Controlling malaria: competition, seasonality and ‘slingshotting’ transgenic mosquitoes into natural populations

Abstract

Forty years after the World Health Organization abandoned its eradication campaign, malaria remains a public health problem of the first magnitude with worldwide infection rates on the order of 300 million souls. The present paper reviews potential control strategies from the viewpoint of mathematical epidemiology. Following MacDonald and others, we argue in Section 1 that the use of imagicides, i.e., killing, or at least repelling, adult mosquitoes, is inherently the most effective way of combating the pandemic. In Section 2, we model competition between wild-type (WT) and plasmodium-resistant, genetically modified (GM) mosquitoes. Under the assumptions of inherent cost and prevalence-dependant benefit to transgenics, GM introduction can never eradicate malaria save by stochastic extinction of WTs. Moreover, alternative interventions that reduce prevalence have the undesirable consequence of reducing the likelihood of successful GM introduction. Section 3 considers the possibility of using seasonal fluctuations in mosquito abundance and disease prevalence to ‘slingshot’ GM mosquitoes into natural populations. By introducing GM mosquitoes when natural populations are about to expand, one can ‘piggyback’ on the yearly cycle. Importantly, this effect is only significant when transgene cost is small, in which case the non-trivial equilibrium is a focus (damped oscillations), and piggybacking is amplified by the system's inherent tendency to oscillate. By way of contrast, when transgene cost is large, the equilibrium is a node and no such amplification is obtained.

Keywords:

• bifurcations
• epidemiology
• mathematical models
• malaria
• transgenic mosquitoes

Where the Italian boot meets the island of Sicily lies the region called Calabria, the character of which, Douglas 24 observed, is incomprehensibly absent with reference to malaria. ‘Malaria’, he wrote, ‘is the key to a correct understanding of the landscape; it explains the inhabitants, their mode of life, their habits, their history’.

That was in 1915. Thirty years later, the American army, in its drive on Rome, responded to endemic malaria with marsh draining and DDT (Figure 1), then a secret weapon. Eradication efforts continued after the war, and in 1970, the World Health Organization (WHO) declared the country malaria-free 75. The Italian success story was replicated in other parts of southern Europe, to a lesser extent in South America and southern Asia and hardly at all in Africa 9 21 25 33 64. Today, a century after Douglas wrote his travel book, worldwide infection rates are on the order of 300 million souls of whom at least a million die every year 78. Unsurprisingly, the social and economic costs are enormous 37 58.

Figure 1. Home disinfection by American medical personnel, Pontine marshes, 1944. Painting by Sgt. Fred Toelle. Image used with permission from Central University Libraries, Southern Methodist University. Available in colour online.

Especially in Africa, a major-cause of resurgent malaria has been societal dysfunction: war, poverty and the collapse of health care systems following abandonment of the WHO's Eradicate Malaria Campaign 9 21 25 33 64 78 18 in the late 1960s. Other factors have also contributed. These include changing patterns of land use, expanding human populations 69, the evolution of Plasmodium resistance to anti-malarial drugs, 9 34 46 89 99 and the evolution of mosquito resistance and avoidance behaviour to insecticides 70 73. Changing public health policies 18 21 33 64 78 – a shift in emphasis from vector control to treatment and the effective banning of DDT – have also taken their toll 8 9 25 69 73 74 99.

These developments have stimulated a search for new solutions: insecticide-treated bed nets (ITNs) 19 57 94 95, entomopathogenic fungi 12 85, new anti-malarials (or the use of old ones in sequence or combination 31 51 67), vaccines 35 36 48 65 and genetically modified (GM) mosquitoes 16 41,61–63.

With regard to all of these approaches, there are population level considerations 20 86 in addition to the medical and molecular ones. For example, in the case of GM mosquitoes, De Roode and Read 20 suggest that ‘the technological challenges of manipulating genes in vivo are trivial compared to the ecological challenges …’. Likewise, Scott et al. 86 stress the ‘need to study gene flow, … mosquito population size and structure, mechanisms of population regulation, …,’ etc., while Hancock and Godfray 29 emphasize ‘the interplay of mortality … at different stages of the lifecycle …’.

In the present paper, we explore the interacting consequences of seasonality and fitness to GM mosquito introduction. With regard to seasonality, it is well known 7 10 68 that over much of the malarious tropics, mosquito densities fluctuate in response to alternating wet and dry seasons (see also, 102). With regard to transgene fitness, the overall picture 15 16 41 62 86 96 is one of cost, i.e., of reductions in reproduction or survival. From this derives much of the current interest 30 42 71 98 101 in ‘driving mechanisms’ to force transgenes into natural populations.

1. Eradication in the abstract

Before turning our attention to GM mosquito introduction, we review alternative control strategies from the vantage of parametric sensitivity. Generally speaking – but see 17 – a disease's ability to persist is captured by its basic rate of reproduction, R ₀ 3 23 50. Essentially, R ₀ is the number of secondary infections that result from a single contagion. With R ₀>1, there is a persistent, endemic level of infection, whereas with R ₀<1, the disease dies out. In such cases, R ₀=1 corresponds to a transcritical bifurcation, whereas the ‘no-disease’ and endemic equilibria collide and exchange stability 50 82. The practical consequence 2–5,22 is that one need not immunize all potential hosts, or, in the case of vector-transmitted infections, kill all of the vectors. Sufficient unto the task is reducing R ₀ to a value that is less than 1. In the case of vaccination programmes, this condition is called herd immunity. For application to malaria, see 7; for a more general discussion including transient immunity and the evolution of vaccine resistance, see 84.

As discussed, for example, in 66, R ₀ can be estimated from historical notifications, although what is obtained is not necessarily suggestive of a constant fuzzed out by the measurement error. R ₀ can also be calculated from mathematical models, in which case the formula depends on the natural history of the disease – more accurately, on the equations used to model its dynamics. For example, in Ross’ original formulation 76 77,

Here, the symbols and their units are as given in Table 1. In particular, N and M are the numbers (assumed to be constant) of humans and female mosquitoes, respectively, and A is the bite rate. The terms, (ABM/r) and (ACN/μ), are thus production rates of new infections (human and mosquito) discounted, respectively, by the human recovery rate, r, and the adult mosquito death rate, μ.

Table 1. Determinants of R ₀ for Malaria.

Symbol	Quantity	Units
N	Humans	Humans
M	♀ Mosquitoes	Mosquitoes
A	Mosquito bite rate	Bites/(mosquito-y)
B	New human infections	Humans/(humans-bites)
C	New mosquito infections	Mosquitoes/(humans-bites)
μ	Mosquito death rate	y^−1
r	Human recovery rate	y^−1
τ	Latency period	y

Likewise, with the addition 59 60 of latency (infected mosquitoes not yet infectious) 7,

where the exponential, e^−μτ, is the fraction of newly infected mosquitoes that survive to become infectious – i.e., τ, is the time (on the order of a week) required for the parasite to migrate from the mosquito's gut to its salivary glands – for extensions, see 29 46 92.

The parameters in Table 1 enter Equations (1) in different ways. Hence, sensitivity of R ₀ varies from one parameter to the next. Following 29 60, and others, we note the following:

1.	The mosquito death rate, μ, enters twice, first in proportion to (1/μ), and second, as e−μτ, which is the probability that an infected mosquito survives long enough to infect a human host. It follows that reducing adult mosquito longevity provides the surest approach to reducing R 0. This explains the historical success 8 9 64 of residual spraying programmes in which houses, stables and other structures were dusted with DDT once or twice a year. For the same reason, entomopathogenic fungi 12 85 and ITNs, which kill mosquitoes in addition to protecting sleeping humans, are attractive control strategies; but see 95 for discussion of the downside of ITNs.
2.	The bite rate, A, also enters twice – in proportion to A 2 – reflecting the fact that completion of the parasite's life cycle requires two bites. Thus, a 50% reduction in the number of bites divides R 0 by a factor of 4, etc. Even without killing mosquitoes, control strategies (screens, pesticides, netting) that keep mosquitoes out of peoples’ houses, or at least out of their beds, decrease the bite rate; likewise, covering up in the evening and the use of insect repellents.
3.	The remaining parameters enter once. Vaccines effectively reduce N. Anti-malarials, a principal focus of the current WHO practice [9,99 , increase the recovery rate, r. Interrupting the parasite's life cycle via the introduction of GM mosquitoes reduces the number of mosquitoes, M, in effect; oiling, spraying and draining marshes, introducing larval predators 11 83, etc., does so in fact. Reducing local human population density (relocation programmes) reduces N and, to the extent that breeding sites are eliminated, M.

In sum, control strategies that focus on the death rate of adult mosquitoes and the bite rate are inherently more efficient than ‘non-A/μ’ strategies 29 60 92. This is not to deny that interrupting the plasmodium's life cycles at any point would end the scourge. For example, Koenraadt et al. 47 attribute the general dearth of malaria in the Kenyan highlands to retarded larval development and survival. Likewise, Baird 9 observes that ‘attacking the specific breeding sites of known vectors’ was a major contributor to the successful campaign against malaria during construction of the Panama Canal. Moreover, as Smith and McKenzie 92 remind us, ‘Single control measures often affect more than one aspect of malaria transmission, as does integrated malaria control. The important point,’ they continue, ‘long understood but often forgotten, is that the benefits are multiplicative’ [emphasis added]. At the same time, these authors point out that, with the incorporation of mosquito population dynamics, ‘increased mosquito mortality has an effect even larger than was proposed by Macdonald in the 1950s’.

2. Competition between GM and wild-type mosquitoes

In this section, we model competition between wild-type (WT) and transgenic mosquitoes. To do this, we must first of all introduce some form of density dependence. Following 29, we suppose that mosquito populations are regulated by density-dependent larval mortality and further that WT and GM larvae are equally affected by crowding. We proceed as follows:

1.	The real world being intermediate between assortative mating and panmixia 86, we replace genes 54 56 87 with strains 53 for simplicity.
2.	Likewise, we use a modified version of Ross’ original equations, even though they omit important biological particulars – parasite development time, details of the gonotrophic cycle, acquired host immunity, super-infection, etc. 7 29 32 93
3.	We use adult mosquito density as a proxy variable (see discussion) for the density of larvae and model mosquito population dynamics as

Here m is the maximum per capita birth rate, and F(M, t), captures the (possibly time-dependent) effects of crowding on larval survival.1 In a constant environment, the equilibrium abundance, M*, of mosquitoes is the solution, M*, to

As argued in the discussion, we believe that the most biologically plausible form of F(M, t)is a reverse sigmoid (Figure 2), essentially because larval resource consumption capacities are finite. In the following, we use

We also allow K(t) to vary as

Here is the magnitude of the seasonal effect, and ω=1 y⁻¹ is the frequency. In constant environments, , and K(t)=K ₀.

Figure 2. Larval survival, F(M, t), according to Equation (4a). M* is the equilibrium density of mosquitoes in a constant environment.

Putting all this together yields

As in Ross’ original equations, X is the number of infected hosts, and Y is the number of infected mosquitoes. The new variables, W and G, denote WTs and transgenics, as do the subscripts, w and g. Thus, F(W, G, t), the larval survival function, is given by Equations (4) with W+G replacing M. Equations (5) generalize the model (no transgenics, constant numbers of mosquitoes) studied by Tumüne et al. 97; for alternative approaches, see 55 79.

2.1. Costs and benefits

As noted above, most studies of transgenic fitness report a fitness cost to GM mosquitoes. In this regard, a recent paper by Marrelli et al. 61 reporting a fitness advantage for hemizygous GM mosquitoes feeding on infected hosts is of interest. Citing Hurd and Cowarker 2 38, Marrelli et al. suggest that transgene advantage reflects the fact that ‘even [a] Plasmodium burden of less than five oocysts can significantly reduce the fecundity of anophelines in experimental or natural systems’.2, 3 Under this assumption, the benefit enjoyed by transgenic mosquitoes is prevalence-dependent,4 in which case, transgenes can never completely eliminate malaria, absent stochastic extinction. This is because as prevalence declines, so too does transgene advantage, which becomes a disadvantage (inherent transgene cost) before the disease is eradicated.5 It follows either that a drive mechanism indifferent to transgene fitness, for example, one involving homing endonucleases 101, will be required or that genetic engineers will have to come up with a transgene that is inherently superior. However, as discussed in Section 3, it may still be possible to use seasonal variations in prevalence to ‘slingshot’ inferior genotypes into natural populations by appropriate timing of their release.

In the analysis that follows, we assume:

1.	GM mosquitoes enjoy a fitness advantage – a consequence of reduced parasite load – relative to WTs feeding on infected hosts.
2.	Absent malaria, GM mosquitoes manifest a fitness deficit relative to WTs, i.e., .

Corresponding to these assumptions, we substitute

and

in Equation (5), where c _w and c _g are the respective costs.6 From a biological perspective, c _w and c _g cannot exceed 1, as this results in negative reproduction – always in the case of transgenics, when (X/N) is sufficiently large for WTs. Note also that negative values of these constants have the effect of converting ‘costs’ into benefits, i.e., the respective fecundities would be increased.

2.2. Dimensionless equations

Let

Substituting into Equations (5) and (6) yields Equation (8)from which all units save time have been eliminated:

Here, f(·) is the dimensionless equivalent of F(·), while

2.3. Reduced system

Before studying Equation (8), we consider the reduced system that is obtained when the number of infected hosts, x, is set to a fixed value. In this case, the derivatives, d w/d t and d g/d t, are independent of y. This allows us to work in the w−g−xspace, with x playing the role of a parameter.7 Then, the transgenic strain prevails if

and the WT if

This is shown in Figure 3, wherein we plot zero-growth isoplanes, w˙=0 and ġ=0, for the two strains. Note the following:

1.	For fixed prevalence, the corresponding isoclines parallel the −45° line, a consequence of assuming that the two strains are equally susceptible to crowding. Generically, i.e., provided , there are three equilibria: the origin, which is unstable, and two boundary equilibria, (w, 0) and (g, 0), of which one is stable and the other, not. At equilibrium, the mosquito population is consequently composed entirely of WTs or transgenics.
2.	The transgene isoplane, ġ=0, is independent of prevalence, whereas the WT isoplane, w˙=0, slopes downwards with increasing x. If , the case shown, the isoplanes intersect at xcrit. Here, there is a line of neutrally stable, and structurally unstable, equilibria connecting the two boundary points. For x>x crit, ġ=0 lies above w˙=0, and the transgenic strain outcompetes the wild type. For x<x crit, ġ=0 lies below w˙=0, and the WT prevails.
3.	If (date not shown), ġ=0 lies below w˙=0 for all x=0 [0,1]. In this case, the transgene can never invade.

Figure 3. Zero growth isoplanes for the two strains with constant prevalence. Competition along the light black dotted line is indeterminate, there being an infinite number of neutrally stable fixed points. Solid circles (nodes) are attractors; half-filled circles, saddles.

In sum, given

the mosquito population transitions from (w, 0) to (g, 0) at x=x _crit.8

2.4. Full system

We now return to Equation (8). The results that follow were obtained by numerical integration (Adams or Gear's method) of the differential equations and by continuation of equilibria and their bifurcations 49 using secant prediction and chord-Newton correction with adaptive step control. As described in 44 45, the required Jacobian matrices were computed using concurrent integration of the equations of first variation. Pitchfork (PF), saddle-node (SN) and transcritical bifurcations were identified by the application of standard 42 49 criteria.

Recall that the reduced system admits to three structurally stable equilibria: the origin, (0, 0), the boundary equilibria, (w, 0) and (0, g), as well as the aforementioned line of structurally unstable equilibria at x=x _crit. In the full system, the last is replaced by a unique (for fixed parameters) structurally stable, interior equilibrium (w*, g*). The origin and stabilization of this equilibrium are detailed in Figure 4. Here we display curves of equilibria and the bifurcations that are obtained when the transgene cost, c _g, is varied from −1 (transgene advantage) to +1 (transgene disadvantage), while fixing the WT cost, c _w=1. Regarding this figure, we note the following:

1.

Top panel. A pair of unstable boundary equilibria, w=0, emerge from the origin, (w, g)=(0, 0), via a PF bifurcation that occurs when the transgene is inherently disadvantageous −c g>0. One of these equilibria corresponds to positive numbers of transgenics; the other, to negative numbers.9 Both acquire stability via transcritical bifurcations (TC1 and TC3) at c g=0. We conclude that the exclusion of WTs by transgenics requires c g<0, which is to say that the transgene is inherently advantageous.

2.

Bottom panel. A pair of interior equilibria originates via a SN bifurcation, which, like PF bifurcation discussed above, occurs when the transgene is inherently disadvantageous −c g>0. At the bifurcation point, both interior equilibria correspond to negative numbers of transgenics. Both are also unstable, one with a single degree of instability, the other with two. With decreasing c g, the former becomes positive and acquires stability at a value of c g>0 via a transcritical bifurcation (TC2) with the boundary equilibrium, g=0. At this point, the boundary equilibrium loses stability, and transgene exclusion gives way to coexistence. Subsequently, i.e., at c g=0, stability of the interior equilibrium is lost via a second transcritical bifurcation (TC1), at which point coexistence gives way to WT exclusion.10

Figure 4. Curves of equilibria and bifurcations attendant on varying transgene cost, c _g. TC1 – TC3 are transcritical bifurcations; PF and SN, pitchfork and saddle-node bifurcations. Top. Origin and stabilization of the boundary equilibria (w=0). Bottom. Origin and destabilization of the interior equilibrium. ‘Unstable 1’ and ‘Unstable 2’ refer to the number of unstable directions in phase space – equivalently, the number of eigenvalues in the right half of the complex plane. Parameters (Equation 8) as follows: a=20; b=1.0; q=2.0; r=4.0; μ=50; s=1.0; m=100, ϵ _K =0. Available in colour online.

In sum, as transgene cost declines, populations transition from all WTs to a mixture of WTs and transgenics and finally to one composed entirely of transgenics.

2.5. WT-transgene coexistence

By continuing 42 49 the transcritical bifurcations, TC1 and TC2, against WT cost, c _w, we can delineate the region, Γ, in the parameter plane corresponding to coexistence of WT and GM mosquitoes. This is shown in Figure 5. Note that the curve of TC1 bifurcations is simply to y-axis and that both curves emerge from the origin, , which is a co-dimension 2 bifurcation corresponding to the identity of the interior and boundary equilibria. Elsewhere, we will document the dependence of Γ on the remaining parameters. The bottom-line result, however, is worth stating here: Any parameter change reducing prevalence, moves the TC2 curve to the left, thereby reducing the maximum transgene cost, c _g, consistent with successful GM introduction.

Figure 5. The region (stippled) of WT–transgene coexistence in the c _w−c _g parameter plane is bounded by curves of transcritical bifurcations, TC1 and TC2 – see Figure 4. For points to the right of the TC2 curve, the boundary equilibrium (g, w)=(0, w*) is stable, and the transgene is outcompeted by the WT. For points to the left of the TC1 curve, which is also the y-axis, the transgene prevails. Parameter values as in Figure 4.

2.6. Stability of the interior equilibrium

In part, the response to transgenic introduction (below) depends on equilibrium stability. In Figure 6, we show how the eigenvalues of the interior equilibrium change in response to varying transgene cost. Only the values of the two small magnitude eigenvalues are shown. The remaining eigenvalues are real, negative and an order of magnitude greater in absolute value. Hence, to first approximation, the dynamics are played out in two dimensions.

Figure 6. Change of small magnitude eigenvalues of the interior equilibrium in response to varying transgene cost, c _g. WT cost c _w=1.0. Other parameters as in Figure 4.

Regarding Figure 6, we note the following:

1.	For c g<0 (points to the left of TC1 in Figure 4), the two small amplitude eigenvalues are both real, one being positive, the other, negative (not shown). In this case, the interior equilibrium, which corresponds to negative values of w, is a saddle.
2.	With increasing c g, the small amplitude eigenvalues approach the origin of the complex plane. At c g=0, there is a double zero. This corresponds to TC1, at which point w*=0.
3.	With further increases in c g, now positive, the small amplitude eigenvalues separate, moving away from the real axis, Re(λ), and to the left. Thus, neutral stability gives way to weakly damped oscillations of a large period.
4.	With still further increases in c g, the eigenvalues continue to move left. Initially, the imaginary parts become larger in magnitude, but this trend is subsequently reversed. Eventually, there is a second double eigenvalue – this one to the left of the imaginary axis, Im (λ). At this point, the eigenvalues become real, and the approach to equilibrium, nodal.
5.	Further increases in c g cause the eigenvalues, now real, to separate, with one approaching zero, and the other becoming more negative. Eventually, the larger eigenvalue equals zero, and we are at TC2, at which point g*=0.
6.	Beyond TC2, i.e., for still larger values of c g, the interior equilibrium, now corresponding to negative numbers of transgenics, is a saddle (not shown).

2.7. Transgenic introduction

The foregoing suggests the following consequences of varying transgene cost to transgene introduction.

1.	For small values of c g, GM mosquitoes should dominate, but with the approach to equilibrium being by way of large amplitude, slowly decaying oscillations.
2.	As c g increases, oscillations should become less pronounced and eventually disappear, even as the equilibrial abundance of transgenics declines.
3.	With c g sufficiently large, introductions should fail.

These expectations are confirmed in Figure 7 for the case of a constant environment. With decreasing transgene cost, c _g, the long-term transgene abundance increases, whereas WT abundance declines. This, of course, is to be expected. What is unexpected, absent the preceding analysis, is the emergence of low-frequency, large-amplitude oscillations as c _g goes to zero.

Figure 7. Competition between WT and transgenic mosquitoes in a constant environment. Transgenic cost, c _g, decreases from left to right and top to bottom. (a) c _g=0.5; (b) c _g=0.25; (c) c _g=0.10; (d) c _g=0.05; (e) c _g=0.01; (f) c _g = −0.001. In all cases, c _g=1. Other parameters as in Figure 4. Note the increasing amplitude and period of oscillations as c _g approaches 0. For c _g<0, oscillations are abolished, and displacement of WTs (now inherently advantageous) by transgenics is monotonic. Available in colour online.

In seasonal environments (not shown), these fluctuations are superposed on the yearly cycle. For the parameter values studied, there was no suggestion of a more complex behaviour 82, though this remains a possibility, especially were the model to be extended, for example, by the inclusion of larval densities as an additional state variable.

3. ‘Slingshotting’ transgenics into a population of WT mosquitoes

The production of transgenic mosquitoes being both time-consuming and costly 72, and laboratory amplification, a frequent source of inbreeding depression 15 16 62 86 96, the question of applied interest is how to maximize the salutary effect of small, infrequent introductions. In seasonal environments, a ‘slingshotting’ technique may be available. Here, one uses the seasonal cycle (Figure 8) much as a space ship uses the gravitational field of a large celestial body to accelerate its velocity. The technique is most effective when seasonality is pronounced and intrinsic transgene cost, small, in which case, post-introduction malaria resurgence can be delayed for significant periods. Figure 9 gives some examples. Here we plot disease prevalence following single small introductions of transgenics at times corresponding to the four phase values indicated in Figure 8. As expected, WT abundance and post-introduction prevalence decline with decreasing transgene cost. Beyond this is the timing of GM introduction relative to the calendar year, which can result in what we call a phase effect. In this regard, we put time (measured in years) on the unit circle and define the phase, φ, of introduction as

where t ₀ is the time of introduction.11 Figure 8 relates φ to the yearly cycle of vectors and disease, i.e., mosquitoes minimal and prevalence intermediate at φ=0.25, mosquitoes intermediate and prevalence close to maximal at φ=0.5, etc. In terms of Equation (4b), φ=0.25 and φ=0.75, respectively, correspond to maximum and minimum larval survival.

Figure 8. Annual cycle of mosquito abundance, w, and malaria prevalence, x, as modelled by Equation 8. Phase values, ϕ, are defined relative to the calendar year. Here, ϵ _K =0.6, and c _g=.01. Other parameters as in Figure 7.

Figure 9. Simulated response to introduction (arrow) of transgenic mosquitoes in seasonal environments. The phase, φ, refers to timing of the introduction relative to the yearly cycle of malaria and mosquitoes as shown in Figure 8. (a–c) Seasonality fixed at ϵ _K =0.6. Phase effect, i.e., time until the first outbreak following GM introduction, increases with decreasing transgene cost, c _g. (d–f) Transgene disadvantage fixed at c _g=0.01. Phase effect increases with increasing seasonality. Remaining parameters as in Figure 7. Available in colour online.

Returning to Figure 9, we note that panels a–c explore the effect of decreasing transgene cost, c _g, for fixed seasonality, . For c _g=0.05, the timing of GM introduction is of little consequence. For smaller costs, choosing the correct phase can delay post-introduction resurgence by as much as 8–9 years. Likewise for fixed transgene cost (d–f), phase effects become more pronounced with increasing seasonality.

In all cases, the best results were obtained with φ=0.25, i.e., when mosquito abundances are approximately minimal. Thereafter, mosquito populations expand, and introduced GMs can ‘piggyback’ on the natural increase. Conversely, the worst (of the four values shown in Figure 9) phase is φ=0.75, which is when mosquito densities are close to maximal and about to decline. Our recommendation for real-world seasonal environments is thus to introduce GM mosquitoes when mosquito populations are about to increase, e.g., after the onset of the rainy season. Lest this appear intuitively obvious, we note that the phase effect is only important when transgene cost is small. In this case, the non-trivial equilibrium is a focus (damped oscillations), and piggy-backing is amplified by the system's inherent tendency to oscillate. By way of contrast, when transgene cost is large, the equilibrium is a node (Figure 6), and no such amplification is obtained.

4. Discussion

4.1. Competition and larval survival

In the preceding analysis, we used the number of adult, female mosquitoes as a proxy variable for larval density. We also introduced a larval survival function which is incompatible with the more common assumption 7 43 52 80 81 of a constant, density-independent rate of anopheline emergence from larval habitats. In short, there are two issues.

4.1.1. Adult mosquito density as a proxy variable for numbers of larvae

The number of larvae, and hence the intensity of larval competition for resources, is a function of the numbers of mosquitoes in the past. One approach to this situation involves the introduction of time delays. Another is to make the number of larvae an independent variable. For example, one might write

where α is the emergence rate, and the remaining parameters are as in Equation (5). Making larval survival dependent on M thus corresponds to the fast time scale assumption, . Then, if L* (M) is the equilibrial number of larvae for fixed M, we have to 0th order,

In fact, larval development time and adult longevity are both on the order of 1–2 weeks. So the separation of time scales is a poor assumption, especially in the case of seasonal environments, this may be important. Consequently, we believe that the inclusion of larval density as an additional state variable – see, for example, 32 – would be a useful extension of Equations (5) and (8).

4.1.2. Shape of the Survival Function

An alternative to Equation (4b), and one compatible with the more common assumption of constant adult emergence, would be to assume . Certainly, this is cleaner than Equation (4a).12 But is it more reasonable biologically? Imagine a body of water in which mosquitoes lay eggs. After the eggs hatch, larvae begin to feed and develop. Both their rate of development and their survival depend on the availability of resources – algae, bacteria, fungi, etc. We submit that, except in the most unproductive situations, the abundance of resources will exceed the capacity of a small number of larvae to consume them. So, adding a few more larvae should have minimal effects on growth and survival. However, if we continue to add larvae, there will come a point at which there is an effect. Larval growth rates will slow, and some individuals will die between molts. Ephemeral habitats may dry up before the larvae complete their development or may so deteriorate that many, perhaps all, die. At this point, survival rates will drop significantly. This gives the initial ‘dip’ in the reverse sigmoid. If one continues to add larvae, survival approaches zero at a decreasing rate. This gives the flattening out of the reverse sigmoid.

By way of contrast, assumes that it is the first larval additions that have the most pronounced effect.

There is also the question as to whether or not it is reasonable to assume a maximum larval density beyond which larval survival is zero. Because larval development times increase with decreasing abundance of resources, because small bodies of water can dry up before slowly developing larvae emerge and because successful emergence requires some minimal availability of resources, it seems reasonable to imagine a larval density above which the emergence rate is zero. An alternative view is that earlier emerging larvae in one way or another 88 ‘out-muscle’ their younger competitors, so that some larvae emerge, no matter how many eggs are laid. The problem is that infinity is a large number. With an arbitrarily large number of larvae in a finite body of water, the putrid, writhing mass of dying and decaying ‘wrigglers’ would prevent those still alive from gaining access to the air to breathe.

4.2. Extensions

The obvious direction in which to extend the analysis here developed would be to replace strains with genes. After all, it is bits of DNA, not populations that one is trying to substitute. Still, we suggest that before embarking down the genetical path, it would be worth extending Equation (5) by inclusion of at least one larval stage as a state variable. This would repair the aforementioned separation of time scales assumption, which, as noted above, is dubious. In this regard, the observation 32 of autonomous cycling in models with explicit larval dynamics is worth noting. This raises the possibility of intrinsic fluctuations, even in the absence of seasonality. More generally, epidemiological systems subject to seasonal forcing can manifest subharmonic resonances of large amplitudes and long periods 82. From the viewpoint of ‘slingshotting’, such cycles could be of considerable importance, extending the technique's applicability to non-seasonal environments.

4.3. Boots on the ground

In the real world, small populations are subject to stochastic extinction. Thus, there will be a minimum inoculate size below which transgenics are likely to die out. Correspondingly, successful transgene introductions that reduce prevalence to low levels may result in pathogen extinction and then, by virtue of its selective disadvantage, extinction of the transgene itself. Combining these expectations with small migratory influxes of infected hosts in the case of small transgene cost converts the system studied here into an excitable medium: stimulate and fire, with subsequent slow restoration of excitability. This brings us back to the population-level phenomena discussed in 20 86: mosquito population size and structure, gene flow within and between populations, etc. Beyond this is the conflict between transgenic introduction and other control measures. As noted above, any intervention that reduces prevalence reduces the likelihood of getting transgenes into natural populations. While here deduced for a simple, cartoon-like model, this result will likely be obtained for a wide class of models, the principal requirement being the assumption of the cost-benefit structure – inherent transgene cost, prevalence-dependant WT advantage – analysed here.

4.4. Meanwhile

Defeating malaria with genetic engineering would be a great achievement. Still it is worth remembering that ‘substantial malaria control is possible by extending the coverage of existing technologies to impoverished households and communities’ 78. Moreover, the successes 11 14 83 of Reed and Gorgas in Panama and Cuba remind us that the disciplined application of imperfect techniques can yield favourable results, even in the holoendemic tropics. Of course, the pursuit of high-tech strategies will continue, as well it should. But for now, there is suffering and death on a massive scale, a consequence of both primary infections and their sequellae e.g., 13 37 40, to say nothing of interaction with other diseases such as AIDS 1 6 40. The accusation 8,25–27,91 that the application of the precautionary principle in the case of malaria amounts to an assault by the ‘Haves’ on the ‘Have Nots’, may or may not be justified. But failure to vigorously implement proven control strategies most certainly is not.

Acknowledgements

This paper is dedicated to Jim Cushing, long-time colleague and friend. We thank W.L. Green for counsel on the fine points of differentiability and JBD’s diligent reviewers for considered and helpful comments.

Notes

In terms of Table 1, m and μ have units of y⁻¹, while F is dimensionless. The units of Equation (2) are thus mosquitoes/y.

It has been argued 2 39 that Plasmodium resistance induced by up-regulating mosquitoes’ immune response in WT mosquitoes does not produce a fitness advantage because immune system activation is costly. By way of contrast, the gene studied by Marrelli et al. inhibits Plasmodium development before such a response is mounted.

The reality of Plasmoduim-induced costs to infected WT mosquitoes has been questioned by Ferguson et al. 28 who suggest that significant adverse effects are only observable in unnatural host–parasite combinations.

Less comprehensible is the observation of reduced homozygote fitness as inferred from cage experiments in which transgene frequencies stabilized at p<1. Marrelli et al. suggest several possible explanations, including ‘hitch-hiking’ by deleterious alleles. But the perplexing earlier finding 62 is that transgenic mosquitoes fed on non-infected hosts manifest Hardy–Weinberg frequencies when introduced into populations of WTs.

While acknowledging that transgene advantage depends on prevalence, Marrelli et al. dismiss the principal implication of this fact, which is that transgenics can never drive WT mosquitoes to extinction, absent stochastic effects. They write, ‘In the field, where infection prevalence is lower than in laboratory systems and only a relatively small proportion of mosquitoes become infected, gene introgression is predicted to be considerably slower and possibly not of sufficient magnitude to establish the transgene in the population. However, once established, transgenic mosquitoes that interfere with parasite development should make more difficult the reintroduction of the parasite after eradication of malaria from the target area’. [Emphasis added].

Alternatively, wild type mosquito fecundity (Equation 6a) be modelled as . Preliminary analysis suggests that the resulting equation manifest behaviour similar to those studied here.

Mathematically, this is equivalent to interspecific competition (between w and g) in the presence of extrinsic mortality, x, that affects one species but not the other; see, for example, 90.

In greater detail, we require (1) that WTs outcompete transgenics in the absence of malaria, i.e., that c _g >0 (inherent cost to transgenics), and (2) that, in the presence of sufficient malaria, transgenics outcompete WTs (prevalence-dependent cost to WTs). Since the maximum possible prevalence is x=1, transgenic superiority for this value requires or . From these requirements, Condition (10) follows directly.

Equilibria corresponding to negative numbers of transgenics are, of course, biologically meaningless. Being cognizant of their existence, however, allows one to understand the origination and stabilization of the positive equilibria, which are biologically significant. Moreover, experience [17 suggests that there may be circumstances, for example, perturbations of the model, in which negative equilibria move into the positive quadrant.

The second interior equilibrium also gains a degree of stability at c _g=0 via a transcritical bifurcation – this one (TC3) involving the negative w=0 boundary equilibrium. We also note that TC1 and TC3 are ‘at a distance’, by which we mean that equilibria exchange stability but do not collide. By way of contrast, TC2 is generic – collide and exchange.

Thus, if we introduce GM mosquitoes on 1 January 2008, t ₀=2008.0, and φ=0; if on 31 January 2008, , and , etc.

Despite the apparent discontinuity at M=K, our function, F(M), is differentiable and therefore continuous. That is, the left- and right-hand difference quotients both go to zero, as M→K. As a matter of practicality, we note that in none of our calculations did mosquito densities exceed M=K, the point at which the second half of Equation (4a) kicks in.