The impact of changes in vaccine coverage in 1960 (introduction of routine vaccination) and 1990 (introduction of SIAs) can be seen in time series of disease prevalence of all three diseases in all three countries under the simultaneous strategy (Figure ). Baseline parameter values (Table ) are used for this simulation as well as subsequent simulations, except where otherwise noted. In 1960, the introduction of routine vaccination induces transient oscillations in infection prevalence until the model dynamics "settle down" to a new equilibrium prevalence that is lower than the pre-vaccine era prevalence in all three countries. In 1990, the introduction of SIAs with a total annual budget of US $180 million for all three SIA programs in all three countries results in further reduction in prevalence for all three infections: Disease B and Disease C remain endemic in all three countries, and Disease A is eliminated from India and Afghanistan although it remains endemic at very low levels in Nigeria. Hence, under the simultaneous strategy for baseline parameter values, no disease is eradicated for the foreseeable future. Therefore, a total annual budget of US $180 million is used as our baseline parameter value, to provide a meaningful contrast to outcomes under the sequential strategy versus the simultaneous strategy. This value is on the same order of magnitude as immunization program budgets in some countries [16
Figure 1 Time series of prevalence under simultaneous strategy. Disease prevalence in India (a), Nigeria (b), and Afghanistan (c) according to simulations, where routine vaccination is introduced in 1960 and SIAs are introduced in 1990. Cumulative incidence for (more ...)
The differences in epidemic patterns between the three countries observed in Figure are due to differing birth rates (Table ) and vaccine unit costs (Table ). The time between outbreaks is longer in India (Figure ) than in Nigeria (Figure ) and Afghanistan (Figure ) because India has a lower birth rate than either Nigeria or Afghanistan (Table ). Similarly, Disease A persists for longer in Nigeria (Figure ) than in India (Figure ) or Afghanistan (Figure ) because the vaccine for Disease A costs significantly more for Nigeria than for India or Afghanistan (Table ). The way that elimination is defined may also be relevant: we considered a disease as eliminated in a country if prevalence fell below one person, and this benchmark is harder to reach in a larger population such as that of India.
In the case of the sequential strategy with the same annual budget of US $180 million, the time evolution is the same as for the simultaneous strategy until budget reallocation begins in 2010. Under the sequential strategy, the model predicts a very different time evolution of future incidence (Figure ). All three diseases are eradicated by 2030: Disease A is eradicated first in 2021, followed by Disease B in 2024, and finally Disease C in 2030 (by comparison, under the simultaneous strategy for the same budget, no disease is eradicated in the foreseeable future). Figure shows a time series of SIA vaccination coverage in each country under the sequential strategy, and the changing vaccine coverage due to budget re-allocation is apparent.
Figure 2 Time series of prevalence under sequential strategy. Predicted disease prevalence in India (a), Nigeria (b), and Afghanistan (c), such that routine vaccination is introduced in 1960, SIAs begin in 1990, and reallocation occurs according to the sequential (more ...)
Time series of vaccination coverage under sequential strategy. SIA vaccination coverage in India (a), Nigeria (b) and Afghanistan (c) under the baseline sequential with an annual SIA budget of $180 million.
This example demonstrates how budget reallocation strategies can take advantage of herd immunity to speed eradication of all three diseases. However, there are tradeoffs: in the period where SIA funds are being dedicated entirely to Disease A, the average prevalence of Diseases B and C increases for a period after budget reallocation begins in 2010. This is seen in Figure versus Figure in the decade after 2010, where recurrent epidemic spikes are present in both figures but the average prevalence of Diseases B and C is higher in Figure than in Figure . This is also seen in plots of cumulative incidence over time, which also appear in Figures and .
We considered a disease to be eliminated in a country once prevalence in that country fell below 1 person. To test the impact of this assumption, simulations were also conducted using elimination thresholds of 2, 0.1 and 0.01 persons for the baseline sequential strategy. We found that the year of eradication of all three diseases changed from 2030 (for the baseline scenario where the threshold is 1 person) to 2024, 2060 and 2066 under these two thresholds, respectively. Hence, the cutoff for defining elimination in a country has some impact on model predictions although the differences are not large enough to change our conclusion that the sequential strategy outperforms the simultaneous strategy, at least for this level of annual budget.
In the following subsections we explore a wider range of possibilities, including what happens under: (1) variation in the total annual SIA budget; (2) variation in the frequency of SIAs; (3) variation in the order of eradication; (4) interruption in SIA budgets; (5) increasing routine vaccination over time; and (6) varying case importation rates.
Impact of varying total budget
We varied the annual SIA budget to understand the impact of budget on time to eradication under the simultaneous strategy versus the sequential strategy. For the sequential strategy, there is a minimum annual budget of approximately $170 million below which eradication of all three diseases never occurs. Above this threshold, the time required to eradicate all three diseases drops sharply. Beyond an annual budget of $190 million, the time to eradication stabilizes and does not decrease significantly as annual budget increases (Figure ).
Year of eradication as a function of annual budget, 3-year SIA intervals. Year of eradication as a function of annual SIA budget under a 3-year SIA interval, for the sequential strategy (a) and the simultaneous strategy (b).
We define an annual budget to be optimal if it requires the least total expenditure, i.e. the sum of annual budgets from the time SIAs are implemented (2010) to the time of eradication is minimized. For the sequential strategy, the optimal annual budget is US $185 million, under which eradication of Diseases A, B and C occurs by 2018, 2021, and 2027, respectively for a total of $3.14 billion between 2010 and 2027. For larger annual budgets, the time to eradication remains the same, such that eradication of all three diseases can be achieved by 2027 under annual budgets of $190, $195, and $200 million. However, the total expenditure required for each of these annual SIA budgets is higher at $3.23, $3.32, and $3.40 billion respectively, between 2010 and 2027.
In summary, under the sequential strategy there is a minimum threshold annual budget required for eradicating all three diseases, and there is also slightly larger optimal annual budget where all three diseases are eradicated for the least aggregate cost over time. Beyond the optimal annual budget, the incremental gains of increased annual budget on time to eradication are small, under our model assumptions.
The impact of annual SIA budget on time to eradication is qualitatively similar under the simultaneous strategy, with both a minimum threshold annual budget and an optimal annual budget. However, the time to eradication is considerably longer for the same or higher annual budget (Figure ). For instance, under the minimal threshold annual budget of $520 million, eradication of Diseases A and B occurs by 1997 and 2003 respectively, but Disease C is not eradicated until 2297. For a higher annual budget of $560 million, eradication of Diseases A, B and C occurs by 1997, 2000 and 2031, respectively (Figure )-hence, the time to eradication under the simultaneous strategy is similar to the time to eradication under the sequential strategy but it requires a much higher annual budget ($560 versus $180 million). If the annual budget under the simultaneous strategy is increased further to $580 million, we obtain the (non-historical) result that all three diseases are eradicated by 1997, requiring a total of $4.1 billion from 1990 to 1997. This total value of $4.1 billion from 1990 to 1997 under the optimal simultaneous strategy is actually less than the total value of $6.8 billion from 1990 to 2027 (year of eradication) under the optimal sequential strategy. Hence, a simultaneous strategy could eradicate all three diseases more quickly and for less money expended overall than a sequential strategy, but the annual budget must be three times higher than usual ($580 million versus $180 million), which may be difficult to achieve.
Impact of varying SIA scheduling
We next analyze the impact of time interval between SIAs on the time and budget required to eradicate all three diseases, under the sequential strategy. If SIAs occur every two years instead of every three years, there is still a minimum threshold budget ($220 million) below which eradication never occurs. Above this threshold there is very little variation in time to eradication, as occurred when SIAs were held every three years. For the optimal budget of $220 million, eradication of Diseases A, B and C, occur in 2017, 2021, and 2027 respectively (Figure ). Utilizing this optimal budget, a 2-year SIA schedule would require a total funding of $3.74 billion between 2010 and 2027.
Year of eradication as a function of annual budget under 2-year and 4-year SIA intervals. Year of eradication as a function of annual SIA budget for the sequential strategy if SIAs occur either every 2 years (a) or every 4 years (b).
If SIAs occur every 4 years, the minimum threshold budget is $160 million. Above this threshold, there is a sharp decline in the number of years required for eradication, but beyond an optimal budget of $180 million, the time to eradication does not significantly decrease. This was also a similar pattern as observed with the 3-year schedule. At the optimal budget of $180 million, we see that the eradication of Diseases A, B and C occurs by 2017, 2021, and 2025, respectively, requiring a total of $2.70 billion.
The optimal annual budgets are $3.74, $3.14, and $2.70 billion for the 2, 3, and 4-year SIA schedules respectively; optimal annual budgets are higher for more frequent SIAs. This occurs because annual budgets are fixed and hence SIA coverage is a function of the interval between SIAs; for the same budget, SIA vaccine coverage is higher when SIAs are less frequent. A higher SIA coverage is more likely to push prevalence below the eradication threshold just after the SIA is completed, and if SIAs are coordinated and occur simultaneously in the three countries, then it is more likely that the disease will be eliminated in all three countries. In contrast, for a shorter SIA interval (e.g. every 2 years), fewer individuals are covered during a given SIA and so it is less likely that incidence will fall below the eradication threshold immediately following the SIA. This effect may not occur if SIAs are un-coordinated and occur in different years in each country, which is the actual practice. The effect of SIA interval is likely also a function of how natural immunity develops in the intervals between SIAs. Finally, these differences apply at the optimal annual budget: for higher annual budgets such that very high coverage can be achieved even when SIAs are held every two years, we expect the time to eradication to be shorter for more frequent SIAs.
Order of eradication
Our baseline sequential strategy eradicates in the order Disease A-B-C, partly on the grounds that Disease A has the lowest basic reproductive number, R0 = 6, meaning that lower vaccine coverage would be required to eradicate it. Hence, Disease A might represent the "low hanging fruit" to policy makers. Here we explored five alternative orders to understand the implications of ordering for time to eradication: (1) A-C-B, (2) B-C-A, (3) B-A-C, (4) C-A-B, and (5) C-B-A.
We found that the minimum threshold budget remains $180 million for each of the five alternative orderings. The year of eradication for the three diseases under all six possible orderings under a budget of $180 million is shown in Table . From this table it is clear that Disease A is eradicated fastest under the four following orderings: A-B-C, A-C-B, B-A-C, and C-A-B; that is, any order attempting to eradicate Disease A either first or second in the series. Disease B is eradicated fastest under either of the two orders where funding is first allocated towards its eradication (B-A-C and B-C-A). Under these two orders, eradication of Disease B can occur in 2015. Similarly, Disease C is eradicated fastest under the orderings beginning with Disease C: C-A-B and C-B-A. Interestingly, eradication of all three diseases occurs fastest (by 2027) for the order B-C-A or B-A-C, i.e., for strategies where Disease B is eradicated first (not Disease A which has the lowest R0). Eradication of all three diseases takes longest (by 2186) for the order A-C-B and also takes very long (until 2177) for the order C-A-B, i.e.--these are strategies where Disease B is eradicated last.
Time to eradication for alternative orderings under an annual SIA budget of $180 million.
For budgets between $180 and $190 million, the time to eradication increases a great deal if Disease C is eliminated before Disease B (Figure ). Beyond $190 million, the time to eradication is roughly the same under all six orderings, such that eradication of all three diseases is possible by 2027 (Figure ). The large difference in time to eradication for budgets between $180 and $190 million for different orderings may be due to the high transmissibility of Disease B, which requires higher coverage to eradicate than Disease A or C. Similarly, any order that aims to eradicate Disease B last requires the greatest amount of total funding, even under their respective optimal annual SIA budgets.
Figure 6 Change in order of eradication. Year of eradication under different reallocation orders (A-B-C; A-C-B; B-C-A; B-A-C; C-A-B; and C-B-A) for the sequential strategy with SIA budgets between $180 and $190 million (a) and budgets greater than $190 million (more ...)
It is important to note that while both Disease B and C are highly transmissible, the basic reproductive number of Disease B is slightly higher than that of Disease C (15 and 14, respectively); Disease A has the lowest basic reproductive number, 6. Thereby, we propose that transmissibility accounts for the difference in time to eradication seen under each order: eradication occurs more quickly if the disease with highest transmissibility is eradicated first; eradication takes longest in the reverse scenario, where the disease with the lowest transmissibility is first to eradicated. We have not explored the impact of natural history and transmissibility assumptions systematically, hence other conclusions may emerge if these factors are explored more exhaustively.
Discontinuity in SIA Funding
We examined any possible consequence of an interruption in SIA funding for the success of the sequential strategy. We examined three different durations of interruption-3, 6, and 9 years-each beginning in 2016. During these periods, SIA coverage was 0% for all three diseases in all three countries.
As in previous simulations, there is a minimum threshold budget, below which eradication is not possible within the near future. As for the sequential strategy without budget interruption, the minimum threshold budget is $180 million and there is a sharp decline in the number of years required for eradication above this threshold. Beyond a budget of $200 million, increases in budget do not translate into significant differences in time to eradication. In these respects, the results under budget interruption are qualitatively similar to the results under no interruption. However, there are quantitative differences: the time to eradication generally increases under the budget interruption scenario. For budgets greater than the optimal budget of $200 million, interruptions of N years translate to delayed eradication by approximately N years: the baseline case with no interruption achieves eradication by 2024 under a $220 million budget; the case with a 3-year break requires 3 extra years for the same $220 million budget, the case with a 6-year break requires 6 more years for a $230 million budget, and the case with a 9-year break requires 9 more years for eradication for a $230 million budget (Figure ). In stark contrast, below the optimal budget of $200 million, the time to eradication increases enormously for relatively small interruptions (Figure ). For example, a 3-year break at a budget of $180 million delays the time to eradication of all three diseases from 2030 to 2080, and the setbacks are even greater for 6-year and 9-year breaks.
Figure 7 SIA funding hiatus. Effect of a break in SIA funding on year of eradication under different annual budget scenarios. Funding is entirely cut (SIA vaccination coverage is 0%) for 0, 3, 6 and 9 years in either all three countries (a) or Nigeria only (b). (more ...)
Short-term interruptions in annual SIA budgets can also have large implications for total required expenditures from 2010 to eradication. For instance, under a $180 million budget, a 3-year interruption would require a total of $11.70 billion over 68 years (2010-2078) to eradicate all three diseases. Similarly, a total of $7.38 billion is required to eradicate over 47 years under a 6-year interruption in SIA funding, and a total of $25.20 billion is required over 149 years under a 9-year interruption. Without any interruption in funding, it only takes $3.6 billion over 20 years (2010-2030) for the same annual budget of $180 million.
A similar pattern is seen when funding is interrupted in just one country. For instance, in the case of interruptions only in Nigeria when the usual annual budget is $230 million, a 3-year interruption again requires 3 extra years, a 6-year interruption again requires 6 more years, and a 9-year interruption again requires 9 more years (Figure ). This occurs because continued endemic infection in Nigeria can act to seed case imports in other countries, necessitating continued control efforts in those countries. For annual SIA budgets below $200 million, a small interruption translates into very long delays in the time to eradication of all three infections in all three countries, as was observed in Figure for the case of interruptions in all three countries. Hence, budget interruptions in SIA funding for any one country can have significant implications for time to global eradication, especially when budgets are close to the optimal or minimal thresholds. These results echo experiences with polio elimination in Nigeria, where there was a cessation of polio immunization between 2002 and 2003 due to a vaccine "scare". As a result, polio spread to other countries and polio eradication was detrimentally affected in the whole region [33
Increasing routine coverage over time
For the results reported thus far in this paper, we assumed routine vaccination began in 1960 for each of the three diseases and remained constant at 50% in all three countries. However, vaccine coverage has been increasing steadily over the past few decades and are high for most vaccines in most countries, with a few exceptions such as Nigeria, India and Afghanistan [23
]. To understand the impact of our baseline assumption of constant vaccine coverage, we repeated the analysis presented in Figure , except that in 2010 routine vaccine coverage begins to increase additively by 1% per year in all three countries until coverage reaches 95% in 2055, after which it remains at 95%.
When routine coverage increases over time, the minimum threshold budget under the sequential strategy is only $40 million instead of $170 million. As annualbudget increases beyond this threshold value, the time to eradication of each disease decreases steadily, and for an annual budget of $200 million all diseases are eradicated by 2035 (Figure ). By comparison, under the simultaneous strategy, the minimum threshold budget is much larger ($110 million) and the time to eradication decreases only slightly as the annual SIA budget is increased (Figure ). The time to eradication remains very different under simultaneous versus sequential strategies under the scenario of rising routine vaccine coverage: for an annual budget of $150, million, Disease A, B, and C are eradicated by 2038, 2041 and 2047 respectively under the sequential strategy, but these events occur much later, at 2059, 2074 and 2083 respectively, under the simultaneous strategy. Hence, even when routine coverage is increasing over time, budget reallocation strategies such as the sequential strategy evaluated here can significantly accelerate eradication of all three infectious diseases compared to a simultaneous strategy. We note that these results do not include the cost of ramping up routine coverage, although accounting for this would not change our qualitative conclusions since the resulting costs would be the same for the sequential and simultaneous strategy, and routine immunization would probably continue after eradication. We also note that exploring whether to allocate funding to improving routine immunization or implementing more SIAs in real-world immunization programs would require a country-specific and disease-specific model.
Increasing routine vaccination coverage. Year of eradication for a range of annual SIA budgets while routine coverage increases by 1% each year starting in 2010, until it reaches 95%, for the sequential strategy (a) and the simultaneous strategy (b).
Case Importation Rate
Here we examine the effects of case importation on the time to eradication for the baseline sequential strategy with an annual SIA budget of $180 million. Case importation has two competing effects on the time to eradication. On one hand, moderate case importation can prevent local elimination of a disease: an ongoing epidemic in one population can "re-seed" other populations where the disease might otherwise have disappeared due to low prevalence, and thus delay elimination in those populations. On the other hand, sufficiently large rates of case importation could also synchronize epidemics in connected populations. As a result, all populations would experience an epidemic trough at the same time, meaning that "re-seeding" effects from other populations would not prevent local elimination in a given population. Because all three populations experience epidemic troughs at the same time and re-seeding is not possible, the chances of eliminating the infection in all three populations are higher in this scenario [34
Because of these two competing effects, the impact of case importation on time to eradication in our model is complex. If the case importation rate between the three countries is higher than our baseline value (m > 0.0001/year), we observe that all three diseases are eradicated by 2025 (Figure ). Synchronization may contribute to this effect and is apparent for some of the diseases in Figure . However, for smaller values of the case importation rate (m < 0.0001/year) the time to eradication increases significantly and eradication tends to be delayed until 2100 or 2125. When m = 0, there is no case importation, and hence there can be no rescue effect due to re-seeding, making it easier for eradication to occur, hence, the time to eradication fall back to 2025 when m = 0 (Figure ).
Figure 9 Impact of case importation rate. Year of eradication of all three diseases versus case importation rate, under the sequential strategy. In subpanel (a), importation rates are varied uniformly across all countries. In subpanel (b) case importation rate (more ...)
The patterns are even more complex when case importation is varied for one country at a time. For example, when the India-Afghanistan and India-Nigeria case importation rates are changed while the Afghanistan-Nigeria case importation rates are held constant at baseline values, the year of eradication is highly variable across the range of case importation values explored (Figure ). This variability in year of eradication is even higher across a range of case importation rates into and out of Afghanistan (Figure ), although the same variability is not observed across a range of case importation rates into and out of Nigeria (Figure ).
To understand how small changes in case importation rate can lead to large changes in the time to eradication, we contrast the scenarios where case importation rates into and out of India occur at 0.05% per year versus 0.06% per year while other case import rates are held constant at baseline values (Figure ). In Figure we observed that eradication of all three diseases occurs by 2060 at a rate of 0.05% per year, but it does not occur until 2168 at a rate of 0.06% per year. In a plot of disease prevalence over time corresponding to these two scenarios (Figure ), we observe that the dynamics of Disease B are driving these contrasting outcomes. When case importation occurs at a rate of 0.05% per year, outbreaks of Disease B in Nigeria and Afghanistan are highly episodic and appear to be subject to local extinction (Figure ). Hence, eradication occurs by 2060. In comparison, when case importation occurs at 0.06% per year, outbreaks of Disease B become more regular and less episodic due to rescue effects, such that whenever prevalence is low in Nigeria, prevalence is often high in Afghanistan and vice versa (Figure ). As a result, all three diseases are not eradicated until 2168.
Figure 10 Time series for similar case importation rates. Prevalence of the three diseases over time in the sequential strategy, given a case importation rate of 0.05% per year into and out of India, for India (a), Nigeria (b) and Afghanistan (c), and given a case (more ...)