The model described here is the first model to mechanistically address the potential for a vector-borne pathogen, such as YFV, to spread around the world through infected airline travelers. It was built using our best understanding of the dynamics of *Ae. aegypti* mosquitoes, YFV infection, and global travel and was designed to assist in assessing the probabilities of spread of YFV in the event of an urban epidemic. To put this model into a real life context, we applied it to an actual outbreak that occurred in Asunción, Paraguay, in 2008. Below, we discuss our estimation of YFV transmission dynamics, what the models suggest about the outbreak in Asunción, our findings regarding the probability of introduction and autochthonous transmission of YFV, the effect of existing vaccine coverage, and the limitations of the data and models.

YFV transmission dynamics

We assessed three transmission scenarios representing drastically different estimations of YFV virus transmissibility and pandemic potential (). Although all three scenarios incorporate plausible estimates for individual parameters, it is likely that the moderate parameter set represents the most realistic scenario. The lowest estimate that we evaluated for *R*_{0} was 0.42, too low to reliably cause epidemics even under the most favorable environmental conditions. The highest estimate for *R*_{0} was 90, extremely high compared with related dengue viruses for which estimates range from 0 to 103 but with median estimates in the range of 1 to 6.^{21}^{–}^{27} Moreover, given that most YFV epidemics are small or progress slowly,^{1} it is more likely that *R*_{0} for YFV is generally much lower, closer to 1 than 90.

Between the low and high *R*_{0} estimates, there is much parameter flexibility. Although our moderate model likely overestimates some parameters, it likely underestimates others, leading to a middle ground. Although this likelihood cannot be explicitly tested, each parameter that we used falls within a reasonable range (more details in Supplemental Information), and the estimated geographic areas where transmission is favored () correspond to the known, historical, and estimated spatial distributions of YFV and dengue virus transmission.^{28} The YFV *R*_{0} estimates from the moderate model are also similar to those estimates in previous studies.^{25}^{,}^{29}^{,}^{30} Note that *R*_{0} is not an absolute determinant of potential transmission; many other factors such as vaccination rates, vector control programs, and personal protective measures may also determine whether transmission occurs.

Further refining estimates of *R*_{0} would be difficult because of the complexity of the underlying components. For example, various studies estimate that the average human to vector efficiency of YFV transmission is much less than 0.5, the estimate under the moderate scenario.^{31}^{–}^{34} At lower efficiency estimates, however, *R*_{0} quickly drops below one, even under ideal environmental conditions. For YFV to cause even occasional epidemics, as it does, either this efficiency has been routinely underestimated or there are other components that have been underestimated.

Asunción

In the actual Asunción outbreak, a total of nine locally acquired infections were confirmed. Using the moderate *R*_{0} parameter set to simulate the introduction of a single infected individual into Asunción, we found that small local outbreaks occurred in 10.8% of the simulations. An outbreak like the one that was reported is, thus, a distinct possibility, although no further transmission was a more common result in simulations (87.2%).

It is possible that we underestimated the probability of local outbreaks by underestimating YFV transmissibility in Asunción. In our high *R*_{0} model, the frequency of local outbreaks was higher, with local transmission occurring in 70.7% of the simulations. However, in 98% of those outbreaks, a pandemic occurred, an eventuality that did not occur in the real outbreak.

We also lack a complete description of the actual outbreak. An infected individual with a travel history to rural areas with ongoing transmission was never identified, and the true number of people infected is likely underestimated, because many infected individuals may be asymptomatic. However, if more than one infected person had arrived, the probability of a local epidemic would have been substantially higher. For example, given that 10.8% of introductions in the moderate *R*_{0} model resulted in local transmission, if six infected people arrived, the probability of local transmission would be almost 50% (1 − [1 − 0.108]^{6}).

The most probable explanation for the short-lived outbreak in Asunción is that it was self-limited because of a relatively inhospitable environment (low local *R*_{0}) and that spread beyond Asunción did not occur, because with so few individuals infected, spread is unlikely to occur. Using Equation 3, with a total of nine infected individuals and average duration of infection of 8 days, the probability of at least one infected individual leaving Asunción is approximately 0.01.

Probability of introduction by travelers

The first event of interest relative to the potential spread of YFV by travelers is the appearance of an incubating or infectious individual in a population where YFV is absent. The simulations presented here can be used to directly estimate the probability of spread under the assumptions that we have presented. In the moderate *R*_{0} model, international introduction from Asunción was rare, occurring in 2.2% of simulations. However, in 90.9% of those simulations, YFV-infected travelers eventually reached every city in the model, leading to a pandemic. Thus, although the probability of spread is low, the consequences may be drastic. In the high *R*_{0} model, both of these events were more common, with 69.0% of simulations resulting in international spread and 99.9% of spread resulting in pandemics.

Focusing on the simulations in which pandemics did occur in the moderate *R*_{0} model, the median time to spread was 259 days, but spread occurred as soon as 14 days after the initial case was introduced to Asunción. At the time of the earliest spreading events, the median outbreak size in Asunción was just over 1,000 people and spread occurred with as little as 3 people infected. This timing, in terms of both actual time and the number of people infected, shows that outbreaks could quickly spread to other locations before being recognized.

The probabilistic models were highly sensitive and specific for the prediction of introduction in the simulations and tended to predict introduction before actual introduction (). As Equation 3 makes clear, the cities with the highest rates of travel are the ones where the first introductions are expected. In our model, Asunción had the highest rates of travel to Paris, London, and New York, the cities where introduction occurred earliest in the simulations. After the initial spread, the situation becomes more complicated, because there are multiple sources of infected individuals.

In the midst of an ongoing outbreak, precise data on the number of people infected and the timing of their infectious periods is generally not available. Therefore, it may be of more use to estimate the risk of spread using an estimate of cumulative infected person-days. As presented in Equation 3, this estimate and an estimate of travel rates are sufficient to estimate both the probability of infected travelers leaving a given city and the probability of infected travelers arriving in a given city.

Probability of introduced autochthonous transmission

Assessing the risk of introduction is only the first step. Often more critical is assessing whether introduction will lead to autochthonous transmission. The only additional information needed to estimate the probability of autochthonous transmission after introduction is the transmission components

and

for the time and location of interest (Equation 5). As discussed above, we have estimated

*R*_{0} and its subcomponents mechanistically, with reassuring concordance with historical observations and environmental suitability models.

Using probability generating functions to estimate the probability of one or more autochthonous infections, we reliably predicted our simulations of these events (). We also estimated the probability of autochthonous transmission occurring in other cities based solely on the cumulative number of infectious person-days in a source city, showing that the probability of autochthonous transmission depends on both the probability of introduction and the efficiency of local transmission (). The stochasticity of these processes contributes to the high degree of variability in the city where the earliest autochthonous infections occurred in the simulations.

Prior vaccination

Prior vaccination in Asunción reduced the probability of outbreaks (). This finding is because of both reduced individual susceptibility (direct effect) and reduced rate of vector to human transmission, because some infectious vectors feed on immune humans (indirect effect). Because the number of local infections is a key determinant of the probability of international spread, vaccination in Asunción reduces the frequency of spread (), and the slower growth of those epidemics that do occur leads to a delay in spread ().

Despite the decreased probability of a seed epidemic and slower spread when these epidemics did occur, pandemics still occurred. Overall, the probability of autochthonous transmission in other cities is slightly decreased, reflecting the decreased transmissibility in the cities with high vaccine coverage (). Previous vaccination also contributed to a global reduction in the number of persons affected by approximately 26% or 100 million persons. Thus, although prior vaccination decreases the probability of spread occurring and slows its pace, the potential for a major global health problem persists.

Although pandemics may occur in the presence of prior vaccination, in our simulations, they only occurred in the high *R*_{0} model. Under the more realistic assumptions of the moderate *R*_{0} model, they did not occur, suggesting that previous vaccination in the population where the first infections occur may be sufficient to prevent international spread. We did not assess the critical threshold for vaccination coverage, but optimal coverage rates can be derived based on *R*_{0} values.^{29}^{,}^{35}^{,}^{36} Preventive vaccination may seem a logical control measure, but there are also problems with vaccine supply, cost, and safety.^{7}^{–}^{12} In future work, we will evaluate the potential impact of both preventive vaccination and reactive interventions, such as local vaccination and vector control, vaccination of travelers, and restriction of travel.

Limitations

Two important sources of uncertainty are the parameterizations of the travel network and YFV transmission dynamics (). The former requires more data,^{14} and the latter is partly captured in the different *R*_{0} scenarios. However, even within a given scenario, there is likely more variability than we could reasonably incorporate. Different vector densities and contact rates, for instance, may vary greatly between cities based on housing characteristics and other factors that cannot be reliably assessed on a global scale. It is also not necessarily true that *Ae. aegypti* are present in all of the areas where YFV transmission may occur in our model.^{28} In some areas, *Ae. aegypti* has been replaced by *Ae. albopictus*,^{37} another competent vector.^{32}^{,}^{34}^{,}^{38}^{,}^{39} Because the geographical distributions of the two species are dynamic and imprecisely known and because the relative importance of each species to YFV transmission is not well-understood, we did not attempt to model any differences between them.

Beyond the parameterization assumptions above, one of the most important assumptions that we make is that local transmission is a mass action-based process. There is ample evidence to suggest that virus transmission by *Ae. aegypti* is highly focal,^{40}^{–}^{43} thus treating each city as a single pool of individuals all experiencing equal exposure risk masks significant underlying heterogeneity. However, our primary interest is the probability of spread between populations, and the local heterogeneity is likely of little importance. Perhaps most critical to the subject of interest here is the simple fact that not all travelers are equivalent. It is well-documented that travelers visiting friends and family are more likely to stay longer, stay in homes rather than hotels, and be infected by pathogens while traveling.^{44}^{–}^{49} Unfortunately, adding more local heterogeneity for human and vector interaction would require parameterization beyond the reach of available data, especially when applied globally.

Lastly, we made significant simplifications regarding the immune status of the populations. We assumed either complete susceptibility or partial immunity on the population scale because of vaccination at a level consistent with the reported country-wide rates, which do not necessarily reflect immunity in the cities. Furthermore, vaccines are not the only source of immunity. Some populations have experienced natural exposure, and others may have acquired some degree of cross-immunity because of exposure to other flaviviruses. For example, cross-protection afforded by prior dengue virus exposure is a principal hypothesis for why YFV has not emerged in Asia, where competent vectors and dengue viruses are ubiquitous.^{50}^{,}^{51} Because of these complications and a lack of data to address them on a global scale, more accurate estimation of YFV susceptibility is a formidable challenge.

General conclusions

The models presented here provide general approaches to assessing the risk of vector-borne disease spread by infected travelers. Despite their limitations, these models may serve as useful tools and starting points for future models of vector-borne disease spread and interventions designed to reduce the risk of spread. The models also represent formal hypotheses about the YFV transmission system and travel network, which is detailed in Materials and Methods and Supplemental Information. We found that the most critical predictors of disease spread are the rates of travel, number of infected individuals, general transmission parameters (

and

), and vaccination rates when vaccines are concerned. With estimates of these components, calculation of the probability of introduction and autochthonous transmission can easily be estimated for any ongoing outbreak. Meanwhile, as improved estimates of transmission components and travel rates become available, they can be incorporated into complete mechanistic models, enabling more detailed analyses of a wider variety of potential outcomes.