The role of vaccination coverage, individual behaviors, and the public health response in the control of measles epidemics: an agent-based simulation for California

To determine which parameters were the most important in determining the number of secondary cases, the outbreak size, and the probability of uncontrolled outbreak in a one year of measles transmission, we selected a Latin Hypercube sample of 1024 different parameter sets from the parameter ranges indicated in Table 1. From each, the number of secondary cases, the number of outbreak size, and the probability of uncontrolled measles outbreaks (over one year) were computed from 256 replications. For definiteness, we chose Humboldt County in California (population size 118,261). The simulation was started by introducing one index case who was randomly selected according to the observed age distribution of measles cases in California. We then computed the partial rank correlation coefficient (PRCC) [40,41] between each input parameter and each of three simulation outcomes (the number of secondary cases, the total outbreak size and the probability of uncontrolled measles outbreaks in 365 days). When the PRCC is close to zero, the value of the parameter has little relation to the simulation output; when the PRCC is close to +1 or −1, the value of the parameter is highly important in determining the simulation output.

We chose 4 parameters (PRCC ≥ 0.1, in Table 2), which are (1) vaccination coverage, (2) level of immunity clustering (Ω), (3) intervention delay, and (4) contact finding probability to explore factors which contribute to the successful control of measles (i.e., the probability of uncontrolled measles outbreak). In this section, we examine (1) the combinations of vaccination coverage and the level of immunity clustering, (2) the combinations of intervention delay and contact finding probability based on different levels of immunity clustering and vaccination coverage, (3) the combinations of vaccination coverage and contact rates in the neighborhood, and (4) the combinations of vaccination coverage and contact rates in all places. We report the probability of escape, or the mean outbreak size (when no scenarios showed a total number of cases exceeding 500 by one year after measles introduction). Due to the critical threshold level for the fraction of susceptible individuals below which introduction of infections can only lead to minor outbreaks and above which minor and major outbreaks can occur [42], we present the probability of uncontrolled outbreaks for scenarios in which minor and major outbreaks occurred and total outbreak size for scenarios that only led to minor outbreaks [43].

For combinations of vaccination coverage ranging from 0.8 to 1 and the level of immunity clustering ranging from 0 to 1 (all other parameter values are shown in Table 1), the results are shown in Figure 1. When at least one escape scenario occurred (out of 256 trials), the result is shown in red; when there were no escape scenarios, the mean outbreak size is shown in blue; the border between an escape scenario and a no escape scenario is shown in black. Figure 1 shows that the vaccination coverage required to prevent measles outbreaks was approximately 95% for all levels of immunity clustering. For the scenario with the lowest level of immunity clustering, the threshold vaccination coverage was 80%. For the combinations with which there were no measles outbreaks, Figure 1 also shows that the outbreak size dramatically decreased with an increase in vaccination coverage for a given level of immunity clustering. These simulations simply illustrate the importance of high vaccination coverage and that clustering of unvaccinated individuals is unfavorable for disease control. The results assumed the contact rates, self-report delays, intervention delays for contacts of the index case, and the probability that a contact can be traced were constant (Table 1).

To expand further on the relationship between vaccination coverage and immunity clustering and their role in the control of measles outbreaks, we examined the influence of two parameters related to contact investigations, intervention delay for contacts of an index case and the contact finding probability, on the occurrence of measles outbreaks. In Figure 2, we present the results of our findings for three levels of vaccine coverage (V = 0.85, 0.90, and 0.95 in the three columns) and three levels of immunity clustering (Ω=0, 0.5, and 1.0 in the three rows). For each coverage-clustering combination, we show the results of our simulations across values of the contact finding probability (x-axis ranging from 0.50 to 1.0) and the intervention delay for contacts of the index case (y-axis ranging from 1 to 7 days). In the setting with the lowest vaccination coverage and the highest level of immunity clustering (V=0.85 and Ω=1.0, Figure 2(A)), we found that escape scenarios could occur for all values of the intervention delay and the contact finding probability, and thus report the probability of escape scenarios denoted in red. In the setting with the highest vaccination coverage and the lowest level of immunity clustering (V=0.95 and Ω=0.0, Figure 2(I)), we obtained no uncontrolled outbreaks, and report the total outbreak size for all values of the intervention delay and the contact finding probability (shown in blue). The settings where the levels of the intervention delay and the contact finding probability become important were in Figure 2(C), (E), and (G), when the coverage-clustering combinations were either both high (Figure 2(C)), both moderate (Figure 2(E)), or both low (Figure 2(G)). For these three settings, the level of intervention delay for contacts of the index case and the probability a contact can be found determines whether we have controlled or uncontrolled measles outbreaks. Even with stochastic simulation error, higher contact finding probabilities with lower intervention delays lead to controlled measles outbreaks.

However, Figure 2 does not show the effects of the speed of contact investigation on the uncontrolled probability and the outbreak size. To study the effects of the speed of contact investigations, we varied the self report delay, the intervention delay and the contact tracing delay and found that the uncontrolled outbreak probabilities and the outbreak sizes were reduced by increasing the speed of contact investigation (as shown in Table 3). In general, the speed of contact investigations does affect how well it contributes to disease control.

The PRCC value for the neighborhood contact rate (in Table 2) indicates that the transmission in the neighborhood setting is highly important in determining the probability of uncontrolled measles outbreaks and the total outbreak size because of the assumption that contact tracing does not happen in the neighborhood (i.e., people are not likely to identify the people they encounter in the neighborhood setting). We used the parameters’ values in Table 1 and combinations of vaccination coverage (ranging from 0 to 1) and the neighborhood contact rate (ranging from 1 to 7) to study their effects on the uncontrolled measles outbreak probability and total outbreak size. The simulation for each combination was run under three scenarios: (1) without contact investigation, (2) with contact investigation and little intervention delay for contacts of the index case, and (3) with contact investigation and more intervention delay for contacts of the index case. For the scenario without contact investigation, Figure 3(A) shows that more neighborhood contacts caused more outbreaks; increasing vaccination coverage could reduce the number of measles cases, but even with the highest vaccination coverage (100%), it still could not completely prevent the measles outbreak for high contact rates in neighborhood. For the scenario with contact investigation and less intervention delay for contacts of the index case, as shown Figure 3(B), with lower contact rates in the neighborhood, measles could be controlled even with 80% vaccination coverage; and 100% of vaccination coverage could prevent measles outbreaks for all contact rates in the neighborhood. For the scenario with contact investigations but with more intervention delays for contacts of the index case, Figure 3(C) shows that measles outbreaks were prevented by the combinations of low contact rates in the neighborhood with all vaccination coverage levels, and the combinations of the highest vaccination coverage level with all contact rates in neighborhood. Most of the uncontrolled outbreak probabilities and outbreak sizes were higher than those of the scenario with contact investigation and less intervention delays for contacts of the index case, i.e., there is less of a chance for a traced contact to receive PEP MMR or PEP IG with more intervention delays. By comparing the same combinations of the scenario without contact investigation in Figure 3(A), the probabilities of uncontrolled outbreaks and the outbreak sizes among the other two scenarios in Figures 3(B) and (C) were dramatically reduced by contact investigations. These simulations suggest: (1) contact investigation plays an important role in preventing measles outbreaks and reducing the outbreak size; (2) with contact investigations, reducing the contact rate in the neighborhood may lower the vaccination coverage required to prevent measles outbreaks; (3) with contact investigation and the highest vaccination coverage, measles outbreaks may be prevented even with very high contact rates in neighborhood; and (4) less intervention delays for contacts of the index case may help contact investigations reduce the probability of uncontrolled measles outbreaks and the outbreak size.

To study further the effects of contact rates in all places and vaccination coverage on the uncontrolled measles outbreak probability and outbreak size, we used a parameter ranging from 0 and 1 to scale contact rates in school, daycare, workplace, household and neighborhood settings (i.e., when the scale parameter is 0, each place has the lowest contact rate; when the scale parameter is 1, each place has the highest contact rate; when the scale parameter is between 0 and 1, the contact rate in each place is between its lower and upper bounds shown in Table 1). Similarly, for each combination of vaccination coverage (ranging from 0 to 1) and the scale parameter (ranging from 0 to 1), we ran the simulations under three scenarios (all other the parameters’ values are shown in Table 1): (1) without contact investigation, (2) with contact investigation and less intervention delay for contacts of the index case, and (3) with contact investigation and more intervention delay for contacts of the index case. For the scenario without contact investigation, Figure 4(A) shows that the more contact rates in all places the more outbreaks and the higher number of outbreak sizes; increasing vaccination coverage could reduce the number of measles cases; when scale of contact rates in all places was very high, even the highest vaccination coverage (100%) could not completely prevent the measles outbreak; however, for the lowest scale of contact rates, even 80% of vaccination could prevent measles outbreak. For the scenario with contact investigation and less intervention delay for contacts of the index case, Figure 4(B) shows that with lower scale of contact rates in all places, measles could be controlled even with 80% of vaccination coverage; and 100% of vaccination coverage could prevent measles outbreak for all scales of contact rates in all places. For the scenario with contact investigations but with more intervention delays among contacts of the index case, Figure 4(C) shows that measles outbreaks were prevented by the combinations of low contact rates in all places and all vaccination coverage levels, and the combinations of the highest vaccination coverage and all contact rates in all places; most of the uncontrolled outbreak probabilities and the outbreak sizes were higher than those of the scenario with contact investigations and less intervention delays for contacts of the index case (in Figure 4(B)) because the effects of intervention delays. By comparing this to the same combinations of the scenario without contact investigations in Figure 4(A), the probabilities of an uncontrolled outbreak and the outbreak sizes in the other two scenarios in Figures 4(B) and (C) were dramatically reduced by the contact investigations. These simulations suggest: (1) contact investigations play an important role in preventing measles outbreaks and reducing the total outbreak sizes; (2) without contact investigations but with the lowest scale of contact rates in all places, even an 80% vaccination coverage may prevent measles outbreaks; (3) with contact investigations, reducing the contact rates in all places may lower the vaccination coverage required to prevent measles outbreaks; (4) with contact investigations and the highest vaccination coverage level, measles outbreaks may be prevented even with very high contact rates in all places; (5) less intervention delays for contacts of the index case may help contact investigations reduce the number of measles cases.