Optimal control of a tick population with a view to control of Rocky Mountain Spotted Fever

: In some regions of the Americas, domestic dogs are the host for the tick vector Rhipicephalus sanguineus , and spread the tick-borne pathogen Rickettsia rickettsii , which causes Rocky Mountain Spotted Fever (RMSF) in humans. Interventions are carried out against the vector via dog collars and acaricidal wall treatments. This paper investigates the optimal control of acaricidal wall treatments, using a prior model for populations and disease transmission developed for this particular vector , host, and pathogen. It is modiﬁed with a death term during questing stages reﬂecting the cost of control and level of coverage. In the presence of the control, the percentage of dogs and ticks infected with Ri. rickettsii decreases in a short period and remains suppressed for a longer period, including after treatment is discontinued. Risk of RMSF infection declines by 90% during this time. In the absence of re-application, infected tick and dog populations rebound, indicating the eventual need for repeated treatment.

Rocky Mountain Spotted fever is caused by one of several Rickettsia species which, together, comprise multiple pathogens, have worldwide distribution, and are carried by several vector species [7][8][9].
The resulting diseases range from mild to, in the case of RMSF, fatal [10,11].
In general, vector borne pathogens cause a large burden of disease and mortality worldwide, and are principally controlled by vector suppression, either alone or as part of a holistic approach that may include wall treatments, dog collars or other interventions [25][26][27].As vector control carries a cost as well as benefits, it is possible to approach the questions of how much and when to apply interventions via the use of optimal control methods [28,29].This approach requires a system of differential equations describing the life cycle of Rh. sanguineus and disease transmission of Ri. rickettsii.Such a model was developed for populations of ticks and dogs in a community in Sonora, Mexico, and is used as the basis for the control problem solved for that model [30,31].The population/transmission model takes into account the multiple stages in tick development, including temperature and humidity dependence [12,32].It incorporates insecticidal wall treatment and the resulting death rate for questing ticks.
In the Sonora intervention, a small number of houses were treated with insecticidal wall treatments [31].The model created for this intervention indicated that it was likely that sufficient wall coverage would remove the need for the primary intervention in that situation, which was treated dog collars [30].Wall treatments in general have a short half-life, requiring re-application.Without re-application it is possible to compare limitations of treatment using models, as was done in the case of malaria.In this study we omit decay of the treatment and instead use an optimal control approach to ask how long and at what level of effectiveness the treatment must remain in place to suppress the vector population.

Materials & methods
The numerical approach includes a system of ordinary differential equations describing the life cycle of Rh. sanguineus, dog population growth and transmission of Ri. rickettsii between these, shown in equations (2.1-refhumidity).The control problem is framed by a system of ordinary differential equations for the adjoint variables, in equations (2.44-2.69).Results of numerical simulations are shown in Figures 2-4.

Model details
The process based model for Rh.sanguineus life cycle and disease transmission, on which optimal control is based, is taken from Álvarez-Hernandez et al, parameterized to reflect local conditions in a town in Sonora, Mexico [30].The default death rate due to treatment is set at 0.7 to reflect 70% coverage of walls.That model includes temperature dependent maturation rates for non-questing stages as well as a temperature and humidity dependent death rate for questing nymphs.A condensed version of the compartment model is illustrated in Figure 1.Like other hard-bodied ticks, Rh. sanguineus passes through maturation stages (larva, nymph, adult).These stages conclude with relatively short questing (searching for a host) and feeding intervals that provide all the needed food and water for the tick to complete its lifecycle, after which it ovipositions (if female) and dies [32].During the relatively long intervals between feeding on a host and the next questing event, these ticks sequester in cracks in walls, floors, and other peri-domestic areas, making them difficult to observe.Parameters are therefore adjusted to match data for questing and feeding ticks.
During the feeding stages disease transmission can take place from an infected tick to an uninfected dog, or from an infected dog to an uninfected tick.The disease transmission model is based on the Ross-Macdonald model for mosquito borne disease, with the additional feature that transmission may occur during any of the three feeding stages of the vector.There is evidence that vertical transmission may also occur, as it does in closely related diseases [33][34][35].Little, however, is known about the rates involved or whether it is indeed a characteristic of this particular vector/disease pair, so this aspect of transmission is omitted from the model.

Eggs
Young, hardening larvae, Uninfected engorged maturing larvae/young nymphs, NU 1 Infected engorged maturing larvae/young nymphs, NI 1 Questing uninfected nymphs, NU 2 Questing infected nymphs, NI 2 Uninfected nymphs feeding on uninfected hosts, Uninfected nymphs feeding on infected hosts, Infected nymphs feeding on uninfected hosts, Infected nymphs feeding on Infected hosts, Uninfected engorged maturing nymphs/young adults, AU 1 Infected engorged maturing nymphs/young adults, AI 1 Questing uninfected adult, AU 2 Questing infected adult, AI 2 Uninfected adults feeding on uninfected hosts, Uninfected adults feeding on infected hosts, Infected adults feeding on uninfected hosts, Infected adults feeding on infected hosts, Engorged adults, A 4 Gestating adults, Uninfected hosts (dogs), Percent available space per uninfected host weighted by probability (q L ) of larvae finding any host, Percent available space per infected host weighted by probability (q L ) of larvae finding any host, Percent available space per uninfected host weighted by probability (q N ) of nymph finding any host, Percent available space per infected host weighted by probability (q N ) of nymph finding any host, Percent available space per uninfected host weighted by probability (q A ) of adult finding any host, Percent available space per infected host weighted by probability (q A ) of adult finding any host, Total number of hosts of all types, S S = U + I, Fraction of hosts that are uninfected, Fraction of hosts that are infected, Transmission term for host infection,

Properties of the Population Dynamics & Disease Transmission Model
In this section we will obtain the existence, uniqueness, nonnegativity, and boundedness of solutions to our model in a single theorem.
Theorem 2.1.For nonnegative initial conditions, the model (2.1-2.25) has a unique solution which exists for all time and is nonnegative in each component.
Proof: Local existence and uniqueness is standard via arguments in [36].A supersolution argument establishes that the solutions are bounded on their interval of existence [37].A subsolution argument proves that the solutions are bounded below by zero.

Optimal control
We wish to minimize the tick population during the questing life stages, L 2 , NU 2 , NI 2 , AU 2 and AI 2 , while also minimizing the death rate caused by the wall treatment intervention, represented by the coefficient w in equations (2.3, 2.8, 2.9, 2.16,2.17).
over the set of admissible controls We use quadratic terms in the cost function, J(w), as is typical for epidemiology control problems, because linear control does not offer closed-form solutions for the optimal control [38][39][40][41].Often a linear-quadratic cost function is used as well.These cost functions represent the nonlinear increase in the effect of each quantity in J(w).The cost of increased infective ticks is probably closer to quadratic than linear because at low levels ticks would prefer the dog host, while at high levels they might prefer humans as space on dogs becomes saturated.Similarly, the effects of the wall treatment probably represent a nonlinear function to the system because after the more willing participants have treated their walls it becomes increasingly expensive to convince the holdouts.
The quadratic term is multiplied by a coefficient, K, which allows for the relative importance of the term to be varied.The final time T determines the size of the interval of existence for the optimal control.

Existence of optimal control
Theorem 2.2.Given the objective functional (2.42), subject to the system given by Eqs.(2.1-2.25) with nonnegative initial conditions, and the admissible control set (2.43) then there exists an optimal control w * (t) such that min w∈V J(w) = J(w * ).
Proof: In order to apply the theory of Fleming and Rishel, [42], we must show that the following conditions are met: 1.The class of all initial conditions with a control fnction w(t) in the admissible control set along with each state equation being satisfied is not empty.
2. The admissible control set V is closed and convex.
3. Each right hand side of the state system is continuous, is bounded above by a sum of the bounded control and the state, and can be written as a linear function of the control function w(t) with coefficients depending on time and the state.4. The integrand of the objective functional (2.42) is convex on V and is bounded below.Since all conditions are satisfied in this case, it follows that there exists an optimal control w * (t) such that min w∈V J(w) = J(w * ).

Characterization of Optimal Control
Theorem 2.3.Given the optimal controls w * and solutions of the corresponding state system, there exist adjoint variables λ 1 , λ 2 , . . ., λ 25 satisfying the following: (2.66) where Furthermore, the analytic representation of the optimal control w * is given by Proof: Suppose w * (t) is the optimal control and that E, Ł 1 , . . ., U, I is the corresponding solution to the system (2.1-2.25).We use standard work in Pontryagin et al. [32] to obtain the result.To find the analytic representation of the optimal control w * (t), begin by forming the Lagrangian.Since the control is bounded, the Lagrangian is where H is the Hamiltonian given by and W i (t) ≥ 0 are penalty multipliers such that To find the analytic representation for w * (t), we analyze the necessary conditions for optimality ∂L ∂w = 0.
By standard optimality techniques for the characterization for the optimal control w * (t), we find that

Algorithm
At the optimum w * , the model differential equations move forward in time from an initial condition, while the adjoint differential equations move backward in time from a final condition.In some cases, it is possible to use Matlab's bvp4c to solve ODE systems with a variety of different types of boundary conditions like this one [43].However, there are often convergence problems with this approach.For this paper, we followed the algorithm developed by Hackbusch [44] and recommended by Lenhart and Workman [45] to solve our optimality system.
The algorithm was implemented in Matlab [43], using ode45 to solve the ODEs and interp1 to pass the solutions from step to step.
Recall that the objective function is where we are simultaneously minimizing the tick populations at the questing stages and the cost of the wall-treatment with the term Kw 2 .

Results
Initial conditions were found by running the original model with an initial number of tick eggs of E 0 = 1, 000, 000, an initial number of uninfected dogs of U = 1248 and one infected dog I = 1.This simulation was run until T = 1000 days and these steady state population values were used as initial conditions for all simulations in this paper.
The model with no treatment results in a seasonally fluctuating steady state, with ticks always present and abundant, seen in Figure2a.Ri. rickettsii prevalence in ticks and dogs reaches steady state will little seasonal fluctuation, seen in Figure2b.In the absence of control, we note that the percentage of infected dogs and ticks remains essentially stable, indicating persistence of the disease in the absence of interventions, seen Figure 2b.With the relatively expensive control at K=100, tick abundance and disease prevalence decline steadily, as in Figure2c and Figure2d, while the optimal control is allowed to decline starting at approximately t=100, seen in Figure2d.
Using the same initial conditions as Example 1, we consider simulations with d WT = 0.7, or 70% of houses treated.Under the assumption that the treatment is more expensive, we set the penalty affecting the cost of treatment in the objective function J to be K = 1, 10, and K = 100 and run the optimal control algorithm for two years, T = 730 days seen in Figure 3a,3c and 3e.Control was discontinued at T = 730, and the subsequent two years tracked in Figure 3b, 3d and 3f for each of the controls respectively.We were unable to obtain convergence of the algorithm for K = 1000.
The risk of human infection with RMSF depends on the likelihood of contact with an infected tick, which in turn depends on the abundance of infected ticks, not just pathogen prevalence in the tick population.The risk of an individual exposure to RMSF depends on the population of infected questing nymphs and adults.Three scenarios are computed, with the optimal control calculated for days 1-730, followed by the rebound of tick populations for the following 730 days.Results are shown in Figure 3 for K=1, 10, and 100.
As the cost of treatment increases from K=1 to K=100, the optimal control goes from always on at full strength with K=1 as seen in Figure3a, to always on for an initial period, then declining to a  lower level, as seen in Figure3c and 3e.In all three cases, Ri. rickettsii prevalence is greatly reduced at the 2-year point, seen in Figure3a, 3c and 3e.When treatment is removed completely the disease prevalence increases, seen in Figure3b, 3d and 3f.The treatment is 100% effective for much of the time interval, although it's efficacy drops later in the time interval.We note that for K = 1, the treatment is 100% effective for the whole time interval.In the presence of control, we note that the percentage of infected dogs and ticks decreases dramatically over time, supporting the theory that the wall treatment is an effective approach to reducing the the number of RMSF infections in both ticks and dogs.
The death rate of questing ticks due to the wall treatment, d WT , can be varied to reflect maximum coverage of houses in the community.Setting K = 1 and varying d WT gives an example of the effect of coverage levels in Figure 4.The death rate of questing ticks at full strength of treatment may vary due to coverage levels or efficacy of the product chosen.The default death rate of 70% of questing ticks per day (at full control (w=1), Figure4c and 4d) was increased to 100% per day (Figure4a and 4b) and decreased to 35% (Figure 4e and 4f).Although the control patterns look similar (Figure4b, 4d, 4f) the resulting decline in pathogen prevalence is more pronounced as the death rate rises (Figure4a, 4c and 4e).Infected questing tick populations during two years of optimal control followed by two years of no control, for various choices of K ((a)control period for K=1, (b)rebound after control for K=1, (c)control period for K=10, (d)rebound after control for K=10,(e)control period for K=100, (f)rebound after control for K=100).Infected nymph and infected adult populations shown with control in blue.Wall-treatment is at 70%.

Discussion
Acaricidal wall treatments have the potential to drastically reduce tick populations as seen in Figure 2. It is clear from both Figures 3 and 4 that an optimal wall treatment must remain at full strength for a considerable period, followed by declining efficacy.Even the shortest duration of high intensity, shown in Figure 3e, is a year in duration.

Disease risk
Figure 3 shows the suppression of disease risk for three choices of optimal control (K=1,10,100 respectively).As K increases, the duration of 100% treatment decreases with subsequent decline in treatment strength for K=10,100, seen in Figure 3c and 3e.This decline is a normal feature of insecticidal treatments [46][47][48].For K=1 treatment is at full strength until day 730 and then is abruptly discontinued.Recall that we assume 70% coverage of surfaces for all three examples.The duration of full strength treatment is 730, 600, and 450 days respectively for K=1,10,100.In Figures 3b, 3d, 3f the rebound of infected tick populations is shown.For all three examples there is an immediate drop in infected tick populations from over 9000 nymphs and 4500 adults on day 1 to less than 10% of this number by day 9, representing a 90% reduction of disease risk.Suppression of infected tick populations persists for 1152, 1123, and 1094 days, for K=1,10,100 respectively (1095 days = 1 year).Taken together, these examples show that if treatment remains at full strength for 450 days or more, with declining partial strength for the shorter treatments, disease risk is reduced by 90% for 1094 days or more.These examples show both the effectiveness of wall treatment and the importance of treatments with effectiveness that persists for a relatively long time.By contrast, note that applications of liquid deltamethrin, must be reapplied every 8 weeks, so a single treatment does not persist that long [30,31].

Coverage
When applying an intervention to an entire community, coverage will always be imperfect.Some households will refuse treatment and some parts of a house may be inaccessible or inappropriate for treatment.
The model expresses coverage levels in a parameter, d WT , which is varied in Figure 4.The tradeoff between the duration of maximum treatment, illustrated in the right hand panel, and rate of reduction of disease prevalence, in the left hand panel, is clear.Interventions that are long lasting are seen to compensate somewhat for lack of coverage.For example, Figures 4e and 4f show a scenario in which disease prevalence in dogs is reduced from 90% to 20% in the course of a year of maximal treatment of 35% of walls.By comparison, with 100% coverage, shown in Figures 4a and 4b, the reduction in disease prevalence is better and the control is allowed to decline at about 700 days.
Studies of the efficacy of insecticidal wall treatments describe the decline in efficacy in terms of a half life, similar to linear toxicokinetics of an organism [46][47][48].Interestingly, the optimal control patterns seen in Figures 3 and 4 show a similar decline after 100% of control is discontinued.The rate of decline of the optimal control seems to be slower than many of the observed rates in the literature, however.

Costs of treatments: an example
Most insecticidal or acaricidal wall treatments lose efficacy within months of application [46][47][48].The cost of repeated application may therefore add substantially to the intervention, on top of the cost of the product used.If a product is inconvenient to apply or must be applied repeatedly, this may have an effect on coverage levels as well, as households may decline to participate in the intervention.
The model on which this optimal control problem is built was based on an intervention in Sonora, Mexico in which two treatments were compared [31].One was application of a liquid acaricide containing 5% deltamethrin, (Bayer K-Othrine WG250, pyrethroid), to the yards and homes of selected houses in the community.Trained personnel were required as well as oversight by licensed pest experts, and reapplied every eight weeks for eight months.Deltamethrin has been used on many species, with emerging resistance in ticks [49].
The other application used in Sonora was a paint developed by Inesfly Corporation that has been used for a range of vector control applications (Safecolor, Codequim, RSCO-USP-39-2016, carbamate) [30].This product contains a slow release formula of 1% Propoxur.It was used to control Triatoma sp., the vector of Chagas disease [50][51][52][53][54].It has proven useful against mosquito malaria and dengue vectors [55][56][57][58][59].It has been used on nets to control Tsetse fly [60] and sand fly [61].However, propoxur itself is not approved for indoor use in the United States.The developer of the slow release formula claims that the insecticidal effect persists for 2 years on interior walls.Because of the long residual effect, it is possible that this product could satisfy the treatment protocol produced by an optimal control problem if shown to be safe for indoor use.If used only on exterior walls, the coverage level would never be above 50%.
These two interventions are an excellent example of the tradeoffs required in cost versus efficacy.The deltamethrin treatment is straightforward and the product is readily available, but there is considerable cost for reapplication every eight weeks by professionals.The propoxur paint has longevity, but the product is likely to be more expensive and, in some locations such as the U.S., coverage might be limited to exterior walls.

Future work
The control problem solved here was approached with the assumption that only questing ticks are susceptible to the acaricide applied to walls.The truth is more complicated, as Rh.sanguineus also sequesters in cracks in the walls (and other locations) when not questing.Whether the acaricide penetrates the cracks, what percent of ticks are not on walls, and other physical and biological uncertainties might change the answer produced here.
The model developed here is based on a community intervention and the tick data that arose from it [30], which was not a controlled experiment.The parameters of the model, in particular, would benefit from a more contained and controlled experiment, perhaps using dogs in kennels that have been colonized by Rh. sanguineus.
The paint formulation of propoxur seems to be a promising intervention for RMSF.Whether it can be shown safe for indoor use would affect the coverage that is possible.In addition there is yet no study of its duration of effectiveness, especially outdoors.Knowing this would allow the model to give a more dependable prediction of the results of any intervention.

Concluding remarks
The numerical experiments in this study demonstrate the value of long lasting acaricidal wall treatments against the tick Rh. sanguineus, vector of Ri. rickettsii , the vector of RMSF.Risk of RMSF declines by 90% with 70% wall coverage and at least 450 of full strength efficacy.This risk reduction persists well beyond the window of effectiveness for the wall treatment.This study highlights the need for further trials of wall treatment interventions against RMSF both in laboratory settings and in the field.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 1 .
Figure 1.Diagram of life cycle and infection status of R. sanguineus and dog hosts.Cross transmission occurs between dogs and feeding nymphs and adults.Transmission arrows are omitted for clarity.
Infection levels with no control for two years.Questing ticks with control under wall-treatment for two years, K = 100 Infection levels and control under wall-treatment control for two years, K = 100

Figure 2 .
Figure 2. Population and disease dynamics for questing tick abundance ((a)with no control, and (c) with optimal control at K=100) and infection prevalence in dogs ((b) with no control, and (d) with optimal control at K=100) Runs are for two years with and without control under 70% wall-treatment.

Figure 3 .
Figure 3. Infected questing tick populations during two years of optimal control followed by two years of no control, for various choices of K ((a)control period for K=1, (b)rebound after control for K=1, (c)control period for K=10, (d)rebound after control for K=10,(e)control period for K=100, (f)rebound after control for K=100).Infected nymph and infected adult populations shown with control in blue.Wall-treatment is at 70%.