Resistance responds to insecticide use in the model in ways that can be explained mechanistically through the process of selection. This mechanistic explanation can help develop a more robust understanding of how resistance is expected to evolve in the field. It is helpful to start by explaining the evolution of resistance to a single insecticide, which is relatively straightforward, and move on to explaining the effect of two insecticides in sequences and mixtures.
When single insecticide use was investigated (Figs.
4,
5) higher values of parameters that increased the selective advantage of resistance all led to more rapid evolution of resistance, whereas higher values for cost led to slower evolution of resistance (Fig.
5b). Mechanisms explaining the evolution of resistance to single insecticide use, are summarised in Table
2. Referring to Fig.
2 can help explain these responses. Exposure and effectiveness act by decreasing the fitness of SS and SR more than RR, dominance of resistance acts by increasing fitness of SR and resistance-restoration acts by increasing fitness of RR and SR. Cost acts by decreasing the fitness of RR and SR among those mosquitoes not exposed.
Two insecticides
Responses to input parameters are different when two insecticides are used either in sequence or in mixtures as compared to when used alone. These differences can help us to understand the dynamics of resistance evolution. In each panel of Figs.
6,
7,
8,
9 and
10 the solid lines, indicating that an insecticide is being used at the same time as another in a mixture, are always shallower than the same colour dashed line indicating the insecticide is being used in isolation as part of a sequence. The shallower curves for the mixture can be explained by each insecticide killing individuals that are resistant to the other insecticide. Because resistant individuals are being killed, the selection pressure for that resistance is lower. Thus, each insecticide reduces the selection pressure for resistance to the other and could be said to ‘protect’ the other. This protection means that evolution of resistance to one insecticide is always slower when it is used in a mixture than when used alone. This protection has been termed multiple intra-generational killing [
10] because individuals have the potential to be killed by more than one insecticide.
Whilst this protection ensures that resistance to each insecticide evolves more slowly in a mixture, it does not guarantee that a mixture strategy is preferable to a sequence once the resistance to both insecticides is taken into account. In a mixture, selection pressure is applied by both insecticides at the same time, whereas in a sequence there is no selection for the insecticide not being used. Resistance can evolve more slowly for both insecticides in sequential use, despite being faster for each, because they occur one after the other. In Fig.
6a, it takes 60 generations for resistance to reach the threshold for each insecticide in the sequence, and 90 when they are applied together in the mixture, so the sequence is still favoured because 2 × 60 is greater than 90.
This advantage of sequence over mixtures can be removed by increasing the effectiveness of one of the insecticides. Increasing the effectiveness of one insecticide (Fig.
6b) decreases the total time for the sequence from 120 to 90 and increases that for the mixture from 90 to 115 thus changing the order and favouring the mixture. The mechanism for this change can be seen by comparing the shape of the curves in Fig.
6a, b. In the mixture, resistance to insecticide 1 (with the increased effectiveness) rises at a similar speed in both figures. However, the increased effectiveness of insecticide 1 causes resistance to insecticide 2 to increase more slowly initially in the mixture. Once insecticide 1 reaches resistance of around 50% at around 50 generations, the curve for insecticide 2 becomes steeper. Once the first insecticide reaches the resistance threshold of 50% it kills fewer individuals that are resistant to the second insecticide, it loses its protective effect and thus stops slowing the rise in resistance to the second insecticide. A similar effect is visible when the effectiveness of both insecticides are increased (Fig.
7c). The identical resistance curves for each insecticide in the mixture are shallower than at the lower effectiveness (Fig.
6a) because more individuals resistant to each insecticide are being killed by the other insecticide. In this case, the ‘protection’ given to both insecticides by the other declines at the same rate so there is no change in slope as there is in Fig.
6b. In both cases increasing the effectiveness of insecticides increases their protective effect when used in mixtures and favours mixtures over sequences.
Figure
6b also demonstrates that when two insecticides have a different effectiveness in a mixture, other parameters being equal, resistance will be expected to increase faster to the more effective insecticide (the red curve in this case). The more effective insecticide prompts both (a) higher selection pressure for its own resistance and (b) greater ‘protection’ reducing the rise in resistance to the other insecticide. The more effective insecticide has a faster evolution of resistance than the less effective one but this is not caused by the presence of the less effective one. Indeed, the presence of the less effective one is still slowing the evolution of resistance to the more effective one, as indicated by the red solid curve for use in the mixture being shallower than the red dashed curve for use alone as part of the sequence.
Difference between exposure and effectiveness
It seems initially counter-intuitive that whereas increasing effectiveness slows the development of resistance for mixtures, increasing exposure speeds it up. Increasing the exposure to both insecticides results in a decrease in time-to-resistance (from 90 generations in Fig.
6a to 40 generations in Fig.
6c) where increasing the effectiveness of both insecticides by the same amount results in a slight increase (from 90 in Fig.
6a to 115 in Fig.
6b). This contrasts with the identical effects that exposure and effectiveness have on a single insecticide (Fig.
4a, b) and on insecticides in sequence (the red dashed lines are the same in Fig.
6b, c). This difference, that exposure reduces times-to-resistance for a mixture and effectiveness increases it, was unexpected. The most likely explanation is that increasing exposure increases overall selection pressure and lowers time to resistance, whereas increasing effectiveness increases mutual protection of the insecticides in a mixture and lowers the speed at which resistance evolves.
Costs of resistance
Fitness costs of resistance have been widely reported (review in [
28]), but it has been suggested that few of these are directly relevant to control failure in the field [
29]. Because of this likely rarity of costs, they are not included in most of the analyses shown here (with the exception of Fig.
10). Costs of resistance favour mixtures in these analyses because the evolution of resistance is slowed more by cost in a mixture than in a sequence (compare the red solid lines for a mixture to the red dotted line for a sequence in Fig.
10c, d). It makes sense that the protective effect of the other insecticide in a mixture combines with the cost of resistance to reduce selection pressure more than in a sequence.
The resistance curves for two insecticides in a mixture that differ only in their cost of resistance, diverge more as the frequency of resistance increases (Fig.
10c). This makes intuitive sense, as the frequency of resistance genes increases the impact of fitness costs associated with those genes also increases. Thus, differences due to cost are less noticeable at low resistance frequencies and become more noticeable at higher resistance frequencies. Thus, if a lower resistance threshold was used cost would have less of an effect and if a higher resistance threshold was used cost would have more of an effect. If costs are set even higher than set here (e.g. above 0.2) they can maintain resistance frequencies below 1 or lead to their decline. However, such high costs are not expected to occur in operational settings [
29].
Declines in resistance frequencies are likely to occur when there are costs of resistance and the insecticide is not being used [
29]. Thus, the frequency of resistance for the first insecticide in a sequence, after it has been replaced, may decline to a level where the first insecticide becomes operationally useful again. Consequently, a repeated sequential strategy could offer benefits over mixtures not considered here. To quantify these benefits it would be necessary to use a more detailed metric than the time-to-resistance (the number of generations until resistance thresholds are first reached). Other measures for quantifying resistance over time can be imagined, for example the mean resistance or proportion of generations below a resistance threshold.
Consistency with previous work
The predicted responses of resistance presented here can be put into the context of previous work and recommendations. A recent review [
10] found that 8 out of 10 empirical studies and 11 out of 14 modelling studies favoured mixtures over sequences. In the 3 other modelling studies the relative performance of mixtures and sequences were dependent on the values of other inputs (as also shown here).
The review concluded that the advantage of mixtures is greatest when seven other conditions are satisfied [
10]: (i) rare starting resistance, (ii) independent loci (i.e. no cross resistance), (iii) high recombination, (iv) high susceptible mortality, (v) resistance is recessive, (vi) similar persistence of insecticides, and vii) some of population remains untreated. Their points (iv) and (vii) agree with our results that mixtures are favoured by high effectiveness and low exposure. Some of the other conditions, notably (i) the starting frequency of resistance and (v) the dominance of resistance did not alter whether mixtures or sequences were favoured in our model [
14].
Earlier modelling work has made sometimes contrasting predictions and these can be explained by the details of their structure and input choices. One model predicted that “the use of mixtures is always more effective in delaying the onset of resistance often by many orders of magnitude” [
12]. That model assumed that SS genotypes were always killed, RR always survived, and that a constant 10% of mosquitoes escaped the insecticide. This would be represented in the model presented here by a very high ‘insecticide effectiveness’ of 1, a ‘resistance restoration’ of 1 and an ‘exposure’ of 0.9. This result is consistent with our prediction that the very high effectiveness is likely to favour mixtures over sequences (e.g. see Fig.
7d and Table
4).
A subsequent model lead to the following more pessimistic conclusion about the potential of mixtures: “As a result of incomplete coverage and residue decay, the mortality of susceptible homozygotes is rarely consistently high enough for pesticide mixtures to be effective” [
30]. This “mortality of susceptible homozygotes” is equivalent to our insecticide effectiveness. Again this points to the importance of effectiveness and the question of whether the effectiveness levels needed to make mixtures better are attainable operationally. This earlier model [
30] (Fig.
3) is also consistent with that presented here in showing that decreasing exposure favours mixtures over sequences.
Caveats
There is evidence of the involvement of multiple genes in resistance mechanisms [
16,
31,
32]. The model presented here, in common with almost all previous ones [
33], assumes a single ‘resistance’ gene per insecticide. Target-site resistance has been shown to be mostly determined by single genes, and control failures in general have mostly been attributed to the effect of single major genes [
13,
16,
31]. One view is that control doses of insecticides in the field are likely to lead to the selection of single major genes whereas the lower doses used to select laboratory strains are more likely to lead to polygenic resistance [
16]. There is also evidence that low insecticide doses in agriculture have led to polygenic resistance [
32]. To represent polygenic resistance a different ‘quantitative genetic’ modelling approach would be required [
34,
35]. A recent review of 187 models of the evolution of resistance [
33] did not include any that represented quantitative multiple gene resistance. However there are a few published studies applying a quantitative genetic approach to assessing insecticide use strategies in the context of polygenic resistance [
34,
36,
37]. If it becomes clearer that polygenic resistance is an issue for vector control then there will be a need for appropriate quantitative genetic models and a comparison of those to single gene approaches like this one.
Poor implementation
One aspect of the results presented here can seem to contradict published advice and has been questioned when the results of this model have been presented. This model predicts that lower insecticide effectiveness or exposure will lead to slower spread of resistance in sole use and sequences (Figs.
4,
6, Table
2). This is consistent with expectations that lower effectiveness and exposure would lead to lower mortality of susceptibles and thus lower selection pressure for resistance [
30]. This can seem to contradict recommendations that poor implementation of insecticide interventions can promote the development of resistance. e.g. from [
5] “Certain pest control practices have consistently been shown to exacerbate the loss of susceptible pest populations and the development of resistance. These include: … - the use of application rates that are below or above those recommended on the label, - poor coverage of the area being treated…”.
Poor implementation would likely cause lower effectiveness of, and exposure to, insecticides, so how can this increase the potential for resistance when our model suggests it would be decreased? There are two main potential explanations for this apparent difference, (a) dominance and (b) polygenic resistance. The effect of dominance is that poor application can reduce the mortality of heterozygotes (SR) thus effectively increasing the dominance of the resistant allele and increasing its rate of evolution [
5,
38,
39] see Fig. 1 of [
14]. The results presented here are consistent with this, showing that increasing dominance of resistance leads to faster spread of resistance (Figs.
4c,
8). So, the effect of poor implementation of an insecticide intervention on resistance could be either positive or negative depending on the relative effects on dominance of the resistance gene, insecticide effectiveness and exposure. Similarly, the effects of poor application on the relative benefits of mixtures and sequences will depend on these trade-offs following the results presented here that higher effectiveness favours mixtures, higher exposure favours sequences and higher dominance favours neither (Table
4). More information on how the mortality of different genotypes (susceptible, resistant and heterozygous) is affected by poor application rates or insecticide degradation over time would help make this clearer. Secondly, if resistance is predominantly polygenic rather than monogenic poor implementation is likely to promote resistance by allowing the build up of many genes of small effect [
32]. Indeed modelling work from 15 years ago [
36,
37] proposes an alternation of low and high doses with the low doses limiting the evolution of monogenic resistance and the high doses limiting that of the polygenic resistance. More information on the relative importance of monogenic and polygenic resistance for vector control would help make this clearer.
Experience from agriculture where single interventions have been associated with a rapid development of resistance have led for calls for a more Integrated Vector Management composed of a series of partially effective tools [
40]. It is suggested that such an integrated approach is more sustainable and ‘evolution-proof’ [
40]. A similar combining of interventions has been advocated to cover mosquitoes exhibiting different behaviours [
41]. Mosquitoes with different behaviours e.g. feeding outdoors and/or on animals are likely to result in lower exposures to insecticides in nets or sprayed on walls. This lower exposure could favour mixtures over sequences as indicated here. In both cases this modelling approach could be used to investigate implications for the development of resistance.
Even if the assumption that single genes are responsible for control failures is correct there is still considerable other uncertainty around the development of insecticide resistance in operational field settings [
2]. This model is able to represent much of that uncertainty, yet in this paper the exploration has been restricted to a limited region of parameter space. Most of the inputs have been held constant at intermediate values while modifying single inputs in isolation. This is a deliberate tactic to develop understanding and communicate the key mechanisms. There is the potential that the system may behave differently in the field. However, the results presented here are supported by an earlier analysis where 10,000 unique scenarios were run, varying more inputs [
14]. This previous analysis also highlighted the relative importance of insecticide effectiveness and exposure in determining the relative performance of mixtures and sequences.
In addition, an on-line user interface to the model (Fig.
3) [
25] is provided, enabling readers with no coding experience to investigate model behaviour beyond that shown in this paper. The user-interface allows inputs to be set for two scenarios and results to be viewed side by side. For example, a higher insecticide effectivenesses could be set to see how this favours mixtures over sequences (Fig.
3) or exposure levels could be increased to see how this favours sequences.
There are other model behaviours that can be changed by modifying model inputs within the computer code. For example, here it is assumed that insecticide interaction in a mixture is multiplicative, e.g. if the probability of surviving exposure to insecticide A alone is 0.3 and of surviving insecticide B alone is 0.2, then the probability of surviving exposure to a mixture of A and B is 0.3 * 0.2 = 0.06. This is consistent with recent work on aphids [
42], although that did also indicate some synergistic effects. Modifying model inputs would allow such synergistic or antagonistic effects of mixtures to be incorporated, i.e. that insecticides used together perform better or worse than when used separately. This would allow the model to represent cross-resistance where a resistance gene provides resistance to more than one insecticide. Similarly, behavioural resistance is likely to offer resistance against more than one insecticide [
1], but until there are more data on whether behavioural resistance is genetic and heritable, representing it in this way is not advisable.
In addition to these uncertainties regarding resistance it is important to remember that this model examines solely the evolution of resistance and not the control of mosquitoes or malaria. Different insecticide use strategies, as well as affecting the evolution of resistance, will also influence both the numbers of vectors and their propensity to transmit the disease [
43]. There are also potential knock-on effects of resistance itself; reduced vector transmission of disease (e.g. by reduced lifespan of the vector) or increased transmission (e.g. by increasing the susceptibility of the vector to the disease) [
43]. This model can explore when a mixture strategy may be favoured over a sequence purely for the evolution of resistance. Other operational factors also need to be considered. On the positive side a mixture is likely to kill more vectors and may also limit their ability to transmit the disease. On the negative side a mixture may be more expensive. Other attributes of insecticides such as their repellency are also relevant. Insecticides with some repellency will reduce exposure to the insecticide and thus, as shown here, are likely to slow the evolution of resistance and favour mixtures over sequences. However, the reduced exposure resulting from repellency also reduces the population protection provided by the insecticide [
19].
This model would predict that the best strategy for limiting the development of resistance in the short term would be to use no insecticide. Of course, using no insecticide is likely to have serious consequences for mosquito and malaria abundance. The challenge is to develop good strategies for delivering mosquito and malaria control in the short term while sustaining their operational capacity in the longer term by restricting the development of resistance.