A Mathematical Model for Tuberculosis: Who should be vaccinating?

Transcription

1 A Mathematical Model for Tuberculosis: Who should be vaccinating? David Gerberry Center for Biomedical Modeling David Geffen School of Medicine at UCLA

2 Tuberculosis Basics 1 or 2 years lifetime New Infection 95% 5% Latent TB Active TB fast progression 5% Active TB slow progression Exogenous reinfection Recovered individuals are susceptible to future infection 1/3 of world s current population has LTBI Major source of global death from infectious disease

3 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 wanes with time (10-55 years of protection) Can interfere with the detection of Latent TB

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

5 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 LTBI BCG efficacy Don t Vaccinate treating LTBI BCG efficacy Vaccinate Can we establish conditions which justify the discontinuation of mass BCG vaccination?

6 ! *!(+# 53)13%((,)$,$*%'+,)$,-& " " # %0)1%$)2(# 3%,$*%'+,)$ (")&# 53)13%((,)$ -!'',$%#&!$,$1 3%'32,+6%$+#,$+) +7%#5)52"!+,)$ %0)1%$)2(# 3%,$*%'+,)$ (")&4# 53)13%((,)$,$*%'+,)$4.,-.,-././ 1,-1 & # *!(+# 53)13%((,)$ S =susceptible V = vaccinated E = latently infected E V = vacc. & latently infected I T = actively infected & treated I T c TB Model = actively infected & untreated R =recovered +3%!+6%$+#)* <./=!""#$%&#'!(%(# )*#!'+,-%#./ (,-( 8%+%'+,)$#)*#./##9()*+ $ %,-0 $ %& %$8# +3%!+6%$+ ' 0 $!+23!"# 3%')-%3:./;,$82'%8# 6)3+!",+:.,-. 3%,$*%'+,)$ c = vaccine coverage p = prob. of fast progression (.05) d = detection rate (active TB) s = treatment success rate (active TB) r = detection/treatment success rate (latent TB)

7 ! -.& & # TB Model <456 " " # ; <456 / -./ -./!"#$ % & /!"#$ % & q 1 =efficacy preventing initial infection q 2 =efficacy preventing fast progression q 3 =efficacy preventing slow progression 0 q 1,q 2,q : ;!""#$%&#'!(%(# )*#!'+,-%#./ ( -.( 7()*+,$ %& $ % -.0 $ %& ' / -./ τ = r (ν + µ) 1 r θ 1 = factor of susceptibility to exogenous reinfection θ 2 = factor of susceptibility to reinfection after recovery ' = 4.' = >456

8 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

9 The Model S 0 =(1 α)(1 c)πα(1 φ)(1 ψ)π + ωv βsi/n µs V 0 = c(1 α)π + απ(1 ψ)φ ωv (1 q 1 )βvi/n µv E 0 =(1 p)βsi/n + απ(1 φ)ψ νe θ 1 pβei/n τe µe E v 0 =(1 p(1 q 2 ))(1 q 1 )βvi/n+ απψφ +(1 p)θ 2 βri/n ((1 q 3 )νe v + θ 1 pβe v I/N) µe v I 0 T = d[νe + θ 1 pβei/n + pβsi/n + p(1 q 2 )(1 q 1 )βvi/n +(1 q 3 )νe v + θ 1 pβe v I/N + pθ 2 βri/n] γ 1 I T µi T + d/2 I T c I 0 T c =(1 d)[νe + θ 1pβEI/N + pβsi/n + p(1 q 2 )(1 q 1 )βvi/n +(1 q 3 )νe v + θ 1 pβe v I/N + pθ 2 βri/n]+(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 + τe θ 2 βri/n µr

10 Basic Reproductive Number R 0 : average number of secondary infections caused by a single infection in a completely susceptible population µs µi µr In a very simple model: S I R βsi/n γi R 0 = average length of infection = β γ + µ average infection rate Disease R 0 Measles Pertussis Diphtheria 6 7 Smallpox 5 7 Polio 5 7 Rubella 5 7 Mumps 4 7 HIV/AIDS 2 5 SARS 2 5 Influenza 2 3

11 Why do we care about R 0? R 0 : average number of secondary infections caused by a single infection in a completely susceptible population R 0 < 1 disease goes extinct R 0 > 1 disease will persist Also tells us how hard we need to work to control a disease - Measles: R 0 =15 need to vaccinate > % of people - Polio: R 0 =6 need to vaccinate > % of people

12 Back to the Tuberculosis Model Model has four disease states: E,E V,I T,I T c. Denote them i =1, 2, 3, 4. Let Θ i = probability that a new infection will be of type i. DFE = (S,V,E,EV,I T,I T,R )= c proportion vaccinated, p V = cµ ω+µ proportion unvaccinated, p S =1 p V (1 c)µπ+ωπ µ(µ+ω) Probability new infection is in unvaccinated individual = Θ 1 = (1 p)βp S βp S +(1 q 1 )βp V = (1 p)[(1 c)µ+ω] (1 cq 1 )µ+ω,, cπ µ+ω, 0, 0, 0, 0, 0. βp S βp S +(1 q 1 )βp V. 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

13 Basic Reproductive Number Transition rate matrix G: G ij = probability per unit time of transferring from j to i for i = j. G ii = probability per unit time of leaving state i. G = ν µ τ µ (1 q 3 )ν 0 0 dν d(1 q 3 )ν γ 1 µ d/2 (1 d)ν (1 d)(1 q 3 )ν (1 s)γ 1 d/2 γ 2 µ µ T x(t) = probability of individual being in the various states at time t, then dx dt = Gx, withx(0) = Θ = x(t) =e Gt Θ and x(t) 0 as t. E[amt of time in the various states] = Infectivity in each state: h T = e Gt Θ dt = G 1 e Gt Θ β(µ µcq 1 +ω) µ+ω 0 β(µ µcq 1 +ω) µ+ω. = G 1 Θ.

14 Basic Reproductive Number R 0 = h G 1 Θ For the TB model: R 0 = A + Bc where A = β [(1 r+rp)ν+pµ] (µ+µ T +γ)(µ+ν) B = βµ [(µ+ν)( pµ+µpq 2+νq 3 ν)q 1 +ν(p 1)(µ+rν)q 3 (µ+ν)(µpq 2 +rν(p 1))] (µ+ω)(µ+ν)(µ+µ T +γ)(ν q 3 ν+µ)

15 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 )

16 With All Protective Efficacies of the Vaccine Considered r =1.0 detect and successfully treat 100% of Latent TB cases r =0.35 detect and successfully treat 35% of Latent TB cases

17 Considering Each Efficacy Separately q 1 only (dashed) initial infection q 2 only (solid) fast progression to active TB 1 r q 3 only (dotted) slow progression to active TB Conclusions: 1. To justify discontinuation you need to detect and treat LTBI at a rate that is at least twice as high as the vaccine s efficacy. 2. It is very unlikely that detection and treatment of LTBI is high enough to justify discontinuing mass vaccination with BCG. 0 q 1

18 So we re done right? Thresholds are based on R 0 R 0 : average number of secondary infections caused by a single infection in a completely susceptible population No country is on the brink of eliminating TB (i.e. near being a completely susceptible population ). Immigration is an important factor in every country that is considering discontinuing mass vaccination. Only tells us that vaccination is likely to be better. Doesn t give any idea of HOW MUCH better. Need a more data-based, country-specific analysis!

19 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

20 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

21 Parameter Values and Distributions Universal 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 Natural cure rate for active TB γ Mortality rate due to TB µ T 0.12 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 Country-specific Country Parameter 1/µ π(millions) α c US UK GER IND ROM GHA MEX BRA

22 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) tim e (y ears )

23 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) Reported Prevalence tim e (y ears )

24 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) Reported Incidence tim e (y ears )

25 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) d =.84 s = inc tim e (y ears )

26 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) d =.83 s = inc tim e (y ears )

27 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) d =.80 s = inc tim e (y ears )

28 Adaptive Simulations connecting WHO data with our model ROMANIA TBode7 Incidence Prevalence (.. NV / - V) Pop = , TB = (3.3% treat), LTB = (98% vac) Sus = (94% vac, 9% rec), β = e-07, Vacc= rate ( per 100,000 ) d =.84 s =.60 d =.83 s =.85 d =.80 s =.61 d =.79 s =.70 d =.82 s =.75 d =.83 s =.73 d =.79 s = tim e (y ears )

29 Calibration Process - Results

30 Calibration Process - Results

31 When is vaccination the better policy? UNITED KINGDOM GHANA q 1 only q 1 only q 2 only q 2 only q 3 only 20 years 50 years 100 years q 3 only 20 years 50 years 100 years Vaccination is always better in high incidence countries. Detecting/treating latent TB can be better in low incidence countries.

32 When is vaccination the better policy if there is no re-infection? UNITED KINGDOM GHANA q 1 only q 1 only q 2 only q 2 only q 3 only 20 years 50 years 100 years q 3 only 20 years 50 years 100 years Without re-infection, low and high incidence countries have the same thresholds. We still don t know how much better vaccination would or would not be.

33 How much better would one policy be? US UK GERMANY <r<0.40 MEXICO INDIA BRAZIL <r< ROMANIA GHANA horizontal axes: overall protective effect q =1 (1 q 1 )(1 q 2 )(1 q 3 ) vertical axes: cumulative proportion of active TB cases prevented over 50 years Vaccination always outperforms treating latent TB in all countries. The proportion of TB cases prevented is very country-specific.

34 So is the CDC policy wrong? Table 1: Vaccinations per case of active TB prevented after 50 years Country Percentile 5 th 25 th 50 th 75 th 95 th US UK Germany Mexico India Brazil Romania Ghana Vaccination is 1-2 orders of magnitude less cost-effective in a low incidence setting. The decision to not vaccinate is made for economic, political and cultural reasons and not purely on epidemiological reasons. Re-infection (both exogenous and after recovery) is very important in making a vaccination decision. The interference of BCG with detecting latent TB is not.

35 Let s get back to some mathematics Forward (supercritical) bifurcation Backward (subcritical) bifurcation Prevalence at Equilibrium Prevalence at Equilibrium 0 1 R R 0 R 0 > 1 disease will persist R 0 < 1 disease goes extinct Very low endemic prevalence if R 0 is slightly greater than 1. R 0 > 1 disease will persist R 0 < 1 it depends High endemic prevalence if R 0 is slightly greater than 1. To eliminate the disease, R 0 < 1 is not enough.

36 How can backward bifurcation happen? Typically: Completely susceptible population provides maximum potential for secondary infections. TB: Exogenous re-infection means latent class can still be infected. TB: Recovered individuals can also be infected. Recovered individuals may even be more susceptible to TB than TB-naïve. In general, backward bifurcation is possible when the at-risk population is divided into different risk groups and the size of those risk groups is affected by infection.

37 1. Rewrite our system as dx dt = f(x, β). 2. Linearize around the DFE at R 0 = 1. J = D x f(x, β )= fi x j (x, β ), WLOG assume x = β = 0. Zero is a simple eigenvalue of J with right and left eigenvectors w and v, respectively, and all other eigenvalues have negative real parts. 3. Center Manifold Theorem. W c = {c(t)w + h(c, β) :c<δ,c(0) = 0,h(c, β) =Dh(c, β) =0} where c(t) E c and h(c, β) E s. Only need to consider dynamics on W c. 4. Center manifold is invariant under the dynamics of dx dt have that d dt (c(t)w + h(c, β)) = f(c(t)w + h(c, β), β). Taylor expansion for h(c, β) 5. Simplified system dc dt = a 2 c2 + bβc, wherea = Sketch of the Proof Idea: Transform to a simple system that has the same bifurcation n k,i,j=1 v k w i w j 2 f k x i x j,b= = f(x, β), so we n k,i=1 v k w i 2 f k x i φ.

38 Backward Bifurcation Theorem 1 The reduced TB model with no treatment of active TB and no vaccine protection against reactivation of LTBI (i.e. d = q 3 =0) has a backward bifurcation at β = β (i.e. R 0 =1)ifandonlyif (ν + µ)(µ + ω)(pµ + νrp + ν(1 r))β S +(ν + µ) p(1 q 1 )(1 q 2 )µ 2 +(ωp +(1 q 1 )ν)µ + ων(1 r + rp) (1 q 1 )β V <µp(µ + ω)([1 p(1 q 2 )](1 q 1 )β V µ +(1 p)β S (µ + νr))θ 1 +(pµ + ν)(µ + ω)[γ(ν + µ)+(1 p)β S νr]θ 2, where β = S = (1 c)µπ + ωπ µ(µ + ω), V = cπ µ + ω and µ(µ+γ+µ T )(µ+ω)(ν+µ) π(µ+ω)[(1 r+rp)ν+pµ]+µπc[(µpq 2 pµ ν)q 1 µpq 2 +rν(1 p)].

39 How do vaccination and treatment of latent TB affect backward bifurcation? 5 θ ! θ θ! 1 1 r =.10,c=0 r =.95,c=0 r =.95,c=.98, ω = θ R Treatment of latent TB increases the likelihood and magnitude of BB. Vaccination could increase the likelihood and magnitude of BB. These statements are fairly misleading! R 0 1 R 0

40 Conclusions Beaten the issue of BCG s interference with the detection of latent TB to death. Detecting and treating latent TB will not outperform the highly imperfect vaccine. By how much? Is it cost-effective? Backward Bifurcation is possible. The public health implications of this are limited.