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Theoretical basis to measure the impact of short-lasting control of an infectious disease on the epidemic peak.

Omori R, Nishiura H - Theor Biol Med Model (2011)

Bottom Line: Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009.Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number.Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Biology, Faculty of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan.

ABSTRACT

Background: While many pandemic preparedness plans have promoted disease control effort to lower and delay an epidemic peak, analytical methods for determining the required control effort and making statistical inferences have yet to be sought. As a first step to address this issue, we present a theoretical basis on which to assess the impact of an early intervention on the epidemic peak, employing a simple epidemic model.

Methods: We focus on estimating the impact of an early control effort (e.g. unsuccessful containment), assuming that the transmission rate abruptly increases when control is discontinued. We provide analytical expressions for magnitude and time of the epidemic peak, employing approximate logistic and logarithmic-form solutions for the latter. Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009.

Results: Our model estimates that the epidemic peak of the 2009 pandemic was delayed for 21 days due to summer holiday. Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number. Peak delay is shown to critically depend on the fraction of initially immune individuals.

Conclusions: The proposed modeling approaches offer methodological avenues to assess empirical data and to objectively estimate required control effort to lower and delay an epidemic peak. Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak.

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A scenario for time-dependent increase in the transmission potential. A. Time dependent increase in the transmission rate. In the absence of intervention (baseline scenario), the transmission rate is assumed to be constant β over time. In the presence of early intervention, the transmission rate is reduced by a factor α (0 ≤ α ≤ 1) over time interval 0 to t1. We assume that the product αβ ileads to super-critical level (i.e. αR(0) >1 where R(0) is the reproduction number at time 0), and t1 occurs before the time at which peak prevalence of infectious individuals in the absence of intervention is observed. B. A comparison between two representative epidemic curves (the number of infectious individuals) in a hypothetical population of 100,000 individuals. R(0) = 1.5, α = 0.90 and t1 = 50 days. The epidemic peak in the presence of short-lasting control is delayed, and the height of epidemic curve is slightly reduced, relative to the case in which control measures are absent.
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Figure 1: A scenario for time-dependent increase in the transmission potential. A. Time dependent increase in the transmission rate. In the absence of intervention (baseline scenario), the transmission rate is assumed to be constant β over time. In the presence of early intervention, the transmission rate is reduced by a factor α (0 ≤ α ≤ 1) over time interval 0 to t1. We assume that the product αβ ileads to super-critical level (i.e. αR(0) >1 where R(0) is the reproduction number at time 0), and t1 occurs before the time at which peak prevalence of infectious individuals in the absence of intervention is observed. B. A comparison between two representative epidemic curves (the number of infectious individuals) in a hypothetical population of 100,000 individuals. R(0) = 1.5, α = 0.90 and t1 = 50 days. The epidemic peak in the presence of short-lasting control is delayed, and the height of epidemic curve is slightly reduced, relative to the case in which control measures are absent.

Mentions: Both questions are addressed by considering time-dependent increase in the transmission rate. Let β be the transmission rate per unit time in the absence of an intervention of interest (or during the mitigation phase in the case of our first question). Due to intervention (or school holiday) in the early epidemic phase, β is initially reduced by a factor α (0 ≤ α ≤ 1) until time t1 (Figure 1A). Though transmission rate abruptly increases at time t1 when the control policy is eased or when the new school semester starts, we observe a reduced height of, and a time delay in, the epidemic peak compared to the hypothetical situation in which no intervention takes place (Figure 1B). More realistic situations may be envisaged (e.g. a more complex step function or seasonality of transmission), but we restrict ourselves to the simplest scenario in the present study.


Theoretical basis to measure the impact of short-lasting control of an infectious disease on the epidemic peak.

Omori R, Nishiura H - Theor Biol Med Model (2011)

A scenario for time-dependent increase in the transmission potential. A. Time dependent increase in the transmission rate. In the absence of intervention (baseline scenario), the transmission rate is assumed to be constant β over time. In the presence of early intervention, the transmission rate is reduced by a factor α (0 ≤ α ≤ 1) over time interval 0 to t1. We assume that the product αβ ileads to super-critical level (i.e. αR(0) >1 where R(0) is the reproduction number at time 0), and t1 occurs before the time at which peak prevalence of infectious individuals in the absence of intervention is observed. B. A comparison between two representative epidemic curves (the number of infectious individuals) in a hypothetical population of 100,000 individuals. R(0) = 1.5, α = 0.90 and t1 = 50 days. The epidemic peak in the presence of short-lasting control is delayed, and the height of epidemic curve is slightly reduced, relative to the case in which control measures are absent.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3040699&req=5

Figure 1: A scenario for time-dependent increase in the transmission potential. A. Time dependent increase in the transmission rate. In the absence of intervention (baseline scenario), the transmission rate is assumed to be constant β over time. In the presence of early intervention, the transmission rate is reduced by a factor α (0 ≤ α ≤ 1) over time interval 0 to t1. We assume that the product αβ ileads to super-critical level (i.e. αR(0) >1 where R(0) is the reproduction number at time 0), and t1 occurs before the time at which peak prevalence of infectious individuals in the absence of intervention is observed. B. A comparison between two representative epidemic curves (the number of infectious individuals) in a hypothetical population of 100,000 individuals. R(0) = 1.5, α = 0.90 and t1 = 50 days. The epidemic peak in the presence of short-lasting control is delayed, and the height of epidemic curve is slightly reduced, relative to the case in which control measures are absent.
Mentions: Both questions are addressed by considering time-dependent increase in the transmission rate. Let β be the transmission rate per unit time in the absence of an intervention of interest (or during the mitigation phase in the case of our first question). Due to intervention (or school holiday) in the early epidemic phase, β is initially reduced by a factor α (0 ≤ α ≤ 1) until time t1 (Figure 1A). Though transmission rate abruptly increases at time t1 when the control policy is eased or when the new school semester starts, we observe a reduced height of, and a time delay in, the epidemic peak compared to the hypothetical situation in which no intervention takes place (Figure 1B). More realistic situations may be envisaged (e.g. a more complex step function or seasonality of transmission), but we restrict ourselves to the simplest scenario in the present study.

Bottom Line: Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009.Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number.Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Biology, Faculty of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan.

ABSTRACT

Background: While many pandemic preparedness plans have promoted disease control effort to lower and delay an epidemic peak, analytical methods for determining the required control effort and making statistical inferences have yet to be sought. As a first step to address this issue, we present a theoretical basis on which to assess the impact of an early intervention on the epidemic peak, employing a simple epidemic model.

Methods: We focus on estimating the impact of an early control effort (e.g. unsuccessful containment), assuming that the transmission rate abruptly increases when control is discontinued. We provide analytical expressions for magnitude and time of the epidemic peak, employing approximate logistic and logarithmic-form solutions for the latter. Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009.

Results: Our model estimates that the epidemic peak of the 2009 pandemic was delayed for 21 days due to summer holiday. Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number. Peak delay is shown to critically depend on the fraction of initially immune individuals.

Conclusions: The proposed modeling approaches offer methodological avenues to assess empirical data and to objectively estimate required control effort to lower and delay an epidemic peak. Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak.

Show MeSH
Related in: MedlinePlus