Optimal control of culling in epidemic models for wildlife

Optimal control of culling in epidemic models for wildlife Maria Groppi, Valentina Tessoni, Luca Bolzoni, Giulio De Leo Dipartimento di Matematica, Università degli Studi di Parma Dipartimento di Scienze Ambientali, Università degli Studi di Parma CNR Milano 27 March 2012

Introduction Culling is the most common strategy to eradicate wildlife diseases when vaccination is impossible or impractical

Introduction Culling is the most common strategy to eradicate wildlife diseases when vaccination is impossible or impractical Aim: reduce the numbers of susceptible and/or infected animals below the minimum threshold for disease persistence. Possible strategies: 1 Reactive culling (Donnelly et al., Nature, 2006): intensified hunting campaigns at the onset of the epidemics (bovine tuberculosis in British badger, rabies in European fox and Canadian raccoon)

Research question When is reactive culling the optimal strategy?

Research question When is reactive culling the optimal strategy?... and when is it better to wait?

Research question When is reactive culling the optimal strategy?... and when is it better to wait? Optimal control theory applied to a SIR Model with the objective to minimize both the number of infected animals and the cost of culling effort

Mathematical model ( Ṡ = r S 1 S + I + R ) + νr βsi c(t) S, K İ = βsi (α + µ + η + c(t)) I, Ṙ = ηi (µ + c(t)) R (1)

Mathematical model Initial conditions: ( Ṡ = r S 1 S + I + R ) + νr βsi c(t) S, K İ = βsi (α + µ + η + c(t)) I, Ṙ = ηi (µ + c(t)) R S(0) = K, I(0) = 1, R(0) = 0 (1)

Mathematical model Initial conditions: ( Ṡ = r S 1 S + I + R ) + νr βsi c(t) S, K İ = βsi (α + µ + η + c(t)) I, Ṙ = ηi (µ + c(t)) R Objective functional to minimize: J(c) = S(0) = K, I(0) = 1, R(0) = 0 T 0 ( I(t) γ + Pc(t) θ) dt, γ, θ {1, 2} in the class of admissible control { } U = c(t) piecewise continuous 0 c(t) c max, t [0, T] (1) (2)

Pontryagin s Maximum principle - Existence of the optimal control follows from standard existence results (Fleming and Rishel)

Pontryagin s Maximum principle - Existence of the optimal control follows from standard existence results (Fleming and Rishel) - The characterization of the optimal control is obtained using the Pontryagin s Maximum principle: Given the problem min c U with x(t) solution of J(c) = min c U t1 t 0 f 0 (x(t), c(t)) dt, dx i dt = f i (x(t), c(t)), i = 1, 2,..., n,

Pontryagin s Maximum principle - Existence of the optimal control follows from standard existence results (Fleming and Rishel) - The characterization of the optimal control is obtained using the Pontryagin s Maximum principle: Given the problem min c U with x(t) solution of dx i dt J(c) = min c U t1 t 0 f 0 (x(t), c(t)) dt, = f i (x(t), c(t)), i = 1, 2,..., n, define the Hamiltonian function H(λ, x, c) = λ f(x, c) = n λ j f j (x, c) j=0

Characterization of the optimal control - Adjoint system dλ i dt = H x i, i = 1, 2,..., n

Characterization of the optimal control - Adjoint system dλ i dt = H x i, i = 1, 2,..., n - Transversality conditions λ i (T) = 0, i = 1, 2,..., n

Characterization of the optimal control - Adjoint system dλ i dt = H x i, i = 1, 2,..., n - Transversality conditions λ i (T) = 0, i = 1, 2,..., n - Optimality condition H c = 0

Characterization of the optimal control - Adjoint system dλ i dt = H x i, i = 1, 2,..., n - Transversality conditions λ i (T) = 0, i = 1, 2,..., n H - Optimality condition c = 0 - Try to determine c = c (x, λ) from the optimality equation H c = 0 - Substitute c in the differential equations of the Hamiltonian system made up of the model differential equations for x and of the adjoint equations for λ

SI model Quadratic costs SI epidemic model (in the limit of vanishing removal rate) ( Ṡ = r S 1 S + I ) βsi c S, K İ = βsi (α + µ + c) I, S(0) = S 0, I(0) = I 0

SI model Quadratic costs SI epidemic model (in the limit of vanishing removal rate) ( Ṡ = r S 1 S + I ) βsi c S, K İ = βsi (α + µ + c) I, S(0) = S 0, I(0) = I 0 min c(t) U T 0 [ I(t) γ + Pc(t) 2] dt, γ = 1, 2

SI model Quadratic costs SI epidemic model (in the limit of vanishing removal rate) ( Ṡ = r S 1 S + I ) βsi c S, K İ = βsi (α + µ + c) I, S(0) = S 0, I(0) = I 0 min c(t) U T 0 [ I(t) γ + Pc(t) 2] dt, γ = 1, 2 Optimal control ( c = max 0, ( ) ) min ĉ, c max, where ĉ = λ 1(t)S(t) + λ 2 (t)i(t). 2P

Linear costs J(c) = [ I(t) + Pc(t) ] dt Optimal control 0 if ψ(t) > 0 c (t) = c sing if ψ(t) = 0 c max if ψ(t) < 0, with ψ(t) = H c = P λ 1S λ 2 I switching function;

Linear costs J(c) = [ I(t) + Pc(t) ] dt Optimal control 0 if ψ(t) > 0 c (t) = c sing if ψ(t) = 0 c max if ψ(t) < 0, with ψ(t) = H c = P λ 1S λ 2 I switching function; R(S, I) c sing = singular control of order 2. Q(S, I) Generalized Legendre Clebsch condition (for the singular control to be optimal) ( 1) 2 [ ] d 4 H = c dt 4 Q 0 c (Pβ 1)SI [ ( r ) ] Q(S, I) = (α + µ + r) βs S + I K + β I.

Optimal control for fast epidemics Ṡ = βsi c S, İ = βsi (α + c) I, Theoretical results for fast epidemic model with linear costs

Optimal control for fast epidemics Ṡ = βsi c S, İ = βsi (α + c) I, Theoretical results for fast epidemic model with linear costs Theorem The optimal control c is bang-bang, µ ( { t [0, T] : ψ(t) = 0 } ) = 0. Theorem Epidemic-transient phase culling cannot occur ψ(0) = H > 0 ψ(t) = H c t=0 c > 0, t.

Optimal culling: numerical simulations quadratic costs Numerical discretization: Forward-Backward Sweep method (Lenhart 2007)

Optimal culling: numerical simulations quadratic costs Numerical discretization: Forward-Backward Sweep method (Lenhart 2007) 600 500 S I capi abbattuti 0.14 0.12 c = 0.1 c = 0 c * 0.1 400 0.08 300 c 0.06 200 0.04 100 0.02 0 0 5 10 15 t 0 0 5 10 15 t Figure: T = 16 years. Initial conditions S(0) = K, I(0) = 1.

Optimal culling linear costs 600 500 S I capi abbattuti 0.14 0.12 c = 0.1 c = 0 c * 400 300 c 0.1 0.08 0.06 200 0.04 100 0.02 0 0 5 10 15 t 0 0 5 10 15 t Figure: T = 16 years. Initial conditions S(0) = K, I(0) = 1.

Sensitivity analysis linear costs The optimal control depends on the parameters of the model

Sensitivity analysis linear costs The optimal control depends on the parameters of the model - Extensive numerical study by varying R 0, culling cost P, virulence parameters α, β

Sensitivity analysis linear costs c c The optimal control depends on the parameters of the model - Extensive numerical study by varying R 0, culling cost P, virulence parameters α, β - Classification of results according to reactive culling ( ), epidemic-transient phase culling ( ), or no culling ( ) 0.1 0.1 ( ) 0 0 t ( ) 0 0 t

Numerical results SI Variation of α and P 18 16 14 12 10 8 6 4 2 0 2 40 50 60 70 80 90 100 110 120 P Figure: T = 6 years. α (0 α 16, step 0.5), P (40 P 120, step 2) and β = R 0 α+µ K with R 0 = 4.6154.

Numerical results SI Variation of R 0 and P 5 4.5 4 3.5 R 0 3 2.5 2 1.5 1 40 50 60 70 80 90 100 110 120 P Figure: T = 5 years. P (40 P 120 step 2), R 0 (1.2 R 0 5, step 0.1) and β = R 0 α+µ K.

Effect of immunity on optimal culling SIR epidemic model ( Ṡ = r S 1 S + I + R ) + νr βsi c(t) S, K İ = βsi (α + µ + η + c(t)) I, Ṙ = ηi (µ + c(t)) R, S(0) = K, I(0) = 1, R(0) = 0.

Effect of immunity on optimal culling SIR epidemic model ( Ṡ = r S 1 S + I + R ) + νr βsi c(t) S, K İ = βsi (α + µ + η + c(t)) I, Ṙ = ηi (µ + c(t)) R, S(0) = K, I(0) = 1, R(0) = 0. Objective cost function to minimize J(c) = T 0 ( I(t) + Pc(t) ) dt, 0 c(t) cmax. Hamiltonian functional: r ( ) H =I + Pc + λ 1 ν(s + R) λ 1 S + I + R S λ1 (µ + c)s K λ 1 βsi + λ 2 βsi λ 2 (α + µ + η + c) I + λ 3 ηi λ 3 (µ + c) R.

Numerical results SIR Variation of η and P 8 7 6 5 4 3 2 1 0 1 30 40 50 60 70 80 90 100 110 120 P Figure: T = 1.5 years. P (30 P 120, step 2), η (0 η 7, step 0.25) and β = R 0 α+µ+η K with R 0 = 4.6154.

Numerical results SIR Variation of R 0 and P 5 4.5 4 R 0 3.5 3 2.5 2 30 40 50 60 70 80 90 100 110 120 P Figure: T = 2.5 years. P (30 P 120, step 2), R 0 (2 R 0 5 step 0.1) and β = R 0 α+µ+η K (η = 1, α = 4).

Conclusions Reactive culling when

Conclusions Reactive culling when Fast course of epidemics

Conclusions Reactive culling when Fast course of epidemics Low culling costs

Conclusions Reactive culling when Fast course of epidemics Low culling costs High immunity

Conclusions Reactive culling when Fast course of epidemics Low culling costs High immunity Wait when

Conclusions Reactive culling when Fast course of epidemics Low culling costs High immunity Wait when High virulence

Conclusions Reactive culling when Fast course of epidemics Low culling costs High immunity Wait when High virulence High culling costs

References L. Bolzoni, G. A. De Leo, A cost analysis of alternative culling strategies for the eradication of classical swine fever in wildlife, Environment and Development Economics 12, 653-671 (2007). S. Lenhart, J. T. Workman, Optimal control applied to biological models, Chapman & Hall /CRC Mathematical and Computational Biology Series, Boca Raton, 2007. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers, Los Angeles, 1962. W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control, Springer-Verlag, New York, 1975.