An exact approach to calibrating infectious disease models to surveillance data: The case of HIV and HSV-2

Transcription

1 An exact approach to calibrating infectious disease models to surveillance data: The case of HIV and HSV-2 David Gerberry Department of Mathematics Xavier University, Cincinnati, Ohio Annual Meeting of the Society for Mathematical Biology University of Utah, Salt Lake City July 19, / 26

2 Model calibration Exact approach Outline of talk Motivation Public health modeling papers often include calibrating a model to disease surveillance data. Williams et al., PLoS Medicine 3(7): e262 Common description: model was fit to data from... If you re lucky: see Supplementary Material for... 2 / 26

3 Model calibration Exact approach Outline of talk Common forms of model calibration Manual calibration - play around with parameter values until things look good Sample and filter - randomly sample parameter sets and filter out those that don t fit the data Sophisticated things - numerical fitting algorithms - statistical approaches 3 / 26

4 Model calibration Exact approach Outline of talk Calibration to HIV prevalence in South Africa (toy example) HIV Prevalence (%) Year S = Λ βsi µs I = βsi µi Goal: calibrate to HIV prevalence at EE of 18% 4 / 26

5 Model calibration Exact approach Outline of talk Calibration to HIV prevalence in South Africa (toy example) HIV Prevalence (%) Year S = Λ βsi µs I = βsi µi Goal: calibrate to HIV prevalence at EE of 18% Algebra: HIV prevalence at EE for model is P = 1 µ/β 4 / 26

6 Model calibration Exact approach Outline of talk Calibration to HIV prevalence in South Africa (toy example) HIV Prevalence (%) Year S = Λ βsi µs I = βsi µi Goal: calibrate to HIV prevalence at EE of 18% Algebra: HIV prevalence at EE for model is P = 1 µ/β Calibration: using β as calibration parameter, β = µ/(1 0.18) matches exactly for any µ 4 / 26

7 Model calibration Exact approach Outline of talk The plan from here How do we do this for more complicated models? Gerberry, Mathematical Biosciences and Engineering, 15(1). SI-type HIV/HSV-2 coinfection model for HIV (with behavioral response) Modified to include HIV/HSV-2 coinfection Granich et al. HIV-only model, including transient features Approximate calibration for Granich et al. HIV/HSV-2 coinfection model, including transient features 5 / 26

8 Model calibration Exact approach Outline of talk The plan from here How do we do this for more complicated models? Gerberry, Mathematical Biosciences and Engineering, 15(1). SI-type HIV/HSV-2 coinfection model for HIV (with behavioral response) Modified to include HIV/HSV-2 coinfection Granich et al. HIV-only model, including transient features Approximate calibration for Granich et al. HIV/HSV-2 coinfection model, including transient features 5 / 26

9 SI-type HIV/HSV-2 coinfection model βs(a+r2c) N Λ S A σs(h+r3c) N σa(h+r3c) N H µs µh C r1βh(a+r2c) N (µ+µ A )A (µ+µ A )C S = Λ βs(a + r 2C) N H = σs(h + r 3C) N A = βs(a + r 2C) N C = r 1βH(A + r 2 C) N σs(h + r 3C) N r 1βH(A + r 2 C) N σa(h + r 3C) N + σa(h + r 3C) N µs µh (µ + µ A )A (µ + µ A )C S = number of susceptible individuals β, σ = transmission coefficients for HIV, HSV-2 H = number of individuals infected with HSV-2 only r 1, r 2, r 3 = disease interaction cofactors A = number of individuals infected with HIV only µ, µ A = natural and HIV-induced mortality C = number of individuals coinfected with HIV and HSV-2 Goal: calibrate to endemic HIV and HSV-2 prevalences of  and Ĥ. 6 / 26

10 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system 7 / 26

11 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Two calibration conditions, Â and Ĥ need two fit parameters. Transmission coefficients α and σ are natural choice. 7 / 26

12 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Let s = S N, h = H N, a = A N and c = C N. Note that s = 1 h a c. System reduces to h = σ(1 h a c)(h + r 3 c) r 1 βh(a + r 2 c) µh a = β(1 h a c)(a + r 2 c) σa(h + r 3 c) (µ + µ A )a c = r 1 βh(a + r 2 c) + σa(h + r 3 c) (µ + µ A )c 7 / 26

13 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Equilibrium conditions (i.e. h = a = c = 0) 0 = σ (1 h a c )(h + r 3 c ) r 1 β h (a + r 2 c ) µh 0 = β (1 h a c )(a + r 2 c ) σ a (h + r 3 c ) (µ + µ A )a 0 = r 1 β h (a + r 2 c ) + σ a (h + r 3 c ) (µ + µ A )c 7 / 26

14 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Calibration conditions (i.e. HIV and HSV-2 prevalences of Â, Ĥ) 0 = a + c  0 = h + c Ĥ 7 / 26

15 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Calibration system = equilibrium conditions + calibration conditions 0 = σ (1 h a c )(h + r 3 c ) r 1 β h (a + r 2 c ) µh 0 = β (1 h a c )(a + r 2 c ) σ a (h + r 3 c ) (µ + µ A )a 0 = r 1 β h (a + r 2 c ) + σ a (h + r 3 c ) (µ + µ A )c 0 = a + c  0 = h + c Ĥ 7 / 26

16 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Newton s Method in n-dimensions: x n+1 = x n [Df ( x n )] 1 f ( x n ) Fast convergence to x if initial approximation x 0 is close enough to x. 7 / 26

17 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system Assume independence to get good enough (h 0, a 0, c 0, β 0, σ 0 ) c 0 = ÂĤ, a 0 =  ÂĤ, h 0 = Ĥ ÂĤ Solve 1 st and 3 rd equilibrium conditions to get σ 0 = µ A + µ (1 Ĥ) (1 (1 r 3) Â), β 0 = µ A (1 (1 r 2 ) Ĥ) r 1 7 / 26

18 Does it work? Parameter ranges for SI-type model calibration Model Parameters Symbol Baseline Range Average time in sexually active population (years) 1/µ Recruitment rate into the sexually active population (/yr) Λ Diseased-induced mortality due to HIV infection (/yr) µ 1 1 A 20, Cofactor for increased HIV susceptibility due to HSV-2 infection r Cofactor for increased HIV infectivity due to HSV-2 infection r Cofactor for increased HSV-2 infectivity due to HIV infection r Calibration Conditions HIV prevalence at endemic equilibrium (%) Â 17, HSV-2 prevalence at endemic equilibrium (%) Ĥ 50, Calibration Parameters Transmission coefficient for HIV β determined by calibration Transmission coefficient for HSV-2 σ determined by calibration 8 / 26

19 Does it work? Randomly sampled model parameters and calibration conditions Sampled parameter values and calibration conditions Calibration results j Λ µ µ A r 1 r 2 r 3 Â Ĥ β σ Â Ĥ m / 26

20 for HIV S µs e P n SI N I 1 I 2 I 3 I 4 I 1 I 2 I 3 I 4 µi 1 µi 2 µi 3 µi 4 S = number of susceptible individuals I 1, I 2, I 3 = number of individuals infected with HIV Behavioral response to HIV e αpn n = 30 n = 9 n = 5 n = 3 n = 2 n = 1.5 n = P, HIV prevalence ds dt = Λ λe αpn SI µs N di 1 dt = λe αpn SI ρi 1 µi 1 N di 2 dt = ρi 1 ρi 2 µi 2 di 3 dt = ρi 2 ρi 3 µi 3 di 4 dt = ρi 3 ρi 4 µi 4 where P = (I 1 + I 2 + I 3 + I 4 )/N 10 / 26

21 for HIV S µs e P n SI N I 1 I 2 I 3 I 4 I 1 I 2 I 3 I 4 µi 1 µi 2 µi 3 µi 4 S = number of susceptible individuals I 1, I 2, I 3 = number of individuals infected with HIV Behavioral response to HIV e αpn α = 50 α = 15 α = 7 α = 3 α = 2 α = 1 α = P, HIV prevalence ds dt = Λ λe αpn SI µs N di 1 dt = λe αpn SI ρi 1 µi 1 N di 2 dt = ρi 1 ρi 2 µi 2 di 3 dt = ρi 2 ρi 3 µi 3 di 4 dt = ρi 3 ρi 4 µi 4 where P = (I 1 + I 2 + I 3 + I 4 )/N 10 / 26

22 for HIV and HSV-2 S H µs S r 1 H µa 1 µa 2 A 1 A 2 µa 3 µa4 A A 1 A 2 A 3 A 3 A 4 4 S A 1 A 2 A 3 A 4 µh Ω = λe αpn A + r 2C N C C 1 1 C C 2 2 C C 3 3 C C 4 4 µc 1 µc 2 µc 3 µc 4 Ψ = κe αpn H + r 3C N Goal: calibrate to endemic HIV and HSV-2 prevalences of  and Ĥ. ds = Λ ΩS ΨS µs, dt dh dt = ΨS r 1ΩH µh, da 1 = ΩS ΨA 1 ρa 1 µa 1, dt da 2 = ρa 1 ΨA 2 ρa 2 µa 2, dt da 3 = ρa 2 ΨA 3 ρa 3 µa 3, dt da 4 = ρa 3 ΨA 4 ρa 4 µa 4, dt dc 1 = r 1 ΩH + ΨA 1 ρc 1 µc 1, dt dc 2 = ρc 1 + ΨA 2 ρc 2 µc 2, dt dc 3 = ρc 2 + ΨA 3 ρc 3 µc 3, dt dc 4 = ρc 3 + ΨA 4 ρc 4 µc 4, dt 11 / 26

23 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters 2 Rescale state variables and reduce dimension 3 Create calibration system of equations 4 Use Newton s Method to solve calibration system 12 / 26

24 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 1 Choose fitting parameters: Ω = λe A + r 2C αpn N, Ψ = H + r 3C κe αpn N use λ and κ to satisfy two calibration conditions 13 / 26

25 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 2 Rescale state variables and reduce dimension Let s = S N, h = H N, a i = A i N and c i = C i N Note that s = 1 h a c = 1 h i a i i c i. System reduces to h = Ψ(1 h a c) r 1 Ωh µh, a 1 = Ω(1 h a c) Ψa 1 ρa 1 µa 1, a 2 = ρa 1 Ψa 2 ρa 2 µa 2, a 3 = ρa 2 Ψa 3 ρa 3 µa 3, a 4 = ρa 3 Ψa 4 ρa 4 µa 4, c 1 = r 1 Ωh + Ψa 1 ρc 1 µc 1, c 2 = ρc 1 + Ψa 2 ρc 2 µc 2, c 3 = ρc 2 + Ψa 3 ρc 3 µc 3, c 4 = ρc 3 + Ψa 4 ρc 4 µc 4, 13 / 26

26 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 3 Create calibration system of equations 0 = Ψ(1 h a c) r 1 Ωh µh, 0 = Ω(1 h a c) Ψa 1 ρa 1 µa 1, 0 = ρa 1 Ψa 2 ρa 2 µa 2, 0 = ρa 2 Ψa 3 ρa 3 µa 3, 0 = ρa 3 Ψa 4 ρa 4 µa 4, 0 = a 1 + a 2 + a 3 + a 4 + c 1 + c 2 + c 3 + c 4 Â, 0 = r 1 Ωh + Ψa 1 ρc 1 µc 1, 0 = h + c 1 + c 2 + c 3 + c 4 Ĥ 0 = ρc 1 + Ψa 2 ρc 2 µc 2, 0 = ρc 2 + Ψa 3 ρc 3 µc 3, 0 = ρc 3 + Ψa 4 ρc 4 µc 4, 13 / 26

27 Calibration to HIV and HSV-2 prevalence at EE Calibration steps 4 Use Newton s Method to solve calibration system Assume independence to get good enough initial approximation h 0 = Ĥ(1 Â) a 1,0 = a 2,0 = a 3,0 = a 4,0 = Â(1 Ĥ) 4, c 1,0 = c 2,0 = c 3,0 = c 4,0 = ÂĤ 4 Solve 1 st and 3 rd equilibrium conditions to get a 1,0 (µ + ρ a 0 µ a 0 ρ c 0 µ c 0 ρ h 0 ρ) e αân λ 0 = ( a 1,0 h 0 r 1 +a a 0 c 0 +2a 0 h 0 +c c 0 h 0 +h 2 0 2a 0 2c 0 2h 0 +1) (r 2 c 0 +a 0 ) h 0 ( a 1,0 µr 1 a 1,0 r 1 ρ + a 0 µ + c 0 µ + µh 0 µ) e αân κ 0 = ( a 1,0 h 0 r 1 +a a 0 c 0 +2a 0 h 0 +c c 0 h 0 +h 2 0 2a 0 2c 0 2h 0 +1) (r 3 c 0 +h 0 ) 13 / 26

28 Does it work? Randomly sampled model parameters and calibration conditions Sampled parameters and calibration condition Calibration results j Λ µ ρ α n  Ĥ λ κ  Ĥ m / 26

29 Including the transient phase Obvious weakness everything calibrated to equilibrium What about transient dynamics? 15 / 26

30 Including the transient phase Obvious weakness everything calibrated to equilibrium What about transient dynamics? Prevalence Calibration # Incidence Prevalence Calibration # Incidence Year Year Goal: calibrate to  = HIV prevalence at endemic equilibrium Π= peak HIV incidence P = HIV prevalence at time of peak incidence 15 / 26

31 using transient features Calibration steps 1 Choose fitting parameters: Ω = λe αpn A + r 2C N use λ, α and n to satisfy three calibration conditions 16 / 26

32 using transient features Calibration steps 2 Rescale state variables and reduce dimension Let s = S N, i 1 = I 1 N, i 2 = I 2 N, i 3 = I 3 N, i 4 = I 4 N. Note that s = 1 i 1 i 2 i 3 i 4. System reduces to i 1 = λe αin (1 i)i ρi 1 µi 1 i 2 = ρi 1 ρi 2 µi 2 i 3 = ρi 2 ρi 3 µi 3 i 4 = ρi 3 ρi 4 µi 4 16 / 26

33 using transient features Calibration steps 3 Create calibration system of equations 0 = λe αin (1 i)i ρi 1 µi 1 0 = ρi 1 ρi 2 µi 2 0 = ρi 2 ρi 3 µi 3 0 = ρi 3 ρi 4 µi 4 0 = i 1 + i 2 + i 3 + i 4 Â 0 = λe α P n (1 P) P Î Incidence: λe αin (1 i)i d di (λe αin (1 i)i) i= P = λe α P n ( α P n n (1 P) P) 0 = 1 α P n n (1 P) 2 P 16 / 26

34 using transient features Calibration steps 4 Use Newton s Method to solve calibration system i 1,0 = i 3,0 = Â (ρ + µ) 3 (2 ρ + µ) (2 ρ ρ µ + µ 2 ), i (ρ + µ) 2 ρ 2,0 = Â (2 ρ + µ) (2 ρ ρ µ + µ 2 ) (ρ + µ) ρ 2 Â (2 ρ + µ) (2 ρ ρ µ + µ 2 ), i ρ 3 Â 4,0 = (2 ρ + µ) (2 ρ ρ µ + µ 2 ), n 0 = numerical solution to e (1 2 P)((Â/ P) n 1) n(1 P) i 1(ρ + µ)(1 P) P (1 Â)ÂÎ = P α 0 =, λ 0 = eαân0 (ρ + µ)i 1,0 n 0 Pn 0(1 P) (1 Â)Â. 16 / 26

35 Does it work? Randomly sampled model parameters and calibration conditions Sampled parameters and calibration condition Calibration results j Λ µ ρ  ΠP λ α n  ΠP m E E E E E E E E E E E E E E E E E E E E / 263

36 Calibration success Calibration #1 Prevalence Incidence Year 18 / 26

37 Calibration success Calibration #2 Prevalence Incidence Year 18 / 26

38 Calibration success Calibration #3 Prevalence Incidence Year 18 / 26

39 Calibration success Calibration #4 Prevalence Incidence Year 18 / 26

40 Calibration success Calibration #5 Prevalence Incidence Year 18 / 26

41 Calibration success Calibration #6 Prevalence Incidence Year 18 / 26

42 Calibration success Calibration #7 Prevalence Incidence Year 18 / 26

43 Calibration success Calibration #8 Prevalence Incidence Year 18 / 26

44 Calibration success Calibration #9 Prevalence Incidence Year 18 / 26

45 Calibration success Calibration #10 Prevalence Incidence Year 18 / 26

46 This approach complements other types of calibration Sample and filter - randomly sample for non-calibration parameters - use exact approach to get corresponding values for calibration parameters - filter out sets with infeasible values for calibration parameters # of samples filtered out will be greatly reduced Fitting algorithms - performance often depends on quality of initial approximation exact approach can be used to get such an initial approximation Calibrating to endemic equilibrium is practical limitation, not technical nothing special about setting a = h = c = 0, for example 19 / 26

47 Extra slides 20 / 26

48 Does it work? Calibrating for HIV and HSV-2 in South Africa and the UK i β i σ i  i Ĥ i precision i South Africa (HIV prevalence 17%, HSV-2 prevalence 50%) United Kingdom (HIV prevalence 0.3%, HSV-2 prevalence 4%) precision i = f (h i,a i,c i,σ i,β i ) 1 5 stopping criterion: max{ β i β i 1, σ i σ i 1 } < / 26

49 Parameter ranges for Granich et al. HIV/HSV-2 calibration Model Parameters Symbol Baseline Range R Background mortality rate (/yr) µ Recruitment rate into the population (/yr) Λ Location of HIV transmission term α Shape of HIV transmission term n Rate of HIV progression (/yr) ρ Calibration Conditions HIV prevalence at endemic equilibrium (/yr) Â HSV-2 prevalence at endemic equilibrium (/yr) Ĥ [? Calibration Parameters Initial value of HIV transmission term (/yr) λ determined by calibration Initial value of HSV-2 transmission term (/yr) κ determined by calibration 22 / 26

50 Parameter ranges for Granich et al. transient calibration Model Parameters Symbol Baseline Range Background mortality rate (/yr) µ Recruitment rate into the population (/yr) Λ Rate of HIV progression (/yr) ρ Calibration Conditions HIV prevalence at endemic equilibrium  Peak HIV incidence Î 10% 50% of  HIV prevalence at time of peak HIV incidence P 5% 95% of  Calibration Parameters Initial value of transmission term (/yr) λ determined by calibration p Location of transmission term α determined by calibration p Shape of transmission term n determined by calibration p 23 / 26

51 Something disturbing Randomly sampled model parameters and calibration conditions Sampled parameters and calibration condition Calibration results j Λ µ ρ  ΠP λ α n  ΠP m E E E E E E E / 26

52 The offending calibrations Calibration #4 Prevalence Incidence Year 25 / 26

53 The offending calibrations Calibration #10 Prevalence Incidence Year 25 / 26

54 The offending calibrations Calibration #20 Prevalence Incidence Year 25 / 26

55 The offending calibrations Calibration #30 Prevalence Incidence Year 25 / 26

56 The offending calibrations Calibration #33 Prevalence Incidence Year 25 / 26

57 The offending calibrations Calibration #40 Prevalence Incidence Year 25 / 26

58 The offending calibrations Calibration #46 Prevalence Incidence Year 25 / 26

59 Behavioral response for offending calibrations Late-peaking epidemic delayed, but severe behavioral response response function step function large value of α 26 / 26