Evaluation of real time methods for epidemic forecasting 2

The advantages of ILI surveillance over the traditionally mented method of laboratory diagnoses to monitor inuenza activity includelower cost, shorter processing time, and more sensitivi

The tracking of an inuenza pandemic can be done by several methods,for instance serological analyses of blood samples, which is an accurate butcostly retrospective method to test whether infection has occurred (Chen etal., 2010) In this study, we choose to track the disease by collecting reportsbased on syndromes from a network of general physicians or family doctors

we set up (Ong et al., 2010) We propose disease modeling as a method topredict the size of the pandemic across time

2.1 Data source

The WHO Inuenza Surveillance Network, established in 1952, serves as aglobal monitoring system for the emergence and activities of inuenza viruseswith pandemic potential The network consists of 135 National InuenzaCentres worldwide, which sample patients with inuenza-like illness (ILI)and submit representative isolates to WHO Collaborating Centres for anti-

15

genic and genetic analyses (WHO, 2009) The WHO will then communicate

or provide advice to member states in accordance with the ndings InSingapore, an existing inuenza surveillance programme relies on sentinelphysicians collecting throat and nasal swabs from patients presenting ILIwith fever ≥ 38◦C, cough, sore throat, headache and muscle ache (Teo etal., 2011) This process of specimen collection does not, however, provide anestimate of the actual number of infected individuals as the denominator isunknown

Other prospective surveillance for inuenza has been carried out in pore by the Department of Pathology in the Ministry of Health, Singapore,since the department was designated a National Inuenza Centre by theWHO in July 1972 (Doraisinghan et al., 1988) Surveillance is done by tak-ing random throat swabs weekly from patients exhibiting symptoms for acuterespiratory infection (ARI) at selected government polyclinics (MOH, 2001).ARIs are dened as patients who present respiratory symptoms of cough,rhinorrhea, nasal congestion and/or sore throat, which may be accompanied

Singa-by fever However, this approach is not specic to inuenza alone as thesymptoms for ARI can be caused by other diseases as well

Because of the aforementioned concerns, as well as data accessibility, wecollected the data used in this study independently We commenced bysending recruitment invitations to e-mails of general practice/family doc-tor clinics compiled from the College of Family Physicians Singapore andthe directory of Pandemic Preparedness Clinics registered under Ministry of

Health to manage inuenza cases The inclusion criteria were that doctorshave to be registered with the Singapore Medical Council and work at leastthree full days a week in a general practice or family medicine clinic Datasubmission was voluntary and participating doctors were given the option towithdraw from the project at any time (Ong et al., 2010) The study designwas approved by the institutional review board of the National University ofSingapore.

Enrolled doctors were requested to submit the number of clinically nosed ARIs by e-mail or facsimile by 2pm on the following day In addition,basic demographics and the temperature at presentation was reported Then,

diag-we extract the number of ILI cases, dened as ARIs which also exhibit fever

of ≥ 37.8◦C, from the data submitted ILI is used in our report because it ismore specic for inuenza infections, and has been used widely as an indica-tor in seasonal inuenza surveillance systems in many countries (Thompson

et al., 2006) The advantages of ILI surveillance over the traditionally mented method of laboratory diagnoses to monitor inuenza activity includelower cost, shorter processing time, and more sensitivity to changes in num-ber of cases at the peak of the pandemic (Lee et al., 2011)

imple-Figure 2.1 shows the inferred average number of patients with ILI perday reported by the doctors (black lines) and 95% credible interval (grey),from declarations in the general practice/family doctor network in this study.The average number of ILIs per GP and the credible interval is generated asthe posterior distribution from a Markov chain Monte Carlo algorithm, on

Figure 2.1: Inferred average number of patients with ILI per day (black lines)and 95% credible interval (grey lines), with weekdays indicated in black solidcircles and weekends or public holidays indicated in hollow circles.

distribu-2.2 Models for disease dynamics

Mathematical epidemiology has grown exponentially starting from the middle

of the 20th century due in part to tremendous improvement in computingpower, thus, a variety of models have now been formulated, mathematicallyanalyzed and applied to infectious diseases

We investigate two classes of models via application to our example ofthe inuenza A-H1N1(2009) pandemic, namely deterministic models andstochastic models Deterministic models are those which use dierence, func-tional or functional dierence equations to describe the changes of each epi-demiological class in time (Hethcote, 2009) In deterministic models, everyset of variable states is uniquely determined by parameters in the model and

by sets of precedent states of these variables (Thrush, 2011) Deterministicmodels are used to describe the outcomes of diseases at a population level,and they always behave identically for a particular set of initial conditionsexcluding numerical overow issues (Lorenz, 1963) Deterministic models arerelatively easy to set up due to availability of software, but do not reect therole of chance in the spread of the disease

Conversely, a stochastic model does not return a set of unique solutionsbut entails stochasticity such that the model will return a range of dierentoutcomes The rates of moving from one epidemiological class to another giverise to dynamic randomness and variation in the model Stochastic modelsare more complicated to set up and may require more computational power

to execute, as they often require Monte Carlo sampling to derive an outcomedistribution (Beissinger and Westphal, 1998).

In the mathematical modeling of infectious diseases transmission, there

is always a trade-o between simple models, which omit most details and aredesigned only to highlight general qualitative or quantitative behaviour, anddetailed models, designed for more specic situations which include short-term quantitative predictions, but are generally dicult or impossible tosolve analytically (Brauer et al., 2008) We have followed the OccamBoxphilosophy of selecting model complexity that is as complex as need to beuseful for our purposes  i.e forecasting aggregate discrete levels  but nomore complicated, and as a result, we have used quite simple models in thisreport In principle more detailed structure can be incorporated using similarapproaches for research questions that demand it These can be extensions

to compartmental models to include more classes or inclusion of demographicfactors in the models

The deterministic models used in this study are described in the nextsection, they are the Richards model, and the deterministic SIR compart-mental model Also, the deterministic version of SIR model is modied to bestochastic to allow for the eects of stochasticity of the inuenza pandemicespecially during the early phases in which there are few cases Therefore wewill also describe the stochastic SIR compartmental model

2.2.1 Richards model

An extension to the logistic growth model was proposed by Richards in 1959

to study the growth process of biological populations; as applied to our

in-uenza A-H1N1(2009) data, we term this the Richards model The model

is developed from a logistic function with a relative growth-rate ing linearly with increasing population size (Richards, 1959) Although theRichards model is not as exible as a generalized logistic function in model-ing growth, it contains fewer parameters and is therefore more parsimonious.The model is dened as follows,

it indicates the end of the current wave of infection, which signies decreasinginfection rate and that control measures, if implemented, might have started

to take eect or that herd immunity might have been established in thepopulation

The solution in (2.2) is a generalisation of logistic function, and is ated with a sigmoid graph shown in Figure 2.2 Figure 2.2 shows the dierenteects of the four parameters K, r, tm, and a on C(t) and I(t); we can seefrom the rst panel that K determines the size of C(t), the daily number ofinfected individuals, the second panel on the left shows that as tm increase,

associ-the time it takes for C(t) to reach its peak also increases From associ-the rst twopanels on the right, we see that r and a change the slope of the C(t) curve.

2.2.2 Deterministic SIR compartmental model

Apart from Richards model, we propose another deterministic model in thisstudy, known as the SIR compartmental model

The basic compartmental models (Kermack and McKendrick, 1927) vide at any point of time the population under study into compartmentsallowing for assumptions about the nature of transfer from one compartment

di-to another Compartments are often labelled with letters M, S, E, I and

R, to indicate the epidemiological classes of passively immune, susceptible,exposed in latent period, infected, removed respectively Examples of exist-ing compartmental models include the SIR, SIS, SIRS, SEIS, SEIR, MSIR,MSEIR and MSEIRS models The specication of compartments to be in-cluded in a model depends on the characteristics of the specic disease, such

as transmission and interactions between classes

We use a basic SIR model for the inuenza A-H1N1(2009) with the lation being divided into three classes: individuals start o in the susceptibleclass S, and move subsequently to the infected class I and terminate in theremoved class R

popu-The deterministic SIR model can be written using ordinary dierential

Figure 2.2: Graphs of C(t) and I(t) from Richards model with parameters

K, r, tm, a = 1 unless otherwise stated

Figure 2.3: The SIR compartmental model with the blue circles indicatingthe compartments Susceptible, Infected and Recovered and arrows indicatingthe per capita rate of transfer from one compartment to the subsequentcompartment.

dt representthe rate of change in the number of individuals in that class at time t In themodel above, the total population N(t) = S(t)+I(t)+R(t) remains constantfor all times t, since we assume negligible rates of birth and death, and thatthe epidemic occurs in a relatively short time and consequently do not giverise to extreme demographic changes which perturb the dynamics of contactbetween individuals (Iannelli, 2005), an assumption that is appropriate forinuenza outbreaks, but not, say, for less explosive epidemics such as AIDS(Alkema et al., 2008) or Chlamydia (Althaus et al., 2010) The unknownnumber of susceptible S(0) and the number of infected individuals I(0) at

the beginning of the epidemic are treated as parameters.

The model implies that susceptible individuals S will contract the diseasefrom infected individuals I at a rate of βSI

N, where the infectivity parameterβ

N is the average number of transmissions per S − I pair in a time period.The model assumes homogeneous mixing in the population, in other words,that individuals interact in the same manner and to the same extent andthere is no clustering of infections in space or social space Therefore, βI/N

is the rate that a susceptible gets infected per unit time The rate of infection

is assumed to be proportional to the fractional of the population currentlyinfected, implying that as population size increases, the rate at which an in-fected individual come into contact with one particular susceptible individualdecreases

Infected individuals will shift to the removed class R at a rate of γ (theaverage infected period being 1/γ) Removal is dened by recovery from thedisease and thus gaining immunity against the virus for the time span ofthe epidemic, or death directly or indirectly caused by the disease We useimportance sampling to look for model trajectories that match the epidemictime series

2.2.3 Stochastic SIR compartmental model

The deterministic version of SIR model outlined in the previous section ismodied to be stochastic While the states S, I and R in a deterministicSIR model are known given their past values, their counterparts in stochasticmodel are allowed to vary randomly according to specic distributions Theparameters β and γ in the stochastic SIR model are analogous to that in thedeterministic SIR model

The model can be described as follows

The number of new infectives between time t and t − 1 is assumed to be

ˆ pinfection(t) = 1 − e−βI(t)/N is the probability of a susceptible beinginfected over a one-day time window;

ˆ precovery(t) = 1 − e−1/γ is the probability of an infected individualbeing removed from the I category;

ˆ At and Bt represent the number of individuals who are newly infectedand removed, respectively, at time t, and we assume them to followbinomial distributions with probability pinfection(t) and precovery(t);

ˆ the data are recorded on a discrete basis;

ˆ the large number of cases makes it prohibitive to search the space ofunobserved, continuous time infection times, which would be necessaryfor a continuous time model;

ˆ we choose to work with a discrete time model because the daily cycleimposes non-constant rates of infection anyway

This discrete model serves as an approximation to the true underlying tinuous time process

con-2.2.4 The observation model

The observation process is supported by observational data which are tained as described in Section 2.1 We use these observational data to makeinferences about the state process which involves the S, I and R categories.Some new parameters introduced here are δwds, the probability that an in-fected individual will visit a doctor for consultation on weekdays and Sat-urdays, δwds× δsph, the probability an infected individual will visit a doctorfor consultation on weekends and public holidays, and φ, the consulting rate

ob-of ILIs which are not caused by the inuenza A-H1N1(2009) The observednumber of ILIs, yt, from our network of doctors is recorded in the observa-tional data, however, yt is not equal to the total number of infected subjects

It present in the population Therefore, we need to incorporate parameters

to address that discrepancy as shown below,

δwds× (φ + It/2084) when t is weekdays or Saturdays,

δwds× δsph× (φ + It/2084) when t is Sundays or public holidays.The estimated number of doctors in Singapore is 2084, and is calculatedfrom the estimated number of general physicians or family doctors in Sin-gapore in 2009, which was 1730 (Ong et al., 2010), and the percentage ofpatients who seek medical attention from these doctors instead of visiting apolyclinic, was 83% (Emmanuel et al., 2004) in 2001

2.3 Statistical methodology

2.3.1 Bayesian paradigm

In the Bayesian paradigm, the process of learning from the data is cally implemented by making use of Bayes' theorem to combine any available