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III. APPLICATIONS TO EPIDEMIC MODELS STRUCTURED BY AGE OF INFECTION

ROCKY MOUNTAIN MATHEMATICS CONSORTIUM SUMMER SCHOOL 2012 MATHEMATICAL MODELING IN ECOLOGY AND EPIDEMIOLOGY UNIVERSITY OF WYOMING JUNE 11-22, 2012. III. APPLICATIONS TO EPIDEMIC MODELS STRUCTURED BY AGE OF INFECTION. Objectives.

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III. APPLICATIONS TO EPIDEMIC MODELS STRUCTURED BY AGE OF INFECTION

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  1. ROCKY MOUNTAIN MATHEMATICS CONSORTIUM SUMMER SCHOOL 2012 MATHEMATICAL MODELING IN ECOLOGY AND EPIDEMIOLOGY UNIVERSITY OF WYOMING JUNE 11-22, 2012 III. APPLICATIONS TO EPIDEMIC MODELS STRUCTURED BY AGE OF INFECTION

  2. Objectives Develop a mathematical model to ascertain the utility of school closures in mitigating the severity of influenza epidemics. Incorporate disease age of infected school children to describe the pre-infectious, infectious, pre-symptomatic, and symptomatic periods of infection. Provide a rational basis for school closings decisions dependent on virulence characteristics and local surveillance implementation, applicable to influenza epidemics such as H1N1 .

  3. Background of Age of Infection Models Modeling approach - track infected individuals through their disease course with a continuum variable corresponding to age of infection. Age of infection models were introduced in 1927 by W.O. Kermack and A.G. McKendrick, Proceedings of the Royal Society of London. A review is given by F. Brauer, 2005, Electronic Journal of Differential Equations. Recent works using age of infection models – Z. Qiu, and Z. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment, 2010, J. Dyn. Diff. Eqs. H. Inaba, H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptotic transmission model, 2008, Math. Biosc.

  4. Model Assumptions • The population consists of school children; initially there are 10,000 susceptible children with a very small number infected. The model is of standard incidence form, so the results are valid for any number of initial susceptibles with a given parameterization. The model is applicable to an individual small school or a large urban school district. • The dynamics of the model are localized to the population of school children. Infection of children outside of the school environment is not incorporated specifically, but can be considered included. • A threshold of the infectious phase, denoted by Ti, is the time during the disease course at which infected children become infectious. The infectious period lasts for time Fi, so that children are infectious from Ti to Ti+Fi.

  5. A threshold of the symptomatic phase, denoted by Ts, is the time during the disease course at which infected children manifest identifiable systemic or respiratory symptoms. The symptomatic phase lasts from Ts to Ti+Fi. We assume that Ts>Ti. • The removal of symptomatic infected children is Rsym % per day after the threshold of the symptomatic period Ts, but 0% per day before Ts. This percentage takes into account time lags in identifying presentation of symptoms and isolation of children from the school, both at home by parents and at school by officials. EXPOSED INFECTIOUS day Ti day Ti+Fi SYMPTOMATIC day Ts day Ti+Fi

  6. The thresholds in these assumptions should be viewed as effective average values for typical infected children, based on available data such as viral shedding and immune response levels. For isolation of infected children in schools, identification of symptoms is largely equated to fever and coughing. Carrat et al. showed an initial advance of viral shedding (peak at 2.0 days) compared to total symptoms score (peak at 3.0 days) in adult volunteer studies. Since the viral shedding measurements range over many orders of magnitude, the assignment of threshold values is justified. F. Carrat et al. (Amer. J. Epid., 2008)

  7. Model Compartments S(t) = the number of susceptibles at time t E(t) = the number of exposed (infected but not yet infectious) at time t I(t) = the number of infectious at time t R(t) = the number of removed at time t

  8. Age of infection Infected individuals are structured by a continuous variable a corresponding to disease age, and the latent, infectious, pre-symptomatic, and symptomatic stages of the disease are modeled by the disease age of infected individuals. Let i(a,t) denote the density of infectives of disease age a at time t. The infection is acquired at age 0. An infective is non-infectious (exposed) from age 0 to age Ti, and infectious for Fi days. The symptomatic period begins at day Ts and lasts until day Ti+Fi. The number of exposed and infectious at time t are The number of presymptomatic and symptomatic at time t are

  9. Equations of the Model The infection age dependent transmission rate of infectious children is a(a) and the infection age dependent removal rate of symptomatic children is bRsym(a), where a is the age of infection.

  10. The infection age dependent transmission rate of infectious children a(a). The infection age dependent removal rate of symptomatic children bRsym(a).

  11. The Baseline Model The parameters of the model at baseline are as follows: The choice of the baseline distributed transmission parameter α(a) is determined by its compatibility with an epidemic reproduction number R0 = 1.29 and an attack rate = 41%, which are within the range of values reported for the early H1N1 outbreak in Mexico (Fraser et al., Science, 2009).

  12. Simulation of the Baseline Model The susceptible population S(t) (Panel A), the cumulative number of cases C(t) (Panel B), the exposed population E(t) (Panel C), and infectious population I(t) (Panel D) in the baseline simulation. The epidemic begins with the introduction of 1 infected child into the school population. The duration of the epidemic is approximately 60 days.

  13. Simulation of the Model for Rsym = 60%per day The susceptible population S(t) (Panel A), the cumulative number of cases C(t) (Panel B), the exposed population E(t) (Panel C), and infectious population I(t) (Panel D) in the modified simulation. The epidemic begins with the introduction of 1 infected child into the school population. The duration of the epidemic is approximately 50 days.

  14. Simulation of the Model for Ti=.75 and Ts=1.25 The infection age dependent transmission rate a(a) for Ti=.75 and Ts=1.25. The infection age dependent removal rate of symptomatic children bRsym(a).

  15. Simulation of the Model for Ti=.75 and Ts=1.25 The susceptible population S(t) (Panel A), the cumulative number of cases C(t) (Panel B), the exposed population E(t) (Panel C), and infectious population I(t) (Panel D) in the modified simulation. The epidemic begins with the introduction of 1 infected child into the school population. The duration of the epidemic is approximately 40 days.

  16. Variable Pre-symptomatic Infectious Periods and Variable Symptomatic Removal Rates We vary the pre-symptomatic infectious period Ts–Ti from 0.0 to 1.0 days, by holding the threshold of infectiousness fixed at the baseline value Ti = 0.75 days and varying the threshold of symptoms Ts from 0.75 days to 1.75 days, and we vary the removal rate of symptomatic children from Rsym = 0% to 100% per day to gain insight into the influence of these elements on the epidemic dynamics. The baseline parameters are otherwise held constant, and we consider R0 and attack rate percentages as a function of these two parameters. It is intuitive that a short pre-symptomatic infectious period or an efficient removal of symptomatic children will depress the attack rate, but it is surprising that both variables must be within a small range to achieve this effect.

  17. R0 as a function of the pre-symptomatic infectious period and the removal rate of symptomatic children 0.0<Ts-Ti<1.0 days, Ti=0.75 days, 0.75<Ts<1.75 days, 0.0%<Rsym<100%. The height of the red plane is at the R0 threshold value 1.0. The green dot is the value of R0=1.29 at the baseline parameter values (Ti=0.75 days, Ts=1.0 day, and Rsym=70% per day).

  18. The attack rate as a function of the pre-symptomatic infectious period and the removal rate of symptomatic children The height of the green plane is 30% and the red plane 60%. Attack rates corresponding to pairings of the control variables above the red plane are not effectively controlled by the removal rates of symptomatic children, whereas those below the green plane are. Pairings between the two planes are marginal for school closings, but enhanced surveillance and removal measures may reduce attack rates to acceptable values. The yellow dot is the value of the removal rate = 41% at the baseline parameter values.

  19. Summary Infection age epidemic models provide a convenient framework for describing the stages of disease progression. It is natural for physicians and public health officials to formulate a disease age time-line for the exposed, infectious, pre-symptomatic, and symptomatic stages. Infection age models also allow transmission, quarantine, and hospitalization rates to be functions of infection age. Infection age models can focus on the relationship of the infectious and symptomatic periods in predicting the severity of the epidemic, and the effectiveness of control measures such as school closings policy. Our model analysis indicates that school closings are advisable when pre-symptomatic transmission is significant or when removal of symptomatic children is inefficient. Our model provides a quantitative basis for school closings decisions dependent on virulence characteristics and surveillance implementation.

  20. For school environments our model provides a rationale for improved surveillance methods. For example, non-invasive temperature monitoring with real-time thermal sensors, as in current use in some airports, offers a potential advance for improved surveillance and rapid isolation of infected students. Another important advance for surveillance methods is to implement antibody-based testing, to see who is immune and no longer susceptible. • Our modeling methodology is deterministic, with a small number of parameters, and can be completely analyzed for parameter sensitivity and parameter dependent outcomes. • Our model is further applicable to other epidemic-at-risk populations, including those at large workplaces, military installations, event (large crowd) gatherings, mass transit sites, and other settings suitable for social distancing policies.

  21. Analysis of the Age of Infection Model Theformula for the epidemic reproduction number R0 is

  22. The attack rate AR as a function of initial condition The dependence of the attack ratio AR on the initial value ARi. The distributed parameters a(a) and b(a) are at baseline values and the initial condition i(a,0)=i0(a) is varied to give ARi. We vary i0(a) from 0 to 250 for the first two days and 0 otherwise. With value 150 we have ARi approximately 0.2, which gives an attack rate 0.62. The initial level of the infected population strongly influences the attack ratio, especially at lower levels corresponding to outbreak. School re-opening policies can be guided by re-setting the initialization to a new level after intervals of school closure.

  23. Extensions of the Model In future work, we will incorporate into the model other important issues of epidemic control, including quarantine policy, vaccination strategy, anti-viral use, and other epidemic sub-population distinctions. References: M. Blaser, Y-H. Hsieh, J. Wu, GW, Pre-symptomatic influenza transmission, surveillance, and school closings published online, April 28, 2010, Math. Mod. Nat. Phen. C. McCluskey P. Magal, GW, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal. Vol.89, No.7 (2010), 1109-1140.

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