Optimal Control of Epidemic Models of Rabies in Raccoons Wandi Ding, Tim Clayton, Louis Gross, Keith Langston, Suzanne Lenhart, Scott Duke-Sylvester, Les Real Middle Tennessee State University Temple Temple University University of Tennessee, Knoxville Departments of Mathematics and Ecology and Evolutionary Biology Emory University Department of Biology and Center of Disease Ecology
Outline Background - disease and control Model with system of differential equations with birth pulse: some control results Model- discrete in space and time: some control results Conclusions Funding by the National Science Foundation and NIMBioS, National Institute for Mathematical and Biological Synthesis discrete paper in Journal of Biological Dynamics, 007 pulse paper to appear in Journal of Biological Dynamics
Rabies in Raccoons Rabies is a common viral disease. Transmission through the bite of an infected animal. Raccoons are the primary terrestrial vector for rabies in eastern US. Vaccine is distributed through food baits (preventative). Medical and Economic Problem - death to humans and livestock and COSTS
Reported Cases of Rabies in Raccoons, 00 Figure: Reported Cases of Raccoon Rabies, 00, http://www.cdc.gov
Costs and Treatment associated with Rabies in USA 0,000 persons/year given rabies post exposure prophylaxis at a cost of $0 million Treatment - one dose of rabies immune globulin (injected near the site of the bite) and- five doses of vaccine over 8 days (injected into upper arm) Symptoms - flu-like at first, about 0-0 days after exposure, later delirium, coma, disruption of nervous system Vaccination and prevention cost $00 million/year In recent years, 8 million baits were distributed over Eastern states.
Deaths due to Rabies Bites by rabid dogs are the source of 0,000 human rabies death each year globally. There are about deaths per year in USA due to rabies. Most of those deaths in USA are attributed to unrecognized exposures to rabid bats. There was one case in USA of a girl (infected with rabies from a bat, not discovered until symptoms occurred) who was treated successfully by putting her into a coma. 00
Simplest optimal control problem for one ODE Find piecewise continuous control u(t) and associated state variable x(t) to maximize subject to x(0) = x 0 and x(t) free T max f (t,x(t),u(t))dt 0 x (t) = g(t,x(t),u(t))
Optimal control and Pontryagin s Maximum Principle Pontryagin and his collaborators developed optimal control theory for ordinary differential equations about 90. Pontryagin s KEY idea was the introduction of the adjoint variables to attach the differential equations to the objective functional (like a Lagrange multiplier attaching a constraint to a pointwise optimization of a function). Converted problem of finding an optimal control to maximize the objective functional subject to DE, IC to maximizing the Hamiltonian pointwise.
Existence comes first! Note that this principle gives NECESSARY Conditions: If u (t), x (t) are optimal, then the following conditions hold... The EXISTENCE of an optimal control must be proven first before using this principle. (using some type of compactness)
Pontryagin Maximum Principle If u (t) and x (t) are optimal for above problem, then there exists adjoint variable λ(t) s.t. H(t, x (t), u(t), λ(t)) H(t, x (t), u (t), λ(t)), at each time, where Hamiltonian H is defined by and H(t, x(t), u(t), λ(t)) = f (t, x(t), u(t)) + λg(t, x(t), u(t)). λ H(t, x(t), u(t), λ(t)) (t) = x λ(t) = 0 transversality condition
Using Hamiltonian Generate these Necessary conditions from Hamiltonian H(t,x,u,λ) =f (t,x,u) + λg(t,x,u) maximize H w.r.t. u at u integrand + (adjoint) (RHS of DE) if H is nonlinear wrt u without bounds on the control H u = 0 f u + λg u = 0 optimality eq. λ = H x λ = (f x + λg x ) adjoint eq. λ(t) = 0 transversality condition
First of Two Epidemic Models GOAL: Model outbreaks of rabies in raccoons considering various features MODEL with Birth Pulse and unusual feature of dynamic equation for vaccine, with systems of ordinary differential equations S susceptibles E exposeds I infecteds (able to transmit the disease) R immune (from vaccine or recovery from disease) V vaccine
Model References Coyne, M. J., Smith, G. and McAllister, F. E., 989, Mathematic model for the population biology of rabies in raccoons in the mid-atlantic states, 989, Amer. Journal of Vet. Research. D. L. Smith, B. Lucey, L. A. Waller, J. E. Childs and L. A. Real, 00, Predicting the Spatial Dynamics of Rabies Epidemics on Heterogeneous Landscapes, PNAS D. L. Smith, L. A. Waller, C. A. Russell, J. E. Childs and L. A. Real, 00, Assessing the Role of Long-Distance Translocation and Spatial Heterogeneity in the Raccoon Rabies Epidemic in Connecticut, Preventative Veterinary Medicine
State System Note birth pulse and natural death rates are included. control u(t)- input of vaccine baits. ( S = βi + b + c ) 0V S + a(s + E + R)χ Ω (t) () K + V E = βis (σ + b)e R = σ( ρ)e br + c 0VS K + V I = σρe αi V = V [c(s + E + R) + c ] + u The birth pulse acts only during the spring time of the year (March 0 to June ) and χ Ω is a characteristic function of the set Ω.
Goal with linear cost Minimize infected population as well as cost of vaccine, the objective functional is min u T 0 [I(t) + Bu(t)]dt, where the set of all admissible controls is U = {u : [0,T] [0,M ] u is Lebesgue measurable} where coefficient B is a weight factor balancing the two terms. When B is large, then the cost of implementing the control is high. NOTE existence of an optimal control can be obtained using compactness arguments.
Hamiltonian H H = (B + λ )u + I ( + λ [ βis + b + c 0V K + V + λ [βis (σ + b)e] [ + λ σ( ρ)e br + c ] 0VS K + V + λ [σρe αi] + λ [ V (c(s + E + R) + c )]. ) ] S + a(s + E + R)χ [t0,t ] ()
Adjoint System λ = H S λ = H E λ = H R λ = H I λ = H V Transversality Condition (final time boundary condition) λ i (T) = 0 ()
Optimal Control Characterization The OC may be obtained by differentiating the Hamiltonian with respect to u: At time t, H u = B + λ. For this min problem, u = 0 when H u > 0 and u = when H u < 0. Singular case: If H u = 0 on some non-empty open interval (a,b ), then λ = B and λ = λ = 0 Substitution into the respective adjoint equation, taking a first and second derivative w.r.t. time and rearranging gives a value for u. To decide if u s is optimal, use gen. Legendre-Clebsch condition.
O C Characterization if λ + B < 0 u = 0 if λ + B > 0 u s if λ + B = 0 ()
Singular Control u s = (K + V ) [βi aχ Ω (t)] () + c 0(K V ) + a (K + V ) (E + R)χ Ω (t) S + V [c(s + E + R) + c ] + (K + V )(λ λ )βi (λ λ ) + + Bc(K + V ) c 0 K(λ λ ) (b + aχ Ω(t)) Bc(K + V ) c 0 SK(λ λ ) [σρe + (b aχ Ω(t))(E + R)]
Numerical Implementation Involves an iterative method with a forward sweep of the state system with Runge Kutta followed by a backward sweep of the adjoint model with a control characterization update afterwards. The iterative method starts with a guess for the control values and then the control is updated after each forward sweep and backward sweep. The forward sweep and backward sweep are repeated until the convergence of the iterates is achieved. reference: book, Optimal Control applied to Biological Models, Lenhart and Workman, CRC Press, 007
I Infection with no birth pulse S 0 0 0 0 0 0 0 0 0 0 0 TIME E 800 700 00 00 00 00 00 00 0 0 0 0 0 TIME 7 70 0 0 0 0 0 0 0 TIME 0 0. 0... R 00 0 V 0.08 0.0 0.0 0.0 CONTROL 0.8 0. 0. 0. 0 0 0 0 0 TIME 0 0 0 0 0 0.8 0 0 0 0 0 TIME Figure: birth pulse not encountered
I Infection, weeks before birth pulse S 0 0 0 0 0 0 0 0 0 0 TIME E 800 700 00 00 00 00 00 00 0 0 0 0 0 TIME 7 70 0 0 0 0 0 0 0 TIME R 0 00 0 V 0. 0. 0. 0.08 0.0 0.0 0.0 CONTROL.. 0.8 0. 0. 0. 0 0 0 0 0 TIME 0 0 0 0 0 0.8 0 0 0 0 0 TIME Figure: Notice two control pulses
Conclusions with Pulse Model Quick response when rabies virus is detected. More vaccine required when birth pulse is encountered. Sometimes a second round of vaccine is required due to the birth pulse. Distribution of vaccine depends on number of days until birth pulse
Model discrete in time and space SIR model with discrete time t and space (k, l). Novel application of the extension of Pontryagin s Maximum Principle to discrete time and space. ALSO note that in a discrete time model, the order of events is crucial.
Basic Assumptions The objective of the problem formulation is to provide a simple, readily modified framework to analyze spatial optimal control for vaccine distribution as it impacts the spread of rabies among raccoons. The epidemiological assumptions: No variance in time from infection to death Random mixing assumed to be the only means of contact and transmission
Temporal Set-up Time scale: There is no population growth or immigration in the model presented here, but is included in a more general model. The scale is assumed to be over a time period (say within a season) over which births do not occur. Mortality occurs only due to infection. The time step of each iteration is that over which all infected raccoons die (e.g. about 0 days).
Spatial set-up Spatial scale: each cell is uniform in size, arranged rectangularly Movement: Raccoons are assumed to move according to a movement matrix from cell to cell, with distance dependence in dispersal.
Vaccine Vaccine/food packets are assumed to be reduced each time step due to uptake by raccoons, with the remaining packets then decaying due to other factors. Then additional packets (CONTROL variable) are added at the end of each time step.
Variables Model with (k,l) denoting spatial location, t time susceptibles = S(k,l,t) infecteds = I(k,l,t) immune = R(k,l,t) vaccine = v(k,l,t) control c(k, l, t), input of vaccine baits
Movement In one time step, if the box size was the size of a home range (about km ), then 9 percent of the raccoons would not leave their box. The percent moving out would be distributed inversely proportional to distance. But a raccoon could not move farther than their home range distance ( km) in one time step. If the box size is smaller, then the percentage moving is changed appropriately.
Order of events Within a time step (about a week to 0 days): First movement: using home range estimate to get range of movement. See sum S, sum I and sum R to reflect movement. Then: some susceptibles become immune by interacting with vaccine Lastly: new infecteds from the interaction of the non-immune susceptibles and infecteds NOTE that infecteds from time step n die and do not appear in time step n +.
Susceptibles and Infecteds Equations v(k,l,t) S(k,l,t + ) = ( e )sum S(k,l,t) v(k,l,t) + K v(k,l,t) ( e )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t),
Susceptibles and Infecteds Equations v(k,l,t) S(k,l,t + ) = ( e )sum S(k,l,t) v(k,l,t) + K v(k,l,t) ( e )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t), v(k,l,t) ( e )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K I(k,l,t + ) = β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t).
Immune and Vaccine Equations v(k,l,t) R(k,l,t + ) = sum R(k,l,t) + e sum S(k,l,t), v(k,l,t) + K v(k,l,t + ) = Dv(k,l,t) max [0,( e(sum S(k,l,t) + sum R(k,l,t)))] + c(k,l,t).
States and Control States: S(m, n, t), I(m, n, t), R(m, n, t), v(m, n, t) for t =,... T (given initial distribution at t = ) Control c(m, n, t), t=,,..., T- Note one fewer time component in the control.
Objective Functional maximize the susceptible raccoons, minimize the infecteds and cost of distributing baits ( ) I(m,n,T) S(m,n,T) + B c(m,n,t), m,n,t m,n where T is the final time and c(m,n,t) is the cost of distributing the packets at cell (m,n) and time t, B is the balancing coefficient, c is the control, t =,,...,T. Use discrete version of Pontryagin s Maximum Principle.
Hamiltonian at time t H(m,n,t) =B c(m,n,t) m,n + [ ( ) LS(m,n,t + ) RHS of S(m,n,t + ) eqn m,n ( ) + LI(m,n,t + ) RHS of I(m,n,t + ) eqn ( ) + LR(m,n,t + ) RHS of R(m,n,t + ) eqn ( )] + Lv(m,n,t + ) RHS of v(m,n,t + ) eqn
Adjoints and Optimal Control LS,LI,LR,Lv denote the adjoints for S,I,R,v respectively LS(i,j,t) = H(t) S(i,j,t), H(t) = Bc(i,j,t) + Lv(i,j,t + ) = 0. c(i,j,t) = c (i,j,t) = Lv(i,j,t + ), B subject to the upper and lower bounds
Numerical Iterative Method Start with a control guess and initial distribution of raccoons Solve the state equations forward Solve the adjoint equations backwards, using LI(k, l, T)=, LS(k,l, T) =-, other adjoints are zero at final time Update the control using the characterization Repeat until convergence
Disease Starts From the Corner: Initial Distribution susceptibles t= infecteds t= 0
Susceptibles, no control susceptibles susceptibles t=, no control t=, no control susceptibles susceptibles t=, no control t=, no control 0 0
Infecteds, no control 0 8 infecteds infecteds t=, no control t=, no control 0 8 infecteds infecteds t=, no control t=, no control
Susceptibles, with control, B = 0. susceptibles susceptibles t=, B=0. t=, B=0. susceptibles susceptibles t=, B=0. t=, B=0. 0 0
Infecteds, with control, B = 0. infecteds infecteds t=, B=0. t=, B=0. infecteds infecteds t=, B=0. t=, B=0... 0... 0.
Immune, with control, B = 0. immunes immunes t=, B=0. t=, B=0. immunes immunes t=, B=0. t=, B=0. 0 8 0 8
Optimal Control, B = 0. control control t=, B=0. t=, B=0. control control t=, B=0. t=, B=0. 8 8
Optimal Control, B = 0.8 control control 0. 0. 0. t=, B= t=, B= 0.8 control control 0. 0. 0. t=, B= t=, B=
Disease Starts From the Center: Initial Distribution susceptibles t= infecteds t= 0
Optimal Control, B = 0., t = control.......8... 0.8 0. 0. 0. t=, B=0. Figure: Optimal Control, B = 0., t =
Inhomogeneous Initial Distribution susceptibles t= infecteds t= 0
Optimal Control, B = 0., t = control........ 0. t=, B=0. Figure: Optimal Control, B = 0., t =
Conclusion for this Discrete Time-Space Model Developed a method and model to determine different optimal distributions of vaccine to control rabies spread; Illustrated the approach using three scenarios; Optimal bait distribution depends on the initial location of the disease outbreak and the distribution of raccoons throughout the grid; The method can be readily extended to evaluate optimal vaccination distribution strategies with other spatially heterogeneous interactions, larger spatial grids and different movement assumptions (including density dependence). Linear objective functional and see the resulting changes.
Currently, we consider PDE models (joint with Rachael Miller Neilan) Other related work Metapopulation Model, system of ODEs, with spatial regions with Asano, Gross, Real (MBE 007) ODE system model investigating the effects of birth pulses on vaccine bait distribution with Clayton AND limited vaccine and density dependent death rates Metapopulation Model with a River in the middle, by Real, Duke-Sylvester