Mathematical Models of Infectious Disease: Very different approaches for very different diseases

Transcription

1 Mathematical Models of Infectious Disease: Very different approaches for very different diseases David J. Gerberry Department of Mathematics, Purdue University Integrative Computational Studies Seminar Series April 24, 2009

2 Outline Model for Childhood Diseases e.g. chicken pox, measles, mumps, rubella, scarlet fever Motivated by recurrent outbreaks (periodic solutions) More theoretical in nature Model for Tuberculosis Motivated by the issue of BCG vaccination for tuberculosis More applied in nature Campaign Promises: Lots of pictures, few equations, life or death issues

3 What causes periodic outbreaks in childhood diseases? * From Stone, Olinky & Huppert, Nature 2007 Various explanations: - autonomous internal forces, stochastic effects, external forcing, school year, age structure Feng& Thieme(Mathematical Biosciences 1995) showed that including isolation into a deterministic autonomous system can produce sustained oscillations. - unsatisfactory agreement with real data - neglects the latent period

4 SEIQR Model for Childhood Diseases Λ S σs I A E γ 1 E I γ 2 I Q ξq R µs µe µi µq µr d dt S = Λ µs σs I A d dt E = (µ + γ 1)E + σs I A d dt I = (µ + γ 2)I + γ 1 E d dt Q = (µ + ξ)q + γ 2I d R = µr + ξq dt S =Susceptible E =Exposed I = Infectious Q = Isolated (Quarantined) R = Recovered A = Active Population (S + E + I + R) Λ =RecruitmentRate µ = Death Rate σ =InfectionRate 1/γ 1 = Ave. Length of Latent Period 1/γ 2 =Ave.LengthofInfectiousPeriod 1/ξ =Ave.LengthofIsolationPeriod

5 Model Simplification Total population size goes to µ as time goes to infinity Λ Assume that total population is equal to µ Eliminate S from our system Λ Rescale time so that New parameters: σ =1 ν = µ σ, θ 1 = γ 1 σ, θ 2 = γ 2 σ, ζ = ξ σ

6 Basic Reproductive Number: Initial Findings R 0 - mean number of secondary infections caused by a single infection in a completely susceptible population R 0 < 1 = disease goes extinct R 0 = R 0 > 1 = disease will persist γ 1 σ (γ 1 +µ)(γ 2 +µ) = θ 1 (θ 1 +ν)(θ 2 +ν) Disease-free equilibrium is unstable for R 0 > 1 DFE is globally asymptotically stable for R 0 < 1 Endemic Equilibrium: Exists in a biologically feasible region when R 0 > 1

7 Hopf Bifurcations Have proven the existence of two Hopf bifurcations. Region in between exhibits sustained oscillations.

8 Hopf Bifurcations Supercritical Hopf bifurcations occur at: ζ 0 (ν) = θ 1θ 2 2 (1 θ 2)(θ 1 +θ 2 ) θ 2 2 (θ 1+1)+θ 1 θ 2 +θ O(ν 1/2 ) and ζ 1 (ν)= r ³ 1 θ θ 1 (1 θ 2 )2 2 (θ 1 +θ 2 ) 2 + θ1 (1 θ 2 ) 2 2 ³ (θ 1 +θ 2 ) 2 1 4θ 2 θ1 (1 θ 2 ) 2 θ 2 1 θ 2 (θ 1 +θ 2 ) 2 1 θ 2 θ 2 2θ 2 ν + O(ν 3/2 ) 1 θ 2

9 Hopf Bifurcations (Cont d) sp sp HB uss HB sss R/A sss Ave. isolation period in days

10 Sketch of the Proof of the Hopf Bifurcation 1. Linearize about endemic equilibrium. 2. Observe that ν θ 1, θ 2, ζ. 3. Note that the coefficients of the characteristic polynomial of the Jacobian are analytic in ν for ν > δ. 4. Compute the roots of the characteristic polynomial when ν =0. 5. Use these to compute the first two terms in the series expansions for each root in ν. 6. Higher order terms can be omitted since ν is very small. 7. w 1 = ζ + O(ν), w 2 = (θ 1 + θ 2 )+O(ν), w 3,4 = ±ai + bν + O(ν 2 ). 8. b goes from negative to positive as ζ becomes smaller than ζ 0 (ν).

11 Parameter Space ζ ζ 0 (ν) θ 2 periodic behavior θ 1 ζ 1 (ν)

12 Intersection of ζ 0 (ν) and ζ 1 (ν) ζ 0 (ν) ζ periodic behavior ζ 1 (ν) θ 2

13 Homoclinic Bifurcation of nearby System Recall that ν θ 1, θ 2, ζ Assume that ν =0 Assume that R 0 = 1 (i.e. θ 2 =1) Center Manifold Theory Normal Form Theory Resulting system: u = v + O(3) v = θ 1 uv + O(3) θ 1 +1

14 Homoclinic Bifurcation of nearby System (Cont d) Biologically reasonable unfolding: u = µ 1 u + v v =(µ 2 µ 1 )v + u 2 uv Hopf bifurcation occurs along H = (µ 1,µ 2 ):µ 2 = µ 2 1,µ 1 > 0 ª Homoclinic bifurcation occurs along HC = (µ 1,µ 2 ):µ 2 = 6 7 µ2 1 + O(µ3 1 ),µ 1 > 0 ª

15 Homoclinic Bifurcation of nearby system (Cont d)

16 Comparison to Scarlet Fever Data Scarlet fever data from England and Wales, Assume: Expect: Mean age at infection = 12 yrs Average life expectancy = 65 yrs Average length of latent period = 1.5 days Average length of effective infectious period = 1.5 days Average length of isolation period between 14 days and 21 days R 0 =6.4 R 0 =7.7 without latent class 22.8 days < D Q < 1.83 yrs 26.7 days < D Q < 1.22 yrs with latent class 18.0 days < D Q < 2.23 yrs 20.8 days < D Q < 1.52 yrs

17 Comparison to Scarlet Fever Data (Cont d) Historical data shows observed interepidemic periods of 3 6 years Without latent class With latent class 6 yrs 3 yrs 27 days < D Q < 57 days 20 days < D Q < 28 days

18 Other Childhood Diseases 1 1 Disease γ 1 γ 2 R 0 Feasible Isolation min(days) max(days) Scarlet Fever days Measles days Mumps days Rubella days Varicella days Parameters used and the isolation periods required to produce sustained oscillations. Scarlet fever is unique among childhood diseases. Isolation is not an important factor for most childhood diseases.

19 Synopsis of the Work Childhood Disease Modeling The Good Lots of nice mathematics Cool ideas The Bad Most of the coolest ideas weren t mine The ugly Nobody really cares Tuberculosis Modeling The Good Data-based modeling Most ideas are mine Contemporary issue in public health The Bad Less interesting mathematically The ugly Will anyone care??

20 Why Tuberculosis? 1/3 of world s population is latently infected with TB TB causes 2 million deaths per year worldwide 2 nd leading cause of global death from infectious disease TB accounts for 1 in 4 preventable adult deaths in the world TB/HIV co-infection is a lethal combination TB causes as much as half of all AIDS-related deaths Estimated that 1/3 of the HIV + population is co-infected with TB Emergence of drug-resistant strains of TB 20

21 Tuberculosis Basics 1 or 2 years lifetime New Infection 95% 5% 5% Latent TB Active TB Active TB slow progression fast progression 21

22 Bacille Calmette-Guérin (BCG) Vaccine Old, inexpensive, safe and well tolerated Highly variable efficacy repeated studies in the UK (60-80%) UK study from (84%) Georgia/Alabama study in 1966 (14%) India in 1979 (0%) Meta-analysis in 2000 (71-83%) Protective efficacy may wane with time (15-55 years of protection) Can interfere with the detection of Latent TB 22

23 Research Question The use of BCG vaccine has been limited because a) its effectiveness in preventing infectious forms of TB is uncertain and b) the reactivity to tuberculin that occurs after vaccination interferes with the management of persons who are possibly infected with M. tuberculosis. - CDC (U.S.) Recommendations and Reports, 1996 treating LTB BCG efficacy Don t Vaccinate treating LTB BCG efficacy Vaccinate Can we establish conditions which justify the discontinuation of BCG vaccination?

24 Should everyone be vaccinated with BCG? 12 of 30 European nations considering changes (as of 2005) 24 Images:

25 TB Model with No Treatment (1 p)βsi S E (1 c)π νe+θ 1 pβei µ T I ωv pβvi pβsi γi I R V cπ (1 p)βvi E V νe V +θ 1 pβe V I (1 p)θ 2 βri pθ 2 βri S susceptible V vaccinated E latently infected (exposed) E V vacc. and latently infected I actively infected (infectious) R recovered π recruitment rate µ natural death rate ( 1/75) 1/ω ave. length of vaccine efficacy 1/ν ave. duration of latent infection 1/γ ave. duration of active infection µ T disease induced death rate c vaccine coverage β transmission rate p prob. of fast progression to Active TB (.05) θ 1 factor of susceptibility to exogenous infection (latent) θ 2 factor of susceptibility to re-infection (recovered) 25

26 TB Model including Vaccine Efficacies V p(1 q 2 )(1 q 1 )βvi [1 p(1 q 2 )](1 q 1 )βvi E V (1 q 3 )νe V +θ 1 pβe V I I q 1 efficacy in preventing initial infection q 2 efficacy in preventing fast progression q 3 efficacy in preventing slow progression 0 q 1,q 2,q

27 TB Model including Treatment E E V I T treated Active TB (1 r)(νe+θ 1 pβei) I T c untreated Active TB r(νe+θ 1 pβei) (1 d)( ) d( ) I (1 s)γ 1 I T Treatment Indicators d detection rate (Active TB) s treatment success rate (Active TB) r detection/treatment success rate (Latent TB) estimated by WHO I Tc d/2 I T c I T Parameters γ 2 I T c sγ 1 I T 1/γ 1 ave. duration of treatment ( few weeks) R 1/γ 2 ave. duration of untreated infection 27

28 The model all the bells and whistles (almost) 1 c recruitment into the population c 1 c π c S infection vaccine waning infection* p 1 p 1 p* p* V S βsi/n ωv (1 q 1 ) βsi/n p 1 p 1 p(1 q 2 ) p(1 q 2 ) V fast progression E E V fast progression E E V slow progression exogenous reinfection exogenous reinfection νe pθ 1 βei/n pθ 1 βe V I/N r 1 r slow* progression r 1 r (1 q 3 )νe all new cases of active TB d 1 d p 1 p all new cases of active TB d 1 d p 1 p I T detection of TB (d/2) end treatment R s 1 s I Tc natural recover y TB induced mortality reinfection I T γ 1 I T (d/2) I Tc 1 s s R I Tc γ 2 I Tc μ T I Tc θ 2 βri/n

29 TB Model including Immigration (1 c)π(1 α) cπ(1 α) απ(1 φ)(1 ψ) S V απφ(1 ψ) απ(1 φ)ψ E E V απφψ Immigration Parameters α percentage of recruitment rate due to immigration φ percentage of immigrants that are vaccinated ψ percentage of immigrants that have Latent TB 29

30 The Model S 0 =(1 α)(1 c)πα(1 φ)(1 ψ)π + ωv βsi µs V 0 = c(1 α)π + απ(1 ψ)φ ωv (1 q 1 )βvi µv E 0 =(1 p)βsi + απ(1 φ)ψ (νe + θ 1 pβei) µe E 0 v =(1 p(1 q 2 ))(1 q 1 )βvi+ απψφ +(1 p)θ 2 βri ((1 q 3 )νe v + θ 1 pβe v I) µe v IT 0 = d[(1 r)(νe + θ 1 pβei)+pβsi + p(1 q 2 )(1 q 1 )βvi +(1 q 3 )νe v + θ 1 pβe v I + pθ 2 βri] γ 1 I T µi T + d/2 I T c IT 0 =(1 d)[(1 r)(νe + θ 1pβEI)+pβSI + p(1 q c 2 )(1 q 1 )βvi +(1 q 3 )νe v + θ 1 pβe v I + pθ 2 βri]+(1 s)γ 1 I T γ 2 I T c (µ + µ T )I T c d/2 I T c R 0 = sγ 1 I t + γ 2 I T c + r(νe + θ 1 pβei) θ 2 βri µr 30

31 Basic Reproductive Number for Compartment Models Diekmann & Heesterbeek - Mathematical Epidemiology of Infectious Diseases Probabilities of New Infection: Transition Probabilities: G = ~Θ = (1 p)[(1 c)µ+ω] (1 cq 1 )µ+ω [1 p(1 q 2 )](1 q 1 )µc (1 cq 1 )µ+ω dp[µ(1 c)+ω+(1 q 1 )(1 q 2 )µc] (1 cq 1 )µ+ω (1 d)p[µ(1 c)+ω+(1 q 1 )(1 q 2 )µc] (1 cq 1 )µ+ω " ν µ µ (1 q 3 )ν 0 0 d(1 r)ν d(1 q 3 )ν γ 1 µ 1/2d (1 d)(1 r)ν (1 d)(1 q 3 )ν (1 s)γ 1 1/2d γ 2 µ µ T # Infectivity: ~ h T =[00 β(µ µcq 1 +ω) µ+ω β(µ µcq 1 +ω) µ+ω ] R 0 = ~ h G 1 ~ Θ

32 Basic Reproductive Number (with No Immigration) where R 0 = A + Bc A = β((1 r+rp)ν+pµ) (µ+µ T +γ)(µ+ν) B = [(µ+ν)( pµ+µpq 2+νq 3 ν)q 1 +ν(p 1)(µ+rν)q 3 (µ+ν)(µpq 2 +rν(p 1))]βµ (µ+ω)(µ+ν)(µ+µ T +γ)( νq 3 +ν+µ)

33 Vaccine Usefulness Vaccine reduces R 0 when B<0 r< (µ+ν)( pµ+µpq 2+νq 3 ν)q 1 +[ν(1 p)q 3 +pq 2 (µ+ν)]µ ν(1 p)(νq 3 ν µ) Stability analysis results in same threshold Vaccines with only one method of protection: Initial Infection (q 1 ): r< q 1(ν+µp) ν(1 p) Fast progression to Active TB (q 2 ): r< q 2µp ν(1 p) Slow progression to Active TB (q 3 ): r< q 3 µ µ+ν(1 q 3 )

34 Calibration to Real Data What data do we have? Incidence and prevalence of Active TB (from WHO) US inc prev UK inc prev Germany inc prev India inc prev Romania inc prev Ghana inc prev Mexico inc prev Brazil inc prev

35 Calibration to Real Data What data do we have? Detection and treatment success rates for Active TB (from WHO) US d s UK d s Germany d s India d s Romania d s Ghana d s Mexico d s Brazil d s

36 Comparison with Real Data US TBode8 Incidence Prevalence (.. NV / - V) Pop = , TB = (8.5% treat), LTB = (20% vac) Sus = (1% vac, 0.45% rec), β = e-08, Vacc= 0 8 MEXICO TBode8 Incidence Prevalence (.. NV / - V) Pop = , TB = (2.5% treat), LTB = (95% vac) Sus = (99% vac, 1.4% rec), β = e-07, Vacc= rate ( per 100,000 ) incidence prevalence rate ( per 100,000 ) Mexico United States time (years) time (years) rate ( per 100,000 ) ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= Romania time (years) rate ( per 100,000 ) BRAZIL TBode6 Incidence Prevalence (.. NV / - V) Pop = , TB = (4.5% treat), LTB = (99% vac) Sus = (99% vac, 3.6% rec), β = 1.135e-08, Vacc= rate ( per 100,000 ) Brazil GERMANY TBode8 Incidence Prevalence (.. NV / - V) Pop = , TB = (11% treat), LTB = (20% vac) Sus = (0% vac, 0.65% rec), β = e-07, Vacc= time (years) time (years) 36

37 Calibration Process Phase 0

38 Calibration Process Phase 1

39 Calibration Process Phase 2

40 Calibration Process Phase 3

41 Calibration Process Phase 4

42 Calibration Process - Results phase 1 phase 2 phase 3 phase 4 phase 5 US β[10, 12] r[0.20, 0.30] q 3 [0.70, 0.80] θ 1 [0.95, 1.00] - UK β[16, 18] r[0.30, 0.40] q 3 [0.40, 0.50] ω[0.08, 0.09] - GER β[12, 14] r[0.20, 0.30] q 3 [0.60, 0.70] - - IND β[12, 14] θ 1 [0.70, 0.80] q 1 [0.50, 0.60] q 2 [0.30, 0.40] r[0.40, 0.50] ROM β[20, 22] θ 1 [0.80, 0.85] q 1 [0.70, 0.80] q 3 [0.40, 0.50] q 2 [0.40, 0.50] GHA β[16, 18] θ 1 [0.85, 0.95] q 1 [0.70, 0.80] q 2 [0.10, 0.20] q 3 [0.40, 0.50] MEX β[20, 22] q 2 [0.50, 0.60] q 1 [0.20, 0.30] q 3 [0.50, 0.60] θ 1 [0.75, 0.80] BRA β[20, 22] q 3 [0.10, 0.20] θ 1 [0.65, 0.75] q 1 [0.70, 0.80] q 2 [0.30, 0.40] Exogenous re-infection and vaccination efficacies much more important for high incidence countries. Treatment of Latent TB more important for low incidence countries.

43 Calibration Process - Results

44 Calibration Process - Results

45 Universal Parameter Values and Distributions Parameter Symbol Range of values Min Max Probability of fast progression p 0.05 Rate of slow progression ν Average duration of treatment for active TB 1/γ yrs NaturalcurerateforactiveTB γ Mortality rate due to TB µ T 0.20 Proportion of immigrants that are vaccinated φ 0.80 Proportion of immigrants with LTBI ψ 0.30 Average duration of vaccine protection 1/ω 10 yrs 55 yrs Efficacy of vaccine in preventing initial infection q Efficacy of vaccine in preventing fast progression q Efficacy of vaccine in preventing slow progression q Factor of protection due to latent infection θ Factor of protection/susceptibility due to previous active infection θ

46 Country-specific Parameter Values and Distributions Country Parameter 1/µ π α d s c β r US [0.80,0.95] [0.60,0.85] 0.00 [10,12] [0.10,0.40] UK [0.78,0.98] [0.60,0.80] 0.00 [16,18] [0.10,0.40] GER [0.75,0.95] [0.60,0.80] 0.00 [12,14] [0.10,0.40] IND [0.57,0.77] [0.55,0.75] 0.74 [12,14] [0.00,0.30] ROM [0.55,0.95] [0.70,0.90] 0.99 [20,22] [0.00,0.30] GHA [0.15,0.35] [0.52,0.72] 0.92 [16,18] [0.00,0.30] MEX [0.55,0.75] [0.65,0.85] 0.99 [20,22] [0.00,0.30] BRA [0.55,0.75] [0.60,0.85] 0.99 [20,22] [0.00,0.30]

47 When is vaccination the better policy?

48 When is vaccination the better policy if there is no exogenous re-infection?

49 Compare with analytic thresholds

50 How much better would one policy be?

51 Is mass vaccination justifiable? Number of vaccinations per prevented case of active TB Country Percentile 5 th 25 th 50 th 75 th 95 th US UK Germany India Romania Ghana Mexico Brazil Vaccination is 1-2 orders of magnitude less efficient in a low incidence setting

52 Conclusions Very unlikely that mass vaccination in low incidence countries would ever lead to increased incidence of TB. Quantifying the contribution of exogenous re-infection is important in making any vaccination policy decision. The efficiency of vaccination in preventing TB is greatly reduced in low incidence countries. Is 4207 vaccinations per prevented case of Active TB justifiable?? I HAVE NO IDEA!!!