Epidemiology. Schedule: Friday, 25 May. 9:30-11:00 Session 1 Basic models and in epidemiology. 11:30-13:00 Session 2 Tuberculosis, pertussis, malaria

Transcription

1 Epidemiology chedule: Friday, 5 May 9:3-: ession Basic models and in epidemiology :3-3: ession Tuberculosis, pertussis, malaria 5:-6:3 ession 3 Multi-strain dynamics: the case of influenza

2 Types of models Mathematical models simple and general: Useful to explore general mechanisms, and to establish basic principles. Further complexity can be incorporated gradually. Traditional and common in biology. Computational models complex and specific: Attempt to describe every detail of the real system as accurately as possible. Weather prediction models are the best examples. Recent and increasingly common in biology.

3 Host classification usceptible nfectious Recovered (with immunity)

4 Host population

5 model Ronald Ross demonstrated that the parasite of malaria is transmitted by mosquitoes and in 9, he received the Nobel Prize of Medicine. He developed mathematical models for malaria transmission, and was a pioneer in mathematical epidemiology. recovery infection

6 model Appropriate for infections that induce no effective immunity (e.g. malaria). τ β Ι d dt = β + τ d dt = β τ β transmission coefficient τ recovery rate

7 model dt d τ β = ) ( As + =, the model is one dimensional and represented by the equation This is a type of growth law known as logistic growth, and it appears commonly in population dynamics models in the form dt d = β τ τ β ) ( The solution can be calculated to give () ( )t e t τ β β τ β τ + =

8 model Given an initial condition, () = 6, the proportion of infectious individuals grows as teady states:. Disease free equilibrium =, =. Endemic equilibrium = τ, = τ β β Parameters: β = 3, τ =

9 The Basic Reproduction Number, R The basic reproduction number is defined as The average number of secondary cases produced by an average infectious individual in a totally susceptible population. The basic reproduction number is calculated as R = (rate of transmission from an infectious individual) x (infectious period) = β x ( / τ) = β / τ R is a nondimensional number, and depends on both the environment and the disease. Disease mallpox Measles Rubella (England and Wales) Rubella (Gambia) R

10 Nondimensional model R R R dt d = = ) ( f time is measured in units of infectious period, D = / τ, then the model becomes The endemic equilibrium is rewritten as, R R = = Epidemic threshold: nfection can invade a susceptible population iff R >

11 tability of the steady states The model is expanded as d dt = R R The stability of a steady state, *, is given by the sign of the linearisation J ( *) = R R *. Disease free equilibrium, =, = : J = J () = R table if R <. Endemic equilibrium, = / R, = / R : J ( ) R = J R = R Possible if R > table

12 Vector field for the model

13 R model Appropriate for infections that induce a highly effective immunity (e.g. measles, mumps, rubella). R β Ι τ d dt = β d dt = β τ dr dt = τ β transmission coefficient τ - recovery rate

14 Epidemic threshold n 97, Kermack e McKendrick fitted the model to various epidemic curves (in particular, the Bubonic plague). They established the theory of the epidemic threshold: the growth of an epidemic requires that, on average, an infected individual infects at least one susceptible. An epidemic falls when the density of susceptibles is below a threshold: < / R. Plague epidemic τ > β τ < β d dt d dt = β = β τ β = 3 τ =

15 Another example - Cholera Cholera outbreak in London, 854: Distribution of cases in a residential area. Contaminated pump

16 Epidemic fad-out The epidemic curve of cholera is typical of an infectious disease that induces protective immunity. Closure of contaminated pump

17 Nondimensional R model f time is measured in units of infectious period, D = / τ, then the R model becomes d dt d dt = R = R and dr dt As individuals become immune, this system always approaches the disease free steady state, =. As the epidemic progresses, the level of susceptibles decreases, and the level of recovered individuals increases. The important question is the final balance between these two classes does the disease die out before all the susceptibles are exhausted? d R = exp R dr = R Using the fact that = R, and at equilibrium =, we get R * = exp R * R =

18 After epidemic R steady state

19 Long-term R dynamics n the long term, the susceptibility pool is replenished by births generating the conditions for new epidemics to occur. e R R e e e

20 teady states: R steady states The new steady states are obtained from the model d = e R e dt d = R dt. Disease free equilibrium: =, =. Endemic equilibrium: = e, = R R The new parameter, e, represents the birth and death rate in units of infectious period. This is equivalent to D / L, where D is the average duration of infection and L is the life expectancy at birth. Assuming that D = month, and L = 7 years, we get e =..

21 tability of the R steady states The linearisation of the R model is the Jacobian matrix J ( *, *) = R R * e * R R * * The stability of a steady state, (*, *), is given by the eigenvalues of the corresponding Jacobian matrix.. Disease free equilibrium, =, =, eigenvalues: e and R table if R <. Endemic equilibrium, = / R, = / R, eigenvalues: er ± e R 4e ( R ) table if R >

22 Vector field and simulation of the R model. 3, = e= R The inter-epidemic period near equilibrium is T = π / ω, where ω is the imaginary part of the eigenvalues of J, and this is ( ) ( ) = = 4 R D L T R e R e R e π ω

23 ntermediate models Temporary immunity and R steady states Partial immunity Temporary-partial immunity -4-5

24 Temporary immunity and Partial immunity Temporary immunity: Decay of immunity over time appears to be an important factor in many diseases for example, pertussis. n the extreme case, when immune efficacy drops from full protection to nothing at a given rate (parameterized by α), the model is d = e R dt d = R dt e + α ( e)( ) Partial immunity: Even more common is that immunity is not fully protective, but rather reduces the risk of further infections by some factor (σ) for example, tuberculosis. This is represented by the model d = e R e dt d = R + ( dt σ )

25 teady states and inter-epidemic period α α σ σ

26 Temporary-Partial immunity When immunity is not fully protective, it is likely to accommodate a specific combination of both temporary (α) and partial (σ) factors. Decreasing degree of protection σ R α Decreasing duration of protection

27 Temporary-Partial immunity model The combined model is represented as d = e R e e dt α ( )( d = R + ( ) dt σ ) Having both α and σ allows better fitting to data sets. West Midlands RV ncidence 5 ncidence (week - ) Date (year)

28 Vaccination mallpox: mmunity is a very old popular concept people who survive certain diseases, acquire protection, and will not get the disease again. t was realised how the same protection could be induced variolation protected against smallpox. n 76, Daniel Bernoulli initiated the development of mathematical techniques to assess the efficacy of such interventions. n 796, Edward Jenner noticed that cowpox caused a less severe disease on people, and also resulted in protection against smallpox. n few years, the new technique of vaccination became very popular. mallpox was declared eradicated from the world in 979.

29 Measles: Vaccination

30 Vaccination Many diseases preventable by vaccination: mallpox Measles Mumps Rubella Diphtheria Whopping cough Meningitis (Hib) Meningitis (Neisseria meningitidis) Tetanus Poliomyelitis Hepatitis B Yellow fever Tuberculosis (BCG) nfluenza And many more are in development.

31 Vaccination R model The collective effect of a vaccination programme is to reduce the pool of susceptible individuals. R R

32 The collective effect of a vaccination programme is to reduce the pool of susceptible individuals. (-v)e ve R R Vaccination R model v: vaccination coverage

33 Vaccination Temporary immunity model Waning of immunity poses a major obstacle on disease eradication. (-v)e α(-e) R R ve σ : susceptibility reduction factor

34 Vaccination Partial immunity model Partial immunity creates a reinfection threshold, R = / σ, above which the prevalence of infection is insensitive to vaccination. (-v)e R R σ R ve σ : susceptibility reduction factor

35 Reinfection threshold controllable uncontrollable

36 Vaccine more protective than natural immunity No existing vaccine confers a level protection that is superior than that induced by natural infection. However, this is a prime goal in vaccine development research. n order to predict the epidemiological impact of such vaccine we need an extention to the partial immunity model. The vaccine is more potent than natural infection iff σ V < σ.

37 Vaccine more protective than natural immunity

38 Back to Grassberger et al. σ R compact growth σ no infection R annular growth R Grassberger, Chaté, Rousseau (997) preading in media with long-time memory. Phys. Rev. E 55, tollenwerk N, Gökaydin D, Hilker FM, van Noort P, Gomes MGM 7 The reinfection threshold is the mean field version of the transition between annular and compact growth in spatial models (submitted). Gomes MGM, White LJ, Medley GF (4) nfection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives. J. Theor. Biol. 8, Breban R, Blower (5) The reinfection threshold does not exist. J. Theor. Biol. 35, 5-5. Gomes MGM, White LJ, Medley GF (5) The reinfection threshold. J. Theor. Biol. 36, -3.

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40 Tuberculosis

41 Estimated TB incidence rate, 3 Rates per, all forms of TB or more No estimate The designations employed and the presentation of material on this map do not imply the expression of any opinion whatsoever on the part of the World Health Organization concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. White lines on maps represent approximate border lines for which there may not yet be full agreement. WHO 4

42 The BCG paradox variable efficacy BCG was introduced in the global immunisation programme in 973, and it is today one of the mostly widely used vaccines. Estimates of its efficacy vary from % to 8%, generating great controversy. TB incidence BCG efficacy

43 Paula Rodrigues, GC A mathematical model (α) Λ α Λ ασ V Λ V L reactivation reinfection τ V reactivation reinfection (α)σ V Λ L V (α) σ Λ C α σ Λ Λ : rate of infection (Λ = β ( + V )) α : proportion progressing to active TB in less than years ω : rate of reactivation of latent TB

44 d = ( v) µ ελ µ dt d = φλ dt dl = dt dc = τ ( σλ + µ ) C. dt ( ε + σ ( C + L) ) + ωl ( τ + µ ) ( φ ) Λ( ε + σc) φσλl ( ω + µ ) Vaccinated A mathematical model dv dt d V dt dl dt V = vµ σ = φσ = V Λ V L ΛV µ V Non vaccinated ( V + L ) + ωl ( τ + µ ) V ( φ ) σ ΛV φσ ΛL ( ω + µ ) L. V V V V V V

45 Model predicts variable efficacy imulation of a vaccination programme in 3 regions: A, B, C. Endemic equilibrium imulations (a) Proportion active TB 3 4 σ V =.5 A B C (b) Proportion active TB 3 4 C B A v = 95% 5 β r 5 5 Transmission coefficient, β Effectiveness of the same vaccination programme in the 3 regions (c) Proportion prevented A B C 3 4 Time (in years) The vaccine is highly effective where incidence is low, but it has no effect where incidence is high.

46 Perspectives for control A major focus of TB research is the development of more potent vaccines. Mathematical models play an important role in the definition of a good vaccine and a good intervention, from the view point of global control. (a) Proportion active TB σ =.5 V C B A β r 5 5 Transmission coefficient, β Vaccine more potent than natural infection Proportion active TB σ =. V C B A β rv 5 5 Transmission coefficient, β Alternative control measures are detection and treatment of both active and latent TB cases, but variable efficacy is also expected. The globalisation of planning is essential.

47 The potential of post-exposure interventions /3 of the world population (~ 9 individuals) is infected, the majority in latent form; t is estimated that % (~ 8 individuals) will develop pulmonary tuberculosis; This immense reservoir can only be controlled by post-exposure interventions: chemoprophylaxis or prophylactic vaccines.

48 Treatment of latent infection usceptible Λ Early Latent L φδ σ R Λ τ nfectious (φ)δ σ Λ τ Recovered R ω R τ ω L Late Latent L

49 Widespread treatment of latents does not benefit all populations

50 Altered susceptibility after treatment

51 ummary

52 ntervention planning

53 ummary. Partial immunity induces a threshold in transmission above which reinfection is high enough to overcome natural immunity;. A vaccine will be ineffective above the reinfection threshold unless it does better than naturally acquired immunity; 3. Post-exposure interventions can have a wide range of outcomes, making it very important to characterise their mode of action; 4. Post-exposure interventions that reduce both the risk of reactivation and the risk of reinfection can have minor of major effects depending on the design of the control programme.

54 Pertussis

55 Pertussis resurgence

56 ncreasing age at infection

57 Basic model Ricardo Águas, GC λ σ λ τ R α τ

58 Basic model Ricardo Águas, GC B (-v)ρ vρ λ σ λ τ R α τ

59 Basic reproduction number, R Pertussis Endemic stability

60 ncreasing disease with decreasing transmission

61 Model equations ) ( ) ( ) ( ) ( ) ( µ τ σλ µ τ λ µ α σλ τ τ µ µ λ α µ + = + = = + + = R dt d dt d R v dt dr R v dt d

62 Age and time equations ) ( ) ( ) ( ) ( µ τ σλ µ τ λ µ α σλ τ τ µ µ µ λ α + = + + = = = + R a t a t R v a R t R v R a t Boundary conditions: (t,) = µ R(t,) = (t,) = (t,) =

63 Malaria

64 Malaria: Data from ub-aharan Africa Proportions Proportions Proportions Proportions Mponda,6,4,,,8,6,4, ,6,4,,,8,6,4,,4,,,8,6,4,,35,3,5,,5,,5 Chonyi Kilifi fakara,6,4,,,8,6,4,,5,,5,,5 Foni Kansala,8,6,4,,,8,6,4, fakara,35,3,5,,5,, Chonyi iaya Age in years Age in years

65 Malaria endemic stability

66 Malaria sensitivity to reporting rate

67 Malaria sensitivity to reduced infectivity

68 Malaria intervention design

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70 Multi-strain dynamics: the case of influenza usceptible nfected - usceptible R R usceptible nfected - Resistant

71 model Appropriate for infections that induce no effective immunity (e.g. malaria). R model Appropriate for infections that induce a highly effective immunity (e.g. measles, mumps, rubella). R Ι R R Ι

72 ntermediate models Temporary immunity and R steady states Partial immunity Polarised immunity R R σ(-e) (-σ)(-e)

73 Partial immunity and Polarised immunity Partial immunity: R e e R e = + = ) ( & & σ Polarised immunity: ( ) R e R e + = = ) ( σ & &

74 teady states Partial immunity Polarised immunity

75 Pathogen diversity and disease epidemiology Combining the epidemiology and evolution of infectious diseases. nfluenza A / H3N 997 Proportion of H3N in the total flu incidence (%) A H3N

76 Why flu is a favourite model system ) Evolutionary and Epidemic time scales coincide. HV Flu Measles + evolutionary change log epidemic change ) Global surveillance provides both epidemic and evolutionary data. 3) mportant recurrent disease major cause of morbidity and mortality.

77 strains with partial cross-immunity Viggo Andreasen, Roskilde Λ ø Λ Force of infection: Λ = R ( Ø + ) Λ = R ( Ø + ) ø ø σ Λ σ Λ

78 strains with partial cross-immunity ( ) ( ) ( ) ( )( ) Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø e e e e e e e e = Λ = Λ = Λ = Λ + = Λ = Λ = + Λ Λ = σ σ σ σ & & & & & & & & Ø Ø Ø = n susceptible classes (in case of n strains) n n- infected classes (in case of n strains)

79 strains with partial cross-immunity and coinfection without coinfection with Graham Medley & James Nokes Warwick ø Force of infection: Λ = R ( Ø + ) Λ = R ( Ø + ) Λ Λ ø with coinfection ø σ Λ σ Λ

80 ( ) ( ) Ø Ø Λ + Λ = Λ Λ + Λ = Λ R R σ σ & & ( ) ( ) Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø e e e e e = Λ = Λ = Λ = Λ + Λ Λ = Λ = Λ Λ = Λ + Λ Λ = σ σ σ σ σ & & & & & & & & strains with partial cross-immunity and coinfection

81 ( ) ( ) ( ) Ø Ø Ø Ø Ø Ø Ø Λ + Λ = Λ Λ + Λ = Λ Λ = Λ Λ = Λ + Λ Λ = R R e e e e σ σ σ σ & & & & & strains with partial cross-immunity and coinfection n susceptible classes n infected classes

82 n strains with partial cross-immunity and coinfection i N J J i i J i J J i J i i J J i i J i i i J J N i i i R e e e Λ Λ = Λ Λ Λ = Λ = σ σ σ σ \ \ & & & Variants indexed by the set N = {,,, n} and ordered by similarity. Variants compete for hosts and interact through cross-reactive immunity. The dynamics of n strains are described by a system of n + n equations.

83 -D strain space

84 -D strain space onset of patterns reinfection threshold σ R

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86 strains with polarised cross-immunity and coinfection... and reduced transmission Julia Gog, Cambridge θ Λ ø Λ θ ø ø σλ σλ (σ) Λ (σ) Λ

87 ( ) ( ) Λ Λ = Λ Λ Λ = Λ Λ Λ = Λ Λ = θ θ θ θ σ θ θ θ θ σ θ θ R R e e e e & & & & strains with polarised cross-immunity and coinfection Ø Ø, + = + = θ θ n susceptible classes n infected classes

88 More tractable, but different no waves σ no reinfection threshold? R

89 Polarised cross-immunity and mutation

90 Polarised cross-immunity and mutation

91 -D strain space

92 Cluster as the modelling unit But despite all differences between models, the cluster appears as a natural modelling unit... shift drift

93 Plotkin, Dushoff, Levin, Princeton Clusters of influenza A virus

94 Derek mith, Cambridge Clusters of influenza A virus Antigenic map of influenza

95 Dinis Gökaydin, GC Model structure

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98 EPDEMOLOGY mathematical models computer models ORGANM-centered biology mathematical models computer models animal models MOLECULAR and CELL biology

99 Flupread A tool for public health policy making José Lourenço, GC Demographic and Geographical nformation

100 Flupread real-time monitoring systems modeller parameter estimation Flupread simulation package user friendly interface LR model Age structure Geo structure mmune profile demographic geographic data policy maker predictions

101 Theoretical Epidemiology Reinfection Threshold