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Figure 1.

Illustration of the concept of optimal control for multiple outbreaks.

We assume that multiple outbreaks can occur, with the intervention only being feasible during the first outbreak. If the intervention is weak (or absent), the first outbreak will be large enough to deplete the number of susceptible people below a critical threshold level (the herd immunity level below which effective reproduction number <1), such that if the infection is re-introduced, its effective reproductive number would be too low to cause a second outbreak (black and cyan lines). If the intervention is very strong, it is possible that after the first outbreak, the number of susceptible people remaining is large enough to support a second (uncontrolled) outbreak upon re-introduction of the pathogen, leading to an overall number of people infected that might be the same as that reached during just one outbreak (red line). In both the “too much” and “too little” intervention scenarios, the number of susceptible people drops below the critical threshold level, which defines the level of herd immunity. The excess drop is termed ‘overshoot’. The optimal intervention is one that minimizes the overshoot by allowing the susceptible population to drop to the critical threshold level during the first outbreak, such that a second outbreak cannot occur (green line). The solid lines represent the susceptible people and the broken lines represent the infected people.

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Figure 2.

Flowchart illustrating the model.

White boxes represent adults, while black boxes represent children. Light grey arrows indicate movements from one stage to another (susceptible to infected to recovered). White arrows represent infection of adults and children by contact with infected adults; likewise, black arrows represent a similar process with infected children.

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Table 1.

Model variables.

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Table 1 Expand

Table 2.

Model parameters.

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Figure 3.

Time series of a simulated epidemic in a population in which there is no transmission between adults (black) and children (grey), in the absence (left) or presence (right) of an intervention.

Broken line: susceptible; solid line: infected. CAAW: Cumulative attack rate (adults, with whole population as the denominator); CACW: Cumulative attack rate (children, with whole population as the denominator). Infection was re-introduced into the population on day 300. The intervention (fAA = fAC = fCA = fCC = 1) started on day 100 (when the children’s epidemic is at its peak) and lasted until the first outbreak was over. βAC = βCA = 0; R0A = 1.25; R0C = 2. 50% adults; 50% children. All other parameters and initial conditions are listed in Tables 1 and 2.

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Figure 4.

Cumulative attack rates (CA) against the start day of intervention, in the absence of inter-group transmission (βAC = βCA = 0).

In the upper row, the denominator in the adults’ and children’s CA is the whole population (SW0 = 1); blue dotted line: CAAW; cyan broken line: CACW; red solid line: CAW = CAAW + CACW. In the lower row, the denominator in the adults’ and children’s CA is their respective proportion in the whole population (SA0 and SC0 respectively); black dotted line: CAAA; grey broken line: CACC; red solid line: CAW = S0A*CAAA + S0C*CACC. Proportion of adults in the population, from left to right: 10%, 30%, 50%, 70% and 90%. Intervention efficacy, fAA = fAC = fCA = fCC = 1. R0A = 1.25, R0C = 2; long intervention; interrupt all routes of transmission. All other parameters and initial conditions are listed in Tables 1 and 2. For the definition of the different cumulative attack rates, please refer to the Materials and Methods section, “Cumulative attack rates”, in the main text.

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Figure 5.

Cumulative attack rates (CA) against children’s population trigger (defined as the cumulative proportion of children infected), in the absence of inter-group transmission (βAC = βCA = 0).

Everything else as described in Figure 4 legend. Note that the x-axis does not go beyond a threshold level of 0.8 since even an unmitigated outbreak does not reach a higher attack rate among the children and therefore an intervention would not be triggered.

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Figure 6.

Cumulative attack rates against children’s population trigger, in the absence of inter-group transmission (βAC = βCA = 0).

Values of R0A are varied: R0A = 1.25 (left); 1.5 (middle); 1.75 (right). For all panels, R0C = 2; 50% adults; 50% children. Everything else as described in Figure 5 caption.

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Figure 7.

Time series of a simulated epidemic in a population in which transmission between adults (black) and children (gray) is 1% of intra-group transmission (βAC = 0.01 * βAA; βCA = 0.01 * βCC; left), or 10% (βAC = 0.1 * βAA; βCA = 0.1 * βCC; middle), or 100% (βAC = βAA; βCA = βCC; right), in the absence of intervention.

Broken line: susceptible; solid line: infected. In the right panel, the black and gray broken lines overlap each other exactly. R0A = 1.25; R0C = 2. 50% adults; 50% children. All other parameters and initial conditions are listed in Tables 1 and 2. The reason why in the right panel, the curves of susceptible adults and children overlapped, while the curves of infected adults and children did not, was that adults and children had different rates of recovery and therefore their average durations in the model compartment of the infected/infectious were different.

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Figure 8.

Cumulative attack rates against the start day of intervention in the presence of inter-group transmission.

Transmission between adults and children was 1% of intra-group transmission (βAC = 0.01 * βAA; βCA = 0.01 * βCC; upper row), or 10% (βAC = 0.1 * βAA; βCA = 0.1 * βCC; middle row), or 100% (βAC = βAA; βCA = βCC; lower row). The fraction of adults in the population was (from left to right) 10%, 30%, 50%, 70%, and 90%. Black dotted line: adults (CAAA); grey broken line: children (CACC); red solid line: whole population (CAW). Like the no-coupling scenario, the CAW curve shifted lower if there were more adults and higher if there were more children. In the lower panels, all three lines overlap with each other. R0A = 1.25, R0C = 2; 50% adults; 50% children; long intervention; interrupt all routes of transmission; intervention efficacy, fAA = fAC = fCA = fCC = 1. All other parameters and initial conditions are listed in Tables 1 and 2.

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Figure 9.

Cumulative attack rates against children’s population trigger in the presence of inter-group transmission.

Everything else as described in Figure 8 caption. Note that again the x-axis does not go beyond a threshold level of 0.8 since even an unmitigated outbreak (even in the absence of coupling) did not reach a higher attack rate among the children and therefore intervention would not be triggered.

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Figure 10.

Cumulative attack rate of the whole population (CAW) under long intervention with different levels of efficacy against different time trigger (the start day of intervention, left column) and population trigger (cumulative proportion of children infected, right column).

βAC = 0.01 * βAA; βCA = 0.01 * βCC (upper row); βAC = 0.1 * βAA; βCA = 0.1 * βCC (lower row). R0A = 1.25, R0C = 2; 50% adults; 50% children (similar patterns were observed in populations with different proportions of adults and children, not shown); long intervention; interrupt all routes of transmission. All other parameters and initial conditions are listed in Tables 1 and 2. The data ranged from intervention efficacy of 0.02 to 1 with intervals of 0.02. This explains why there was no colour in the contour plots from intervention efficacy 0 to 0.02. Note that beyond a threshold level of 0.8, the intervention was not triggered, as CAAA never reached 0.8 even in the absence of any intervention.

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Figure 11.

Cumulative attack rate of the whole population (CAW) under 28-days intervention with different levels of efficacy against different time triggers (start days of intervention, left) and population triggers (cumulative proportion of children infected, right).

The legend colour panel displays CAW. βAC = 0.01 * βAA; βCA = 0.01 * βCC (upper row); βAC = 0.1 * βAA; βCA = 0.1 * βCC (lower row). All other details are the same as Figure 10. Compared to Figure 10, it is notable that the J-shaped contours of Figure 10 are replaced by vertical minima in Figure 11. In other words, weak interventions that started early would not lead to the optimal outcome if its duration was short.

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Figure 12.

Cumulative attack rate of the whole population (CAW) under school closure with short intervention (28 days) with different levels of efficacy against time trigger (different start day of intervention, left) and population trigger (cumulative proportion of children infected, right).

The legend colour panel displays CAW levels. All other parameters follow Figure 11.

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