Estimating the subcritical transmissibility of the Zika outbreak in the State of Florida, USA, 2016
Dinh et al. Theoretical Biology and Medical Modelling
Estimating the subcritical transmissibility of the Zika outbreak in the State of Florida, USA, 2016
Linh Dinh
Gerardo Chowell
Kenji Mizumoto
Hiroshi Nishiura
Background: Florida State has reported autochthonous transmission of Zika virus since late July 2016. Here we assessed the transmissibility associated with the outbreak and generated a short-term forecast. Methods: Time-dependent dynamics of imported cases reported in the state of Florida was approximated by a logistic growth equation. We estimated the reproduction number using the renewal equation in order to predict the incidence of local cases arising from both local and imported primary cases. Using a bootstrap method together with the logistic and renewal equations, a short-term forecast of local and imported cases was carried out. Results: The reproduction number was estimated at 0.16 (95 % Confidence Interval: 0.13, 0.19). Employing the logistic equation to capture a drastic decline in the number of imported cases expected through the course of 2016, together with the low estimate of the local reproduction number in Florida, the expected number of local reported cases was demonstrated to show an evident declining trend for the remainder of 2016. Conclusions: The risk of local transmission in the state of Florida is predicted to dramatically decline by the end of 2016.
Prediction; Zika virus; Epidemic; Mathematical model; Basic reproduction number
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Methods
Patients’ data
In order to keep Florida residents and visitors abreast of the presence of Zika
cases in the state, the Florida Department of Health has maintained up-to-date
counts of the number of Zika cases diagnosed in this state [2] while
strengthening a robust mosquito-borne illness surveillance system. Our study relies on
confirmed cases of Zika virus infection. Before 29 June 2016, except for pregnant
women, only those cases exhibiting at least two of the following symptoms: fever,
rash, joint pain and red eyes and having an epidemiological link (travel history or
sexual contact with travelers from Zika-affected areas or suspected contact with
cases) underwent laboratory diagnosis by serology or rRT-PCR. From 29 June, all
laboratory-confirmed asymptomatic cases were counted. Non-pregnant cases with
recent travel history to an area with widespread Zika virus transmission were
classified as either travel-related or non-travel related cases [3]. Hereafter,
travelrelated cases are referred to as imported cases, while non-travel related cases are
referred to as local cases. We analysed the temporal evolution of confirmed Zika
cases from 1 May to 23 September 2016. Since epidemiological and diagnostic
procedures typically required 7 days [3], we analysed weekly case counts with
week 0 starting on 1 May 2016.
Modelling method
In order to quantify the extent of local transmission, we first set out to estimate
the average and uncertainty of the reproduction number, R, associated with the
Zika outbreak in the state of Florida. R is interpreted as the average number of
secondary “local” transmission events caused by a single primary case. A primary
case can be either a local or an imported case. Assuming that congenital or
sexual transmission cases of Zika are rare in the state of Florida, our modelling
exercise focused on mosquito-borne transmission alone. Let ct and it be local and
imported cases in week t, and let ws represent the probability mass function of
the serial interval of length s weeks, which was obtained by
ws ¼ Gð7sÞ−G 7ðs−1Þ ;
Xs∞¼1Rðct−s þ it−sÞws;
for s > 0 where G(.) represents the cumulative distribution function of the gamma
distribution with mean of 14 days and standard deviation of 2 days [4]. We describe an
expected value of local cases E(ct) as
as discussed elsewhere [5]. Assuming that variation in case counts in week t is
sufficiently captured by the Poisson distribution, in agreement with the
underlying mechanism of the infection process in deterministic models, the
maximum likelihood estimate of R is obtained by minimizing the
Poissondistributed likelihood function that uses the right-hand side of (2) for the
expected value. A constant R is supported by the negligible impact of herd
immunity on the transmission dynamics due to the limited scope of the epidemic in this
area thus far.
Subsequently, we assume that the dynamics of cumulative counts of imported cases,
I(τ), at day τ is sufficiently captured by a logistic curve, i.e.,
where K is referred to as the carrying capacity (i.e., the expected total number of
cases during the outbreak), γ, the steepness of the curve, and t0 the time of the
sigmoid’s midpoint. In addition to the recent interest in using phenomenological models
that generalize the logistic equation [6], the SIR (susceptible-infectious-recovered)
model can be approximated by the logistic curve [7]. Hence, it may not be surprising
that eq. (3) can be useful to capture single-epidemic transm (...truncated)