## APPENDIX

Estimating the probability of ZIKV infection among travellers from FP.

We assumed a closed population of size N and the absence of competitive risks, that is, recovery and mortality rates were neglected.

The probability of ZIKV infection can then be calculated using a two-state model without recovery, that is, an SI type model, in which *S* individuals are susceptible to ZIKV, and *I* individuals have been infected in the past and acquired lifelong immunity. The model is a finite continuous-time Markov chain (CTMC)Y = {*Y*(*t*):*t* ⩾ 0}, where *Y*(*t*) denotes the number of infectives at time *t*, and its infinitesimal transition probabilities are specified as follows:

where *o*(*ϕ*) is a mathematical notation that describes the limiting behaviour of a function when the argument *ϕ* demonstrates a tendency towards a particular value or infinity, and *o*(Δ*t*) → 0 as Δ*t* → 0. This means that a typical infective makes infectious contacts at the points of a Poisson process at a rate of *λ*(*t*) during an infectious period.

First, we calculated the probability that *y* individuals are in state *I* during the period of time between *t* and *t* *+* Δ*t*, as follows (see reference [25] for details):

In equation (A2), the first term refers to the probability that there were *y* individuals at time *t* in condition *I* (prevalence of notified cases) and that no susceptible individuals (*x*) acquired the infection in the period. The second term refers to the probability that there were (*y* *−* 1) individuals at time *t* in condition *I* and that one susceptible individual (*x*) acquired infection during this period.

Taking the limit of equation (A2) when Δ*t* → 0, it is possible to obtain the Kolmogorov Forward Equation, as follows:

Note that in equation (A3), as mentioned before, we are assuming a closed population of size *N* = *x* *+* *y.*

The general equation for the Probability Generation Function (PGF), *G*(*z,t*), *i*s expressed as follows:

For the specific model described in equation (A3), PGF was expressed in the following manner:

The intermediate steps between equations (A4) and (A5) are numerous, but the interested reader may obtain these details in [17].

The average number of y individuals at time *t* can then be calculated by taking the first partial derivative of the PGF with respect to *z* at *z* = 1, as follows:

Hence, the average probability (π_{d}(*t*)) of at least one ZIKV infection at time *t* may be estimated using the following equation:

The variance of the probability distribution (*σ*
^{2}) for the number of infected individuals at time *t* may be calculated as follows:

which results in

To summarise, the risk of ZIKV acquisition at time *t, or Risk*(*t*) [26], may be estimated using the following equation:

where the second term is the 95% CI (assuming a normally distributed error). In addition, we also considered the error propagation derived from the fitting procedures of the force of infection, which may be estimated as follows:

where, again, *λ*(*t*) is the force of infection or incidence density rate. Note that the risk expressed in equation (A10) indicates the risk of infection among travellers that remain in the ZIKV endemic area during the period between *t*
_{1} and *t*
_{2}. For locals, *t*
_{2}−*t*
_{1} is the time interval considered in the risk calculation (e.g., the week-by-week risk calculation).

## REFERENCES

2.
Massad, E, et al.
Estimated Zika virus importation to Europe by travellers from Brazil. Global Health Action
2016; 9: 31669. doi: 10.3402/gha.v9.31669.

3.
Zanluca, C, et al.
First report of autochthonous transmission of Zika virus in Brazil. Memórias do Instituto Oswaldo Cruz
2015; 110(4): 569–72.

4.
Enserink, M. Infectious Diseases. An Obscure Mosquito-Borne Disease Goes Global. United States: Science, 2015, pp. 1012–3.

5.
Lednicky, J, et al.
Zika virus outbreak in Haiti in 2014: molecular and clinical data. PLoS Neglected Tropical Diseases
2016; 10(4): e0004687. doi: 10.1371/journal.pntd.0004687.

6.
Musso, D, Nilles, EJ, Cao-Lormeau, VM. Rapid spread of emerging Zika virus in the Pacific area. Clinical Microbiology and Infection
2014; 20: 595–6.

7.
Tabatabaei, SM, Metanat, M. Mass gathering and infectious diseases: epidemiology and surveillance. International Journal of Infection
2015; 2(2): e22833.

8.
Massad, E, et al.
Dengue outlook for the World Cup in Brazil. Lancet Infectious Diseases
2014; 14(7): 552–3. doi: 10.1016/S1473-3099(14)70807-2. No abstract available.

9.
Massad, E, et al.
Risk of symptomatic dengue for foreign visitors to the 2014 FIFA World Cup in Brazil. Memórias do Instituto Oswaldo Cruz
2014; 109(3): 394–7.

10.
Massad, E, Coutinho, FA, Wilder-Smith, A. The olympically mismeasured risk of Zika virus in Rio de Janeiro – Authors’ reply. Lancet
2016; 13;388(10045): 658–9. doi: 10.1016/S0140-6736(16)31228-4.

11.
Massad, E, Coutinho, FA, Wilder-Smith, A. Is Zika a substantial risk for visitors to the Rio de Janeiro Olympic Games?
Lancet
2016; 388(10039): 25. doi: 10.1016/S0140-6736(16)30842-X.

12.
Faria, NR
et al.
Zika virus in the Americas: early epidemiological and genetic findings. Science
2016; 352(6283), 345–349.

14.
Musso, D, Cao-Lormeau, VM, Gubler, DJ. Zika virus: following the path of dengue and chikungunya?
Lancet
2015; 386(9990): 243–244. doi: 10.1016/S0140-6736(15)61273-9.

15.
Amaku, M, et al.
Estimating the size of the HCV infection prevalence: a modeling approach using the incidence of cases reported to an official notification system. Bulletin of Mathematical Biology
2016; 78(5): 970–90. doi: 10.1007/s11538-016-0170-4. Epub 2016 May 9.

16.
Ximenes, R, et al.
The risk of dengue for non-immune foreign visitors to the 2016 summer Olympic games in Rio de Janeiro, Brazil. BMC Infectious Diseases
2016; 16: 186. doi: 10.1186/s12879-016-1517-z.

17.
Burattini, MN, et al.
Potential exposure to Zika virus for foreign tourists during the 2016 Carnival and Olympic Games in Rio de Janeiro, Brazil. Epidemiology and Infection
2016; 144(9): 1904–6. doi: 10.1017/S0950268816000649.

18.
Massad, E, et al.
Estimation of R_{0} from the initial phase of an outbreak of a vector-borne infection. Tropical Medicine and International Health
2010; 15(1): 120–6. doi: 10.1111/j.1365-3156.2009.02413.x.

19.
Funk, S, et al. Comparative analysis of dengue and Zika outbreaks reveals differences by setting and virus. bioRxiv preprint first posted online Mar. 11, 2016. doi: 10.1101/043265.

21.
Massad, E, et al.
Modeling the risk of malaria for travelers to areas with stable malaria transmission. Malaria Journal
2009; 8: 296. doi: 10.1186/1475-2875-8-296.

22.
Massad, E, et al.
Cost risk benefit analysis to support chemoprophylaxis policy for travellers to malaria endemic countries. Malaria Journal
2011; 10: 130. doi: 10.1186/1475-2875-10-130.

23.
Zinszer, K, et al.
Reconstruction of Zika virus introduction in Brazil. Emerging Infectious Diseases
2016; 2017 Jan [13 September 2016]. DOI: 10.3201/eid2301.161274.

24.
Fergusson, N, et al.
Countering the Zika epidemic in Latin America. Science
2016; 353(6297): 353–4. doi: 10.1126/science.aag0219.

25.
Lopez, LF, et al.
Modeling importations and exportations of infectious diseases via travelers. Bulletin of Mathematical Biology
2016; 78(2): 185–209. doi: 10.1007/s11538-015-0135-z.

26.
Massad, E, Rocklov, J, Wilder-Smith, A. Dengue infections in non-immune travellers to Thailand. Epidemiology and Infection
2013; 141(2): 412–7. doi: 10.1017/S0950268812000507.