We first established three different model parameter sets and characterized them in terms of R0, which indicates the average number of infected humans produced by a single infected human in a completely naïve population. Using low, moderate, and high literature estimates of the parameters for the human infectious period, vector density, vector biting rate, efficiency of human to vector transmission, and efficiency of vector to human transmission, we estimated R0 values to be 0.42, 4.1, and 90 under the respective scenarios at peak transmissibility conditions (). Although these R0 estimates classify transmissibility at 36°C and 100% relative humidity, transmissibility in the model is adapted to reflect temporal and geographic variation of local climate. shows global climate-adjusted estimates of R0 based on the moderate transmissibility scenario for January and July.
Figure 2. Estimated R0 in January and July for an average year. R0 is a measure of transmission potential under idealized contact conditions and does not account for important extant determinants of transmission, such as the prevalence of vaccination or personal (more ...)
Spread without vaccination
Before the outbreak-associated vaccination campaign in 2008, reported vaccine coverage in Paraguay was 34%,17
but vaccination efforts had been concentrated on children throughout the country and people in rural border areas rather than in Asunción.2
We, therefore, assumed that the population of Asunción was 100% susceptible. Furthermore, to simulate a worst-case scenario, we assumed that all other populations were also 100% susceptible. The first cases in Paraguay were reported in January of 2008, and therefore, we ran 1,000 simulations with the introduction of a single incubating individual to Asunción on January 1. Given the local climate at that time of year, the initial R0
values in Asuncion were 0.047, 0.46, and 10 for the low, moderate, and high scenarios, respectively.
In the low transmissibility scenario, only 2.3% of the simulations resulted in local transmission, with a maximum of seven additional cases occurring (). Because transmission under this scenario was so limited and spread to other cities did not occur, it was not considered in later experiments. Under the moderate transmission scenario, a single introduction led to additional human transmission in 128 (12.8%) of the simulations. In 108 (84.4%) of these outbreaks, the outbreak involved only local transmission, affecting a median of 2 persons with a range of 1–981 persons. In two outbreaks, infectious individuals arrived in other cities but did not initiate any additional transmission. In the other 20 (15.6%) simulations, however, large epidemics occurred, affecting 450,000–550,000 people in Asunción, and international spread occurred, resulting in YFV pandemics. shows epidemic curves for a selection of cities under the moderate R0 model. Under the high R0 scenario, nine (0.9%) simulations resulted in only small-scale local transmission (one to three additional cases), one of which resulted in a single infected traveler going to New York (). In 689 (68.9%) other simulations, there were large local outbreaks leading to pandemics.
Occurrence of local YFV transmission in Asunción, infected travelers, and autochthonous transmission in other cities in simulations under different transmissibility scenarios
Figure 3. Pandemic simulation. The lines indicate the number of individuals becoming infectious each day for nine representative cities in a single moderate R0 pandemic simulation.YFV transmission is initiated in Asunción, where transmission follows a seasonal (more ...)
Dynamics were monitored locally for both potential introduction (i.e., the presence of infectious individuals) and autochthonous transmission, which was evidenced by a locally acquired human infection (). In the moderate R0 model, the first international spread of YFV by an infected or infectious traveler from Asunción occurred at a median of 259 days (range = 14–561 days) after introduction into Asunción. At the time of introduction, a median of 1,013.5 infections (range = 3–6,363 infections) had occurred in Asunción. The first international autochthonous transmission occurred after 596.5 days (range = 203–1,310 days) when there had been 10,654 infections (range = 1,045–61,240 infections) in Asunción. In the high R0 model, both introduction and autochthonous transmissions occurred earlier at a median of 53 (range = 3–80 days) and 68 days (range = 27–94 days), respectively. This finding corresponded to a median of 756 infections (range = 5–8,735 infections) occurring before the earliest foreign introduction event and 9,468 infections (range = 28–140,681 infections) before the first foreign autochthonous transmission event.
The first three cities to which YFV was introduced by infected travelers in both the moderate and high transmissibility scenarios were Paris, London, and New York. Autochthonous transmission occurred earliest, on average, in New York, Miami, and Singapore in the moderate R0 model and Miami, Sao Paulo, and Singapore in the high R0 model. This order of initiation of autochthonous transmission varied greatly between simulations. For example, although Miami was, on average, the first city to experience autochthonous transmission in the high R0 model, in 25.7% of the simulations, 10 or more cities experienced transmission before Miami.
Spread with vaccination
We then repeated simulations for both the moderate and high transmissibility scenarios under the assumption that the population of each city had been vaccinated at the last reported coverage rate for its respective country. In Asunción, for example, 67% of the population was assumed to be immune, because the estimated YFV vaccine coverage for Paraguay was 67% after the 2008 vaccination campaign. Under these conditions, local transmission occurred in 69 and 640 of 1,000 simulations in the moderate and high transmission scenarios, respectively (). In the 6.9% of moderate scenario simulations in which transmission occurred, transmission was limited to 1–12 new local infections with a single infected traveler but no subsequent transmission. In the high R0 model, 57.4% of the simulations resulted in pandemics. When pandemics occurred, the total number of infections globally was reduced by ~26% (307.2–307.8 million with vaccination versus 416.4–416.8 without vaccination).
For each simulation, the probabilities of introduction, pINTRO, and introduction leading to autochthonous transmission, pAUTO, were calculated using the theoretical models. shows the increase in pINTRO over the course of a single simulation for three cities. Introduction is predicted when pINTRO = 0.5. In the moderate R0 model, there were 2,802 simulated introductions of a possible 140,000 (1,000 simulations for 140 cities). On average, predicted introduction was 22 days (middle 95% = −253–277 days) ( and C) an 2 days (middle 95% = −23–23 days) before introduction in the moderate and high R0 scenario simulations, respectively. Of the 2,802 simulated introduction events, 2,800 were predicted for a sensitivity of greater than 99% and a negative predictive value (NPV) of greater than 99%. With 20 false positives, both the specificity and positive predictive value (PPV) were also greater than 99%. In the high R0 model, all but 1 of 96,460 introductions were predicted for a sensitivity and NPV of greater than 99%. Specificity was greater than 98% with 689 false positives, and the PPV was greater than 99%.
Figure 4. Probability of introduction events (moderate R0). A shows the probability of introduction (solid line) for three cities as a function of time in a single simulation. For each city, the threshold, pINTRO = 0.5, is indicated by the horizontal dashed line, (more ...)
The onset of autochthonous transmission is predicted at pAUTO = 0.5. In the simulations, autochthonous transmission was predicted on average 33 days (middle 95% = −171–242 days) () and 11 days (middle 95% = −14–33 days) before occurrence in the moderate and high R0 simulations, respectively. In the moderate R0 model, autochthonous transmission was predicted on 2,580 occasions, of which 2,560 had simulated transmission events (PPV > 99%). Specificity was also greater than 99% with no false negatives, and sensitivity and NPV were 100%. The high R0 model also had no false negatives (sensitivity = 100%, NPV = 100%). There were 689 false positives, however, with specificity and PPV approximately 99%.
Figure 5. Probability of autochthonous transmission (moderate R0). A shows the probability of autochthonous transmission (solid line) for three cities as a function of time in a single simulation (the same simulation as ). For each city, the threshold, (more ...)
With preexisting vaccination, only the high R0 model led to autochthonous transmission in other areas (). For introduction and autochthonous transmission, sensitivity, specificity, PPV, and NPV were all greater than 99%. Both the prediction and occurrence of introduction and autochthonous transmission were delayed when vaccination was incorporated ().
Figure 6. The effect of vaccination on simulated and predicted events (high R0). A shows simulated and predicted introduction times (N = 2,000, sampled randomly from the complete set) for the high R0 model with (grey) and without (black) prior vaccination. B shows (more ...)
We also assessed the probabilistic models in the case where complete data on an epidemic is unknown. shows the probability of spread, pSPREAD
, from Asunción using the travel parameters presented here under increasing cumulative infected person-days and the probability of spread resulting in autochthonous transmission, pSPREAD→AUTO
, in at least one other city based on the number of infected person-days and
on January 1. The probability of spread leading to autochthonous transmission is delayed compared with the probability of spread, and it is further delayed with decreased R0
or the presence of preexisting vaccination in other cities. With 10,000 infected person-days, for example, the probability of spread having already occurred is approximately 0.8, and the probability of autochthonous transmission having occurred in another city is approximately 0.5 under the high R0
model and 0.2 under the moderate R0
model. Note that an average human infection results in 7.6 infected person-days (4.6 days incubating and 3 days infectious), and therefore, 10,000 infected person-days is roughly equivalent to a cumulative total of 1,300 people infected.
Figure 7. Probability of spread. A shows the relationship between accumulating infected person-days in Asunción and the risk of spread to at least one other city. The solid line is the probability of an infected traveler departing Asunción, pSPREAD (more ...)
The modified infected person-day equations can also be used to calculate the probability of introduction to a particular city. shows how the cumulative probabilities of introduction and autochthonous transmission in three cities follow the number of infected person-days in Asunción. In the case of Paris, introduction is highly probable, but R0 is so low on January 1 that the probability of autochthonous transmission is virtually zero. Meanwhile, when R0 is high, such as in Miami and Johannesburg in the high R0 model, the probability of introduction leading to autochthonous transmission is nearly equivalent to the probability of introduction.