccination status, we examined the effect of age specific scheduling of vaccination during a pandemic influenza outbreak, when there is a race between the vaccination campaign and the dynamics of the pandemic. Our results agree with some recent studies on that age specificity is paramount to vaccination planning. However, little is known about the effectiveness of such control measures when they are applied during the outbreak. Comparing five possible strategies, we found that age specific scheduling can have a huge impact on the outcome of the epidemic. For the best scheme, the attack rates were up to 10% lower than for other strategies. We demonstrate the importance of early start of the vaccination campaign, since ten days delay may increase the attack rate by up to 6%. Taking into account the delay between developing immunity and vaccination is a key factor in evaluating the impact of vaccination campaigns. We provide a general framework which will be useful for the next pandemic waves as well.
Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza
Olivia Prosper, Omar Saucedo, Doria Thompson, Griselle Torres-Garcia, Xiaohong Wang and Carlos Castillo-Chavez
2011, 8(1): 141-170 doi: 10.3934/mbe.2011.8.141 +[Abstract](123) +[PDF](795.6KB)
The lessons learned from the 2009-2010 H1N1 influenza pandemic, as it moves out of the limelight, should not be under-estimated, particularly since the probability of novel influenza epidemics in the near future is not negligible and the potential consequences might be huge. Hence, as the world, particularly the industrialized world, responded to the potentially devastating effects of this novel A-H1N1 strain with substantial resources, reminders of the recurrent loss of life from a well established foe, seasonal influenza, could not be ignored. The uncertainties associated with the reported and expected levels of morbidity and mortality with this novel A-H1N1 live in a backdrop of $36,000$ deaths, over 200,000 hospitalizations, and millions of infections (20% of the population) attributed to seasonal influenza in the USA alone, each year. So, as the Northern Hemisphere braced for the possibility of a potentially "lethal" second wave of the novel A-H1N1 without a vaccine ready to mitigate its impact, questions of who should be vaccinated first if a vaccine became available, came to the forefront of the discussion. Uncertainty grew as we learned that the vaccine, once available, would be unevenly distributed around the world. Nations capable of acquiring large vaccine supplies soon became aware that those who could pay would have to compete for a limited vaccine stockpile. The challenges faced by nations dealing jointly with seasonal and novel A-H1N1 co-circulating strains under limited resources, that is, those with no access to novel A-H1N1 vaccine supplies, limited access to the seasonal influenza vaccine, and limited access to antivirals (like Tamiflu) are explored in this study. One- and two-strain models are introduced to mimic the influenza dynamics of a single and co-circulating strains, in the context of a single epidemic outbreak. Optimal control theory is used to identify and evaluate the "best" control policies. The controls account for the cost associated with social distancing and antiviral treatment policies. The optimal policies identified might have, if implemented, a substantial impact on the novel H1N1 and seasonal influenza co-circulating dynamics. Specifically, the implementation of antiviral treatment might reduce the number of influenza cases by up to 60% under a reasonable seasonal vaccination strategy, but only by up to 37% when the seasonal vaccine is not available. Optimal social distancing policies alone can be as effective as the combination of multiple policies, reducing the total number of influenza cases by more than 99% within a single outbreak, an unrealistic but theoretically possible outcome for isolated populations with limited resources.
A note on the use of influenza vaccination strategies when supply is limited
Sunmi Lee, Romarie Morales and Carlos Castillo-Chavez
2011, 8(1): 171-182 doi: 10.3934/mbe.2011.8.171 +[Abstract](117) +[PDF](737.0KB)
The 2009 A (H1N1) influenza pandemic was rather atypical. It began in North America at the start of the spring and in the following months, as it moved south, efforts to develop a vaccine that would mitigate the potential impact of a second wave were accelerated. The world's limited capacity to produce an adequate vaccine supply over just a few months resulted in the development of public health policies that "had" to optimize the utilization of limited vaccine supplies. Furthermore, even after the vaccine was in production, extensive delays in vaccine distribution were experienced for various reasons. In this note, we use optimal control theory to explore the impact of some of the constraints faced by most nations in implementing a public health policy that tried to meet the challenges that come from having access only to a limited vaccine supply that is never 100% effective.
A note on the use of optimal control on a discrete time model of influenza dynamics
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez and Carlos Castillo-Chavez
2011, 8(1): 183-197 doi: 10.3934/mbe.2011.8.183 +[Abstract](162) +[PDF](291.3KB)
A discrete time Susceptible - Asymptomatic - Infectious - Treated - Recovered (SAITR) model is introduced in the context of influenza transmission. We evaluate the potential effect of control measures such as social distancing and antiviral treatment on the dynamics of a single outbreak. Optimal control theory is applied to identify the best way of reducing morbidity and mortality at a minimal cost. The problem is solved by using a discrete version of Pontryagin's maximum principle. Numerical results show that dual strategies have stronger impact in the reduction of the final epidemic size.
Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics
Karen R. Ríos-Soto, Baojun Song and Carlos Castillo-Chavez
2011, 8(1): 199-222 doi: 10.3934/mbe.2011.8.199 +[Abstract](154) +[PDF](1519.5KB)
The recent H1N1 ("swine flu") pandemic and recent H5N1 ("avian flu") outbreaks have brought increased attention to the study of the role of animal populations as reservoirs for pathogens that could invade human populations. It is believed that pigs acquired flu strains from birds and humans, acting as a mixing vessel in generating new influenza viruses. Assessing the role of animal reservoirs, particularly reservoirs involving highly mobile populations (like migratory birds), on disease dispersal and persistence is of interests to a wide range of researchers including public health experts and evolutionary biologists. This paper studies the interactions between transient and resident bird populations and their role on dispersal and persistence. A metapopulation framework based on a system of nonlinear ordinary differential equations is used to study the transmission dynamics and control of avian diseases. Simplified versions of mathematical models involving a limited number of migratory and resident bird populations are analyzed. Epidemiological time scales and singular perturbation methods are used to reduce the dimensionality of the model. Our results show that mixing of bird populations (involving residents and migratory birds) play an important role on the patterns of disease spread.
A perspective on the 2009 A/H1N1 influenza pandemic in Mexico
Rodolfo Acuňa-Soto, Luis Castaňeda-Davila and Gerardo Chowell
2011, 8(1): 223-238 doi: 10.3934/mbe.2011.8.223 +[Abstract](85) +[PDF](390.5KB)
In this article, we provide a chronological description of the 2009 H1N1 influenza pandemic in Mexico from the detection of severe respiratory disease among young adults in central Mexico and the identification of the novel swine-origin influenza virus to the response of Mexican public health authorities with the swift implementation of the National Preparedness and Response Plan for Pandemic Influenza. Furthermore, we review some features of the 2009 H1N1 influenza pandemic in Mexico in relation to the devastating 1918-1920 influenza pandemic and discuss opportunities for the application of mathematical modeling in the transmission dynamics of pandemic influenza. The value of historical data in increasing our understanding of past pandemic events is highlighted.

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