# Optimal control for disease vector management in SIT models: an integrodifference equation approach

## Abstract

Vector-borne diseases are a major public health concern inflicting high levels of disease morbidity and mortality. Vector control is one of the principal methods available to manage infectious disease burden. One approach, releasing modified vectors (such as sterile or GM mosquitoes) Into the wild population has been suggested as an effective method of vector control. However, the effects of dispersal and the spatial distribution of disease vectors (such as mosquitoes) remain poorly studied. Here, we develop a novel mathematical framework using an integrodifference equation (discrete in time and continuous in space) approach to understand the impact of releasing sterile insects into the wild population in a spatially explicit environment. We prove that an optimal release strategy exists and show how it may be characterized by defining a sensitivity variable and an adjoint system. Using simulations, we show that the optimal strategy depends on the spatially varying carrying capacity of the environment.

## Keywords

Density dependence Dispersal Mosquito control## Mathematics Subject Classification

92D25 92D40 92B05## 1 Introduction

Vector borne diseases are a major public health concern, causing high levels of morbidity and leading to nearly one million deaths, annually (WHO 2016). Often vector control methods are the only feasible management option with the aim of vector control being to suppress or often eliminate insect vector populations. The sterile insect technique (SIT) is a well established empirical method for reducing population size. This technique has been widely used to suppress or eradicate many insect pest species, through the release of modified insects (lab-sterilised, irradiated and/or novel genetic technologies) (Alphey et al. 2010). Control is achieved as mating between sterile and wildtype mosquitoes reduces the reproductive potential of the wild population.

Eclectic synopsis of different spatial mathematical models developed for understanding mosquito control using SIT methods

Model assumption | Results | |
---|---|---|

Manoranjan and van den Driessche (1986) | Modelled the effectiveness of SIT using a reaction–diffusion equation. Sterile mosquitoes do not disperse while the wild population disperse randomly. 2 | The number of releases required for population elimination depends on the growth parameters of mosquitoes, domain length and the initial population distribution |

Ferreira et al. (2008) | Used a stochastic two-dimensional cellular automata (discrete time and space). Releases based on ratio between sites occupied by sterile and wild mosquitoes. The diffusion is incorporated using Margolus neighborhood | The wild population can not be eliminated for a spatially heterogeneous distribution of mosquitoes |

Yakob and Bonsall (2009) | Developed a spatially explicit model to find the optimal timing and sex specificity of lethal transgene activation for the control of different types of pest population. Sterile mosquitoes are uniformly distributed across the space | Optimal release strategy is influenced by the growth of the population, mosquito stage structuring, competition, and space. High rates of mosquito dispersal reduces the effectiveness of SIT |

Legros et al. (2012) | Applied a ‘Skeeter Buster’ model which is a stochastic, spatially explicit model of | Population elimination is feasible only in small geographical settings, unless the habitat and releases are homogeneous |

However, a constant dilemma is whether model predictions replicate real biological situations as the field data are often incomplete or not straightforward to analyse. In order to achieve successful vector control, we need a much more thorough understanding of how mosquitoes would respond to these control interventions (such as sterile insect releases) by focusing more on their biology, ecology and behaviour (Alphey et al. 2010). Spatial spread is a key element in mosquito reproduction and some studies argue that SIT technique may not be very effective in controlling mosquito populations due to their dispersal and distribution (Ferreira et al. 2008).

To address this issue we present an alternative modelling framework that allows us to consider population dynamic outcomes associated with different dispersal behaviour. Integrodifference equations (IDEs), which are discrete in time and continuous in space, incorporate a dispersal kernel for the spatial distribution of mosquitoes, model populations in which growth and dispersal do not happen at the same time (Kot and Schaffer 1986; Hardin et al. 1990; Kot 1992; Neubert et al. 1995; Kot et al. 1996; Neubert and Parker 2004; Hsu and Zhao 2008; Zhou and Kot 2011; Reimer et al. 2016). Unlike the model approaches discussed in Table 1, IDEs allow a wide range of redistribution kernels for dispersal to be considered. This way of formulating time and space also provides a better approach for determining invasion speeds (compared to the reaction–diffusion PDE approach which can often underestimate patterns of spread; see Kendall 1965; Murray 2001).

The aim of this paper is to use optimal control theory to find the most efficient release strategy under different environmental conditions and dispersal behaviours. This is of critical importance as it can be used as a guideline for the number of sterile mosquitoes that need to be reared as well as aiding best practice for their release. Specifically, we develop a bioeconomic model with the corresponding cost function for the control of the wild mosquito population, where the control parameter *r* describes the release ratio that should be applied.

Previous work (Gaff et al. 2007; Joshi et al. 2007; Martinez et al. 2015) has considered optimal harvesting in an integrodifference population framework, but to the best of our knowledge no one so far has investigated the effectiveness of SIT through an IDE framework. In particular we will analyse the effect of spatial heterogeneity on the sterile release strategies. We find that an optimal control release strategy exists and efficiently suppresses the wild mosquito population. Our approach allows us to find the timing and the intensity of the control that needs to be applied. In Sect. 2 we introduce the integrodifference framework. We present the model in Sect. 3 and derive the characterization of the optimal control in Sect. 4. This characterization corresponds to the most cost-effective release strategy to suppress or eliminate the wild mosquito population. In Sect. 5 we use numerical simulations to illustrate the theory developed.

## 2 Integrodifference equations

*t*, \(f(N_t)\) the growth function where \(f(N_t )=N_t g(N_t)\) and \(g(N_t)\) is the per capita growth. We model the population (in discrete time) with no movement as

*k*(

*x*,

*y*), which is a probability density function (pdf) and gives the probability that an individual starting at point

*y*, will settle at point

*x*by the next time step. The number of individuals moving to location

*x*is found by integrating the dispersal kernel

*k*(

*x*,

*y*) over the domain of interest. Hence, we have \(\int _{\Omega } k(x,y)dy\le 1\) (as we are looking at a population on a finite domain), where \(\Omega \) is the spatial domain. We get the full dynamics of the population by combining the growth function with the dispersal kernel as follows:

## 3 The model

*d*is the density independent death rate, \(e^{-d}\) is the density independent survival probability and

*K*is related to the carrying capacity. Now define \(A=\alpha \beta e^{-d}\) (as the intrinsic population growth rate) and assume that no mating difficulties arise. Therefore, Eq. (3) takes the form:

*t*into the wild mosquito population. The total population size is then \(N_t=W_t+R_t\). The number of offspring produced by a wild mosquito over its life that would make it to adulthood without density-dependence, given that random mating with sterile mosquitoes occurs is \(\frac{AW_{t}}{W_{t}+R_{t}}\). We assume that the life cycle happens in one time unit and that density dependence occurs only at the larval stage (Clements 1992; Lord 1998). This means that the survival probability depends only on the number of wild mosquitoes (and is independent of \(R_t\)). We assume that the number of released mosquitoes is \(R_t=r_{t}W^{*}\), where \(r_t\) is the release ratio at time step

*t*and \(W^{*}\) is the equilibrium population; the release of sterile mosquitoes is proportional to the wildtype equilibrium population size. Under these assumptions the population dynamics in a non-spatial system are governed by the following equation:

*k*(

*x*,

*y*) in a one dimensional domain \(\Omega \). We assume a closed domain, for example an area that is surrounded by unfavourable conditions such that mosquitoes would not travel across. It is important to note that this might not always be the case and different spatial analyses will be needed for these different boundary conditions. To analyse the dynamics of the wild mosquito population under the influence of SIT release, we use:

*S*is the average dispersal success over the domain \(\Omega \), given by \(S=\dfrac{1}{\Omega }\int _{\Omega }\int _{\Omega }k(x,y)dx dy\). See van Kirk and Lewis (1997) and Reimer et al. (2016) for a more detailed analysis on equilibrium solutions.

Description of the variables and parameters of the IDE model

Variable | Description |
---|---|

\(W_t\) | Wild mosquito population at time |

\(R_t\) | Released mosquitoes at time |

\(W^{*}\) | Equilibrium for the wild mosquito population |

Parameter | Description |
---|---|

\(\alpha \) | Number of matings per individual |

\(\beta \) | Number of offspring produced per mating |

\(\gamma (W_t)\) | Survival probability |

| Density independent death rate |

| Strength of negative feedback–proxy for the carrying capacity |

| Intrinsic population growth rate |

\(r_{t}(x)\) | Release ratio at time step |

## 4 Optimal control formulation of integrodifference equation

In this section, we describe the optimal control framework for minimizing the cost of controlling the wildtype mosquito population. We propose to control a vector population over a time period \(t\in [0,T]\). The state variable is \(W(x)=(W_{0}(x), W_{1}(x),\ldots ,W_{T}(x))\) and the control is \(r(x)=(r_{0}(x), r_{1}(x),\ldots , r_{T-1}(x))\), representing the wild mosquito population and the sterile insect release ratio respectively at location *x* and time step *t*, where the initial distribution \(W_{0}(x)\) is given.

*f*is twice differentiable in \(W_{t}(y)\) and that partial derivatives \(\frac{\partial f(W)}{\partial W}\), \(\frac{\partial ^{2} f(W)}{\partial W}\) are \(L^{\infty }\) bounded for any \(W\in L^{\infty }(\Omega )\). We want to find the optimal strategy that suppresses the mosquito population and minimizes the cost of vector control via the release of sterile mosquitoes. Assume there is a linear cost associated with the wild mosquitoes (due to impact on human health, lost tourism, etc.) which we denote by \(m_t\). There is also a cost for producing and releasing sterile mosquitoes and we assume it to be a quadratic of the form,\(( n_t r_{t}W^{*}+s_t r_{t}^{2}(W^{*})^{2})\). The cost function is a nonlinear relationship in r*. This choice of cost function is based on the reality that sterile insect releases, through mating disruption, introduce an Allee effect into the wild population (Bonsall et al. 2010; also see Kirschner et al. 1997 for similar choice of quadratic costs for a HIV chemotherapy application). Using these costs, we define the objective functional:

### Theorem 1

*X*is the solution of

*b*and

*d*are defined by: \(b=\frac{n}{2sW}-1\), \(d=\frac{-f(W_t (x),x)\int _{\Omega } \lambda _{t+1}(y) k(y,x)dy}{2sW_{t}W^{*}}\).

Proof of this Theorem is given in the “Appendix”. We find the characterization of the optimal control by solving Eq. (13). Using the proof of Theorem 1 (see “Appendix”), if we analyse Eq. (32), we find its derivative is positive for all time. This implies that Eq. (32) has precisely one real root. If we obtain a negative real root, we set the control to zero. The solution is zero when \(C>A_{1}\) indicating that the optimal strategy is to not release any sterile mosquitoes. This means that the burden imposed by the wild mosquito population (in terms of disease or nuisance biting) is so low that we do not need to instigate a sterile mosquito release programme. When the real root is positive, we release sterile insects at the release ratio dictated by the root.

## 5 Numerical results

*r*is considered to be optimal.

### 5.1 Homogeneous environments

*K*(related to the carrying capacity) is the same across the whole domain \(\Omega \). Throughout this section, we assume a uniformly distributed wild population at \(t=0\) with \(m=n=1\), \(s=5\) and consider \(\Omega =[0,1]\), unless otherwise stated. We have chosen these values to correspond to a high cost of producing and releasing sterile insects compared to the cost associated with the wild mosquitoes. With this set of costs, we do not expect a complete elimination of the wild population. In the following, we investigate the Laplace kernel for the dispersal, which is described by:

### 5.2 Heterogeneous environments

Very often, a habitat has different attractiveness to mosquitoes in different areas. This can be influenced by factors such as resource availability (e.g. food, mates, breeding sites) and predation. In this section, we analyse how landscape heterogeneity affects the dispersal and the optimal control of mosquitoes. We replicate this behaviour by varying the value of *K* (the strength of density feedbacks related to the carrying capacity). We consider situations (a) when the centre of the domain has more favourable Conditions (see Figs. 3, 4a), (b) when the boundaries of the domain have more favourable conditions (see Figs. 5, 6a). We conclude that through time, the population in situation (a) aggregates more in the centre of the domain, as expected, because the conditions are more favourable there. This is true for populations with and without control. Without control, the wild mosquito population slightly decreases initially at the boundaries until it approaches the stable equilibrium. On the other hand, in the centre of the domain, the wild mosquito population increases until it approaches its equilibrium at \(t=5\). Once the control is introduced, we see from Fig. 4b that releasing sterile mosquitoes greatly suppresses the wild mosquito population. The optimal strategy in this scenario is to release more sterile mosquitoes in the centre of the domain where they are concentrated. Figure 4a gives the optimal number of sterile mosquitoes that need to be released in order to control the wild mosquito population. We release substantially more mosquitoes at the centre of the domain than its boundaries at time step \(t=1\). This is because dispersing mosquitoes are moving more towards the centre and at this level the burden imposed by the wild mosquito population is so high that a large number of mosquitoes need to be released in order to suppress the population. Once the population is suppressed, after \(t=2\), the ratio of releases between boundaries and centre of domain is decreased. We get the opposite behaviour when the conditions are more favourable in the boundaries. Without control, from Fig. 5, we notice that the population increases initially and it approaches an equilibrium at time \(t=5\). As expected, the population size increases more in the boundaries, because mosquitoes will move more towards them as the conditions for mosquito reproduction are better. Using the control, we see from Fig. 6b that the population quickly decreases until it reaches a threshold where they do not impose a high burden. The optimal strategy in this case is to release more sterile insects in the boundaries initially (until \(t=3\)) and after the wild population has reached a critical low threshold (e.g., where the effects are not very harmful), the ratio of release between boundaries and the centre of domain is decreased.

## 6 Discussion

Here, we formulated a novel mathematical model to understand the effects of releasing sterile mosquitoes into wild populations as well as the effect of spatial spread on mosquito population dynamics. The approach described here has not been used before in designing and optimizing sterile insect release strategies.

The model is described by an integrodifference equation, which are used to model populations where growth and dispersal do not happen at the same time. In our model we consider a homogeneous population consisting of wild and sterile mosquitoes with no overlapping generations. The growth function is based on the Ricker model and we assume releases proportional to wild population equilibrium. The spatial spread of the mosquitoes is described by the dispersal kernel *k*(*x*, *y*) in the integrodifference equation.

Using numerical simulations, we considered homogeneous environments with uniform carrying capacity, and heterogeneous environments with different carrying capacity in different areas, where the Laplace kernel describes mosquito movement. One significant finding is that, due to redistribution, applying the optimal control does not eradicate the wildtype mosquito but only substantially reduces population size. In practice, sterile mosquitoes may be released from aircraft resulting in releases that are approximately uniform. Our results highlight that this is not optimal and instead releasing more where the population densities are higher is more efficacious for vector control. Our model predicts a 73\(\%\) suppression of the wild population which is close to observed field estimates, where \(> 80\%\) suppression rates have been reported for *A. aegypti* control (Harris et al. 2012; Carvalho et al. 2015).

In heterogeneous environments, we consider situations where the centre of the domain has more favourable conditions or when the boundaries have more favourable conditions. In both cases the control significantly suppresses the mosquito population. Our results suggest (as expected) that more mosquitoes should be released where densities are higher. Another (expected) result from this model is that the optimal strategy is to release significantly more sterile mosquitoes at the beginning of the vector management control programme until the wild mosquito population is suppressed to a level that the imposed burden is so low.

Here we show that continuous releases predict that complete eradication of the wild population is not an optimal solution. Furthermore, integrating different control options (insecticide knockdowns, pulse or continuous SIT) to achieve cost-efficient control strategies needs more thorough investigation (Hackett and Bonsall 2018). Additionally, we argue that this discrete-time, continuous-space model approach is better than previous ones (described in Introduction) in finding the time and intensity of control, as it can include a variety of dispersal behaviour. This is crucial when modelling mosquito control. However, we emphasize the importance of determining a more accurate dispersal kernel that supports the field data, as the wave speed is very sensitive to the dispersal behaviour. Depending on the specific system parameterization, the results presented here are likely to be sensitive to the cost function form and the parameters used in this function. We have assumed a quadratic form, but other functions can be explored (see Khamis et al. 2018). This will change the objective functional *J*(*r*) and the characterization of the optimal control. Importantly, our results emphasize that optimal control does not necessarily lead to population elimination. Varying the parameters (*m*, *n*, & *s* in Eq. 7) associated with the cost of the wild and sterile mosquitoes will most likely modify this outcome. If we lower the cost of producing and releasing sterile mosquitoes, we can achieve elimination of the wild population. In summary, we showed that the optimal control of the SIT model described by an integrodifference equation exists and that the control can significantly suppress the wild mosquito population.

## Notes

### Acknowledgements

The authors are grateful for financial support from DARPA (Contract: HR00111-16-2-0005).

### Author Contributions

KK, DK, CEM and MBB conceived the study. KK performed the mathematical analysis and drafted the manuscript. DK, CEM and MBB helped draft the manuscript. All authors gave final approval for publication.

## References

- Alphey L, Benedict M, Bellini R, Clark GG, Dame DA, Service MW, Dobson SL (2010) Sterile-insect methods for control of mosquito-borne diseases: an analysis. Vector-Borne Zoonotic 10:295–311CrossRefGoogle Scholar
- Bonsall MB, Yakok L, Alphey NA, Alphey L (2010) Transgenic control of vectors: the effects of interspecific interactions. Isr J Ecol Evol 56:353–370CrossRefGoogle Scholar
- Carvalho DO, McKemey AR, Garziera L, Lacroix R, Donnelly CA, Alphey L, Malavasi A, Capurro ML (2015) Suppression of a field population of
*Aedes aegypti*in Brazil by sustained release of transgenic male mosquitoes. PLoS Neglect Trop D 9:e0003864CrossRefGoogle Scholar - Clements AN (1992) The biology of mosquitoes: development, nutrition and reproduction, vol 1. CAB International, WallingfordGoogle Scholar
- Estep LK, Burkett-Cadena ND, Hill GE, Unnasch RS, Unnasch TR (2014) Estimation of dispersal distances of
*Culex erraticus*in a focus of Eastern Equine Encephalitis virus in the Southeastern United States. J Med Entomol 47:977–986CrossRefGoogle Scholar - Ferreira CP, Yang NM, Esteva L (2008) Assessing the suitability of sterile insect technique applied to
*Aedes aegypti*. J Biol Syst 16:565–577CrossRefGoogle Scholar - Gaff H, Joshi HR, Lenhart S (2007) Optimal harvesting during an invasion of a sublethal plant pathogen. Environ Dev Econ 12:673–686CrossRefGoogle Scholar
- Gratton C, Zanden M (2009) Flux of aquatic insect productivity to land: comparison of lentic and lotic ecosystems. Ecology 90:2689–2699CrossRefGoogle Scholar
- Hackett SC, Bonsall MB (2018) Management of a stage-structured insect pest: an application of approximate optimization. Ecol Appl 28:938–952CrossRefGoogle Scholar
- Hardin DP, Takác P, Webb GF (1990) Dispersion population models discrete in time and continuous in space. J Math Biol 28:1–20MathSciNetCrossRefzbMATHGoogle Scholar
- Harris AF, McKemey AR, Nimmo D, Curtis Z, Black I, Morgan SA, Oviedo M, Lacroix R, Naish N, Morrison N, Collardo A, Stevenson J, Scaife S, Dafa’alla T, Fu G, Phillips C, Miles A, Raduan N, Kelly N, Beech C, Donnelly CA, Petrie WD, Alphey L (2012) Successful suppression of a field mosquito population by sustained release of engineered male mosquitoes. Nat Biotechnol 30:828–830CrossRefGoogle Scholar
- Hsu SB, Zhao X-Q (2008) Spreading speeds and traveling waves for non-monotone integrodifference equations. SIAM J Math Anal 40:776–789MathSciNetCrossRefzbMATHGoogle Scholar
- Legros M, Xu C, Okamoto K, Scott TW, Morrison AC, Lloyd AL, Gould F (2012) Assessing the feasibility of controlling
*Aedes aegypti*with transgenic methods: a model-based evaluation. PLoS ONE 7:e52235CrossRefGoogle Scholar - Isidoro C, Fachada N, Barata F, Rosa A (2009) Agent-based model of
*Aedes aegypti*population dynamics. In: Seabra Lopes L, Lau N, Mariano P, Rocha LM (eds) Progress in artificial intelligence: 14th Portuguese conference on artificial intelligence. Springer, New York, pp 53–64CrossRefGoogle Scholar - Joshi HR, Lenhart S, Lou H, Gaff H (2007) Harvesting control in an integrodifference population model with concave growth term. Nonlinear Anal-Hybrid 1:417–429MathSciNetCrossRefzbMATHGoogle Scholar
- Kendall (1965) Mathematical models of spread of infection. In: Mathematics and computer science in biology and medicine: proceedings of a conference held by the Medical Research Council in association with the Health Dept., Oxford, July 1964. Conference on mathematics and computer science in biology and medicine (1964: Oxford, England). HMSO, London, pp 213–225Google Scholar
- Kirschner D, Lenhart S, Serbin S (1997) Optimal control of the chemotherapy of HIV. J Math Biol 35:775–792MathSciNetCrossRefzbMATHGoogle Scholar
- Khamis D, El Mouden C, Kura K, Bonsall MB (2018) Optimal control of malaria: combining vector interventions and drug therapies. Malaria J 17:174CrossRefGoogle Scholar
- Kot M (1992) Discrete-time travelling waves: ecological examples. J Math Biol 30:413–436MathSciNetCrossRefzbMATHGoogle Scholar
- Kot M, Schaffer WM (1986) Discrete-time growth-dispersal models. Math Biosci 80:109–136MathSciNetCrossRefzbMATHGoogle Scholar
- Kot M, Lewis MA, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77:2027–2042CrossRefGoogle Scholar
- Lenhart S, Workman JT (2007) Optimal control applied to biological models. CRC Press, LondonzbMATHGoogle Scholar
- Li J, Yuan Z (2015) Modelling releases of sterile mosquitoes with different strategies. J Biol Dyn 9:1–14MathSciNetCrossRefGoogle Scholar
- Li X, Zou X (2012) On a reaction–diffusion model for sterile insect release method with release on the boundary. Discrete Cont Dyn-B 17:2509–2522MathSciNetCrossRefzbMATHGoogle Scholar
- Lord CC (1998) Density dependence in larval
*Aedes albopictus*(Diptera: Culicidae). J Med Entomol 35:825–829CrossRefGoogle Scholar - Manoranjan VS, van den Driessche P (1986) On a diffusion model for sterile insect release. Math Biosci 79:199–208MathSciNetCrossRefzbMATHGoogle Scholar
- Martinez MV, Lenhart S, White KAJ (2015) Optimal control of integrodifference equations in a pest–pathogen system. Discrete Cont Dyn B 20:1759–1783MathSciNetCrossRefzbMATHGoogle Scholar
- Murray JD (2001) Mathematical biology. II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics, vol 18. Springer, New YorkGoogle Scholar
- Neubert MG, Parker IM (2004) Projecting rates of spread for invasive species. Risk Anal 24:817–831CrossRefGoogle Scholar
- Neubert MG, Kot M, Lewis MA (1995) Dispersal and pattern formation in a discrete-time predator–prey model. Theor Popul Biol 8:7–43CrossRefzbMATHGoogle Scholar
- Potgieter L, van Vuuren JH, Conlong DE (2013) A reaction–diffusion model for the control of
*Eldana saccharina*Walker in sugarcane using the sterile insect technique. Ecol Model 250:319–328CrossRefGoogle Scholar - Reimer JR, Bonsall MB, Maini PK (2016) Approximating the critical domain size of integrodifference equations. Bull Math Biol 78:72–109MathSciNetCrossRefzbMATHGoogle Scholar
- van Kirk RW, Lewis MA (1997) Integrodifference models for persistence in fragmented habitats. Bull Math Biol 59:107–137CrossRefzbMATHGoogle Scholar
- WHO (2016) World Health Statistics 2016: monitoring health for the sustainable development goals. World Health Organization, GenevaGoogle Scholar
- Yakob L, Bonsall MB (2009) Importance of space and competition in optimizing genetic control strategies. J Ecol Entomol 102:50–57CrossRefGoogle Scholar
- Zhou Y, Kot M (2011) Discrete-time growth-dispersal models with shifting species ranges. Theor Ecol 4:13–25CrossRefGoogle Scholar

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