The thesis describes how transmission of the 2009 A/H1N1 influenza pandemic in the United States varied geographically, focusing on population distribution and age structure. A mechanistic mathematical model is then constructed to describe the transmission of the epidemic between cities.
Biology and epidemiology of influenza
A mathematical modeling analysis identifies key predictors of the spread of the large fall wave of pandemic A/H1N1pdm influenza in the US in 2009 between cities, pinpoints the geographic locations where the outbreak first emerged, and identifies which age groups were most responsible for sustaining its transmission. In pandemic influenza outbreaks, there may be changes in the age groups that suffer the greatest morbidity and mortality, with children and young adults often disproportionately affected [254].
Chronicle of the 2009 A/H1N1pdm influenza pandemic in the United Statesin the United States
A number of intervention strategies were implemented to mitigate the spread of the 2009 A/H1N1pdm influenza pandemic in the United States. The CDC began developing a vaccine for the new A/H1N1pdm strain in April 2009, just six days after the first case was identified in the United States [40].
Key concepts in mathematical epidemiology
Early contributions to mathematical epidemiology
However, the vaccine was only approved in October and was initially administered only to groups considered to be at high risk of infection [40]. Thus, while vaccination may have reduced the overall burden of infection during the pandemic, it did not have a major impact on the transmission of the spring or fall wave of infection.
Geography and age in epidemiological models
Metapopulation models are sometimes criticized for ignoring important aspects of the physical environment, because few populations actually adhere to the strict spatial clustering that the theory assumes [8]. For IBMs, the kernel can be more concretely interpreted as a description of the movement tendencies of individuals in the population [128].
Turning points in the theory of infectious disease
Since the introduction of phylodynamics, a significant body of literature has developed that considers epidemiological and genetic data in conjunction. The development of robust methods to integrate epidemiological and genetic data remains a central priority in infectious disease research [95], and is a primary motivation for the theory developed in Chapter 4 of this thesis.
Mathematical contributions to the study of influenza
Geographic models of influenza transmission in Europe have been developed by Smiesheket al for Switzerland and by Paget et al for the continent as a whole. On an international scale, an early model that takes into account the spread of influenza via airways was also contributed by Rvachevet al.
Summary of thesis
Chapter 4 then reverse-engineers the geographic transmission model to identify the sites, or hubs, of the fall 2009 A/H1N1pdm outbreak in the US. Chapter 5 looks at how the transmission strength of the 2009 A/H1N1pdm influenza pandemic in the US varied by age group.
Background
- A brief history of influenza surveillance
- Modern influenza surveillance in the United States
- Geographic divisions in the United States
- Outbreak onset detection
Medical claims records are not free of bias; a WHO report notes that "the interpretation of data derived from [electronic medical claims-based] systems will depend heavily on the local coding practices, the external forces influencing coding decisions (such as reimbursement), and the clinician's understanding of the coding system ” [248].The ILI data to be considered in this thesis are tagged by the 3-digit postal codes of the outpatient clinics from which they were collected.
Description of the ILI dataset
Data source
Infections in distinct small populations probably all have a common ancestor, and so the time of arrival of the first successful ancestor could be taken as the time of outbreak establishment. The IMS-ILI data, aggregated across all age groups and locations, is depicted in Figure 2.2.
Data validation
Inferring outbreak onset times from the IMS-ILI data
Breakpoint onset detection method with peak adjustment
A bilinear trend is fit for the 17 weeks of the time series before and including the week of peak incidence. The start date is defined as the estimate of the maximum probability of a break in the bilinear trend, rounded to the nearest half-week.
Evaluation of the breakpoint method
However, in the IMS-ILI time series, there is evidence of autocorrelation in the number of ILIs outside the flu season. For each simulated outbreak, the outbreak time is calculated using both versions of the threshold method and the breakpoint method.
Breakpoint outbreak onset times for the autumn wave of the 2009 A/H1N1pdm influenza pandemic in the United
- Estimation of onset times
- Investigation of the 2009 outbreak onset times and their uncer- taintiestainties
- A rough calculation of R at the start of the autumn 2009 pan- demic wave
- Age-stratified autumn 2009 onset times
In the western half of the state, there is evidence of early transmission in the Central Valley of California. Red dots correspond to ZIPs in the eastern US (HHS regions 1-5), and black dots correspond to ZIPs in the western US (HHS regions 6-10).
Geographic data
If two age groups have equal breakout times, both are counted as the earliest onset in that zip. Rides are only shown where a reliable starting estimate for the age group could be obtained.
Schools data
The color scale matches that in Fig 2.8, to allow comparison between school start dates and the time of the outbreak in 2009. However, the pandemic wave spread much more slowly than the opening of schools; in Maine, there is approximately a two-month lag between the school start date and the state's average outbreak start time.
Antigenic data
However, the gap between start times and school start dates widens as the epidemic progresses. As the outbreak progressed, the discrepancy between start times and school start dates increased.
Discussion
A major advantage of the breakpoint method is that it avoids a need to define baseline and threshold levels of ILI. Statistical analysis of outbreak onset times provides insight into the geographic transmission of the fall 2009 A/H1N1pdm influenza pandemic wave in the US.
Summary
The estimate of reproductive numbers of rats at the start of the fall wave of the 2009 A/H1N1pdm pandemic is in close agreement with previous estimates. The model is fit to city-level outbreak timing since the 2009 fall wave of the A/H1N1pdm outbreak in the United States.
Background
The gravity model
Optimal fitting of a Gaussian process reveals a likely region of increased transmission in the southeastern United States at the start of the fall wave of the 2009 A/H1N1pdm pandemic. The model can be formulated in terms of equations describing particle radiation and absorption.
Survival analysis
Integration provides an expression for the survival functions, and thus the overall distribution of the time at which the event occurred, in terms of the hazard function: This provides a way to express the discrete time probability distribution of event occurrence in terms of the continuous hazard functionh .
Gaussian processes
Gaussian processes with a SE covariance function are smooth, with mean-square derivatives of all orders [196]. The exponential covariance function is drunk (see Fig. 3.1), with no guaranteed mean-square differentiation of any order.
Model definition
Motivating the model structure
The proposals are accepted or rejected with probability proportional to the likelihood ratio between the current and most recently accepted proposals, ultimately providing a good estimate of the posterior densities of yyyyandΩ. In both transmission models presented in this chapter, the infectious force consists of two summed parts.
The fundamental transmission model
Outbreak onset times are closely related to the start of the fall school semester in the southeastern US. Compared to real epidemic onsets (Fig 2.8), the simulated epidemics have similar patterns of radial spread and overall duration.
The transmissibility surface transmission model
For comparison, the transmissivity adjustments ξξξT and ξξξSa have been reestimated using (1) a SE covariance function with a spatial length scale of 100 km and a time length scale of 8 half weeks, (2) a SE covariance function with a length scale spatial length scale of 500 km and temporal length scale of 8 half weeks, and (3) an RQ covariance function with spatial length scale of 200 km and temporal length scale of 8 half weeks. We proceed using the mean posterior estimates for ξξξT and ξξξS obtained using an SE covariance function with a spatial length scale of 200 km and a temporal length scale of 8 semiweeks.
Further exploration of the Gaussian process fitting procedure
Second, transferabilityβd is increased in the southeastern US (HHS regions 4 and 6) to check whether the Gaussian process fits can correctly identify authentic differences in transferability. This underestimation of the actual increase in transmissibility may be partly due to the stiffness in the.
Discussion
However, the fundamental model does not fully account for variation in the geographic structure of the outbreak at the regional scale. Second, the temporal process appears to artificially absorb some of the spatial variation in transmissibility in the simulations, reducing the observed spatial variance in the Gaussian process fits.
Summary
This strategy will identify the transmission centers of the 2009 A/H1N1pdm influenza pandemic in the United States and map the total onward transmission caused by each hub. Onward transmission from these and a further six hubs is responsible for 90% of the observed outbreaks at ZIP level in autumn 2009.
Background
Terms for various epidemiological hotspots
In the specific case of influenza, it is widely accepted that many new virus strains originate in East and Southeast Asia [204], and that superspreaders can contribute to disease transmission [138, 191]. Charuet al proposes an empirical hub-finding technique for cities in the United States where outbreak start times are available.
Epidemiology and phylogeography
Mathematical framework
Characterising the forward transmission network
Consider a function that characterizes the strength of the infection at the time location of the start of its outbreak as the sum of the forces exerted by all previously infected sites, plus a background risk of seeding from outside the metapopulation. The partial infection forces λι,j can be visualized as a transmission network, as depicted in the left diagram in Fig 4.2.
Reversing the infection process
In the limit, the i,jth entry of the pde power of the transition matrix gives the probability that the outbreak in location was initially caused by a boring event in location j. Element(M∞)i,j gives the probability that state j was the ultimate source of the outbreak in locationiei.
Hubs of the 2009 A/H1N1pdm influenza pandemic in the United Statesthe United States
- The transmission model revisited
- Calculating hubs
- Accounting for onset uncertainty
- Re-calculating the transmission hubs with the true hubs missing
These groupings are estimates of the larger regions representing the transmission centers listed in Table 4.1. According to the methods developed above, the transmission centers of the 2009 A/H1N1pdm influenza pandemic in the United States were small and medium-sized cities (see Table 4.1).
Simulation-based validation of methods
- Overview of the epidemic simulation methods
- Specifying city coordinates and population sizes
- Commuting
- Model running
- Binning and noise
- Parameters and model validation
- Simulation results
- Comments on model formulation
At the beginning of the simulation, one city is chosen as the epidemic launch site. The sample mean, standard deviation, and skewness of the population sizes of the misidentified zips are and 1.41, respectively.
Discussion
Epidemiological interpretation of the hub-finding procedure
Accounting for the set of observed transmission hubs
This highly random filter between introduction and firm establishment may help explain the unusual spatial invasion pattern of the fall wave of the 2009 A/H1N1pdm outbreak in the US. Elsewhere, the start of the fall school year may have increased the likelihood of pathogen settlement.
Linking with genetic data
Indeed, Shamanet already correctly predicted a third pandemic wave in the southeastern United States based on a spatiotemporal model of the effective reproductive number RE driven by absolute humidity. The results presented in this chapter show that the southeast also played a decisive role in the spread of the second (autumn) pandemic wave, since the two most influential nodes are located in this region.
Summary
STE provides a relative ranking of which age groups dominate transmission, rather than a reconstruction of the explicit inter-age group transmission matrix. The pairwise STE is also calculated between age groups in cities between which infection is likely to have spread, as identified by the geographic transmission model presented in Chapter 3.
Background
Age as a key characteristic in disease transmission
Pandemic outbreaks are often characterized by shifts in the age groups that suffer the highest morbidity and mortality. Differences in health care-seeking behavior between age groups may confound ILI-based estimates of which age groups predominate transmission.
Incorporating age into epidemiological models
Methods that aim to identify age-related differences in transmission should take into account known differences in health care-seeking behavior between age groups, estimating that 40% of adults (defined as individuals over 18 years of age) and 56% of children with ILI during the 2009 health care searched for. influenza pandemic in the United States. These estimates remained fairly consistent during the following 2010–11 influenza season, when 45% of adults and 57% of children with ILI sought health care [ 21 ].
Transfer entropy
Next, the probabilities that make up the sum of the STEs, Equation 5.9, are estimated using the relative frequencies of the symbols in the symbol sets. The probabilities that make up the sum of the STEs, Equation 5.9, are estimated using the relative frequencies of the symbols in the character strings.
Symbolic transfer entropy and epidemiological processes
The contextual STE
However, for disease outbreaks, the amplitude of the underlying process (ie the number of cases) clearly matters. The remaining seven terms in the contextual sum of STE, eq 5.12, can also be expressed in terms of the epidemiological process using similar derivations.
Contextual STE for under various epidemiological scenarios
This is done for all eight terms in the contextual STE expression (cf. 5.12), yielding a contextual STE that is exactly zero. Interestingly, when R<1 the reverse is true; the contextual STE is higher in the group with fewer cases.
Simulations on a two-age-class SIR model
The within-group transmission rate for group J is equal to the transmission rate from group J to group I. When k=1, the within-group transmission rate for age class I is four times the baseline transmission rate, and the transmission rate from group I to group J is twice the base transfer rate.
Simulations on a two-age-class Poisson model
For the simulations, the length of the time steps is assumed to coincide with the generation interval for the disease. The reproductive number is 1.5 for the first eight weeks of the outbreak, then drops to 0.8 by the end of the epidemic.
Simulations on a four-age-class Poisson model with variable re- porting ratesporting rates
Second, we consider how different reporting rates between age groups affect the STE, when the true transmission rates are all equal. 800 epidemics each were simulated for reporting rates between 0.1 and 1 in steps of 0.1, with equal reporting rates across all age groups.
Simulations on a twelve-age-class Poisson model
The average pairwise STEs between all age groups are shown in the left graph in Figure 5.14. The average pairwise STEs across all age groups in this scenario are shown in the right-hand graph in Figure 5.14.
STE to identify dominant age groups in transmission of the 2009 A/H1N1pdm influenza pandemic in the Unitedof the 2009 A/H1N1pdm influenza pandemic in the United
Age-group differences in within-city transmission
In that simulation, only transmission from children is increased, but this causes a moderate increase in STE from adults and infants. School-aged children (5-19 years) have the highest transfer of information to the age-total time series.
Age-group differences in geographic transmission
Again, STE is elevated in school-aged children, with children aged 5–9 years providing the most information on age-aggregated ILI in a nearby infected ZIP. In terms of intra-ZIP transmission, school-age children (ages 5–19) have the highest STE over the total-age time series.
