The simulation results in a collection of thirty plots (ten for each level of MDA coverage in patch 1, repeated for three relative pre-intervention disease intensities between the two patches). Only the last three MDA coverage levels (70%, 80%, and 90%) in patch 1 were focused on here, as the assembly effects in these scenarios are more pronounced for the demonstration purpose. The columns of sub-plots in Fig. 3 represent the MDA coverage in the patch 1 (column 1 = 70%, column 2 = 80%, column 3 = 90%); and the rows represent the relative pre-intervention disease intensities in patch 2 compared to patch 1 (top = higher, middle = identical, and bottom = lower).
Assembly effect between two patches with the same incidence
In the middle row of Fig. 3, both patches have an identical pre-intervention incidence, requiring a baseline MDA threshold of 78% coverage to achieve elimination (when the patches are not connected). In Fig. 3d, there is a negative assembly effect for patch 2 (the grey area above the baseline MDA threshold) because of the increasing connectedness with patch 1, which has a relatively low MDA coverage (70%). However, the increasing connectedness is beneficial to patch 1 (a positive assembly effect). Despite patch 1 having 70% MDA coverage, and not being able to achieve elimination on its own, the increasing connectedness with patch 2 (when patch 2 has more than enough MDA coverage for itself e.g., 94% MDA coverage), makes elimination still attainable in patch 1 (dark blue triangle at the upper right corner).
An opposite effect is seen when patch 1 has higher MDA coverage (80 and 90%) than is necessary to achieve elimination on its own (Fig. 3e, f). In this scenario, patch 2 experiences a positive assembly effect, indicated by the extension of the dark blue areas below the baseline MDA coverage threshold of 78%. However, patch 1 experiences a negative assembly effect; as connectedness increases, elimination in patch 1 is not predicted to occur for low MDA coverage in patch 2 (grey area in the lower right corners) because less-than-optimal coverage in patch 2 prevents patch 1 from achieving elimination at those levels of connectedness.
When the pre-intervention transmission intensities are the same in the two patches, the resulting assembly effects are purely due to differences in intervention coverage. To quantify the total assembly effect in patch 2 in each plot, the area between the “baseline” MDA threshold line (the red line in Fig. 2) and the diverging MDA threshold for increasing levels of connectedness (i.e. area “i” + “ii” in Fig. 2) was integrated. The total effect is assigned positive if it is beneficial to patch 2, and it is assigned negative otherwise.
Figure 4 displays how the total assembly effect in a particular patch is modulated by its connectedness to the other patch for different relative incidence ratios. The total assembly effect in patch 2 increases with increasing intervention coverage in patch 1 (black dots in Fig. 4). The switch from negative to positive total assembly effect occurs at the “baseline” coverage threshold for the particular disease intensity shared by both patches (78% coverage in this case).
The model’s prediction was compared against results from an MDA trial described in Parker et al. [4] where a village failed to achieve elimination presumably due to a cluster of non-participation in the MDA. This scenario was modelled as a set of two contiguous patches with 100% connectedness and with identical incidence. One patch received approximately 80% MDA coverage and the other 64%, with the latter representing the non-participation cluster (details in Additional file 1). The model accurately predicted that neither patch would achieve elimination (the red asterisk in Fig. 3e).
Assembly effect when two patches have different pre‐intervention incidences
Hotspot vs. non-hotspot
In the bottom row of Fig. 3, patch 2 has a 25% lower pre-intervention incidence compared to patch 1. This is analogous to a scenario where a low-incidence community (non-hotspot: patch 2) is connected to a high-incidence community (hotspot: patch 1). For this example, the following definition of malaria hotspots is used: “geographical areas within a wider area of transmission in which the transmission intensity is significantly higher than the average level in the surrounding area of that setting and are widely observed in malaria-endemic regions” [20]. When in isolation (no connectedness between patches), the MDA coverage threshold for elimination is very low at 5% for the non-hotspot, whereas it is 78% for the hotspot.
When MDA coverage in the hotspot is slightly below its required threshold for elimination (70% rather than the required 78%, Fig. 3g), both a negative assembly effect for the non-hotspot and a positive assembly effect for the hotspot are seen (areas similar to negative assembly effect for patch 2: “i” + “ii” and positive assembly effect for patch 1: “ii” + “iii” respectively in illustrative Fig. 2).
This suggests that when MDA coverage in the non-hotspot is high, and when the connectedness between hotspot and non-hotspot is high, elimination could be achieved in both patches despite the hotspot having less-than-optimal MDA coverage. For instance, when there is 60% connectedness, MDA coverage over 30% in the non-hotspot is predicted to result in elimination in both patches.
In panels H and I of Fig. 3, the hotspot has an adequate MDA coverage at 80 and 90% respectively. In those scenarios, the hotspot is predicted to always achieve elimination, regardless of the level of connectedness and the value of MDA coverage in the non-hotspot.
As seen in Fig. 4, non-hotspots (blue circles) will always experience a negative total assembly effect. However, the magnitude of the negative total assembly effect decreases with increasing coverage in the connected hotspot. The opposite is true for the positive total assembly effect gained by the hotspot (i.e., it increases with increasing coverage in the connected non-hotspot as seen in Additional file 1: Fig. S4). These trends suggest that the difference in transmission intensity is the main determinant of what types (positive or negative) of assembly effects can be observed.
In Fig. 3i, the required intervention threshold for the non-hotspot plateaus between 40 and 80% of connectedness. Further increase in the connectedness decreases the required intervention threshold slightly.
Assembly effect when intervention is ineffective for the connected patch
An intervention may not be appropriate if the disease intensity is too high e.g., MDA may not work in a high-transmission setting unless a very high MDA coverage is achieved. This scenario was simulated in the first row of Fig. 3 by setting patch 2 as a high-transmission setting. In isolation, patch 2 would require almost 100% of MDA coverage, while patch 1 would require more than 78% coverage of MDA for elimination to be attainable. As a consequence of being connected to patch 2, the prospects for elimination in patch 1 would be greatly diminished (large negative assembly effect for patch 1 represented by grey areas in Fig. 3b, c).