Mathematical Analysis of Epidemiological Models: Introduction

Transcription

1 Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010

2 1. Introduction. The effectiveness of improved sanitation, antibiotics, and vaccination programs created a confidence in the 1960s that infectious diseases would soon be eliminated. Consequently, chronic diseases such as cardiovascular disease and cancer received more attention in the United States and industrialized countries. But infectious diseases have continued to be the major causes of suffering and mortality in developing countries. Moreover, infectious disease agents adapt and evolve, so that new infectious diseases have emerged and some existing diseases have reemerged [142]. Newly identified diseases include Lyme disease (1975), Legionnaire s disease (1976), toxic-shock syndrome (1978), hepatitis C (1989), hepatitis E (1990), and hantavirus (1993). The human immunodeficiency virus (HIV), which is the etiological agent A Great Reference For Epidemiological Models SIAM REVIEW Vol. 42, No. 4, pp c 2000 Society for Industrial and Applied Mathematics The Mathematics of Infectious Diseases Herbert W. Hethcote Abstract. Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number R0, the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R0 and σ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed. Key words. thresholds, basic reproduction number, contact number, epidemiology, infectious diseases Hethcote, 2000, SIAM Review, 42: AMS subject classifications. Primary, 92D30; Secondary, 34C23, 34C60, 35B32, 35F25 PII. S

3 The General Epidemic, or SIR, Model We divide the population into three groups: Susceptible individuals, S(t) Infective individuals, I(t) Recovered individuals, R(t) λs S γi I R Assumptions Population size is large and constant, S(t) + I(t) + R(t) = N No birth, death, immigration or emigration No latent period Homogeneous mixing Infection rate is proportional to the fraction of infectives (frequency dependent), i.e. λ = βi/n Recovery rate is constant, γ

4 Model Equations A system of three ordinary differential equations describes this model: or ds I(t) = β dt N S(t) di dt = β I(t) S(t) γi(t) N dr dt = γi(t) ds I(t) = β dt N S(t) di dt = β I(t) S(t) γi(t) N R(t) = N S(t) I(t)

5 R 0 How many new infectives are caused by a single infective introduced into a population that is entirely susceptible? In this case the second ODE is di dt (β γ)i(t). So if β γ > 0 then I(t) increases. Define the basic reproductive number R 0 = β γ. If R 0 > 1 then I(t) increases and we have an epidemic.

6 Epidemic Curves N R 0 >1 N R 0 <1 Susceptible Infective Recovered 0 t 0 t

7 Linear Stability Analysis Step 1 Find equilibrium points Step 2 Linearize at each point Step 3 Find eigenvalues of linearized problem

8 Find equilibrium points Equilibria are points where the variables do not change with time: i.e. ds dt = di dt = dr dt = 0. This gives two equilibria: ds dt = 0 = β I N S di dt = 0 = β I N S γi (S = N, I = 0) and (S = 0, I = 0) We ll concentrate on the first, the disease-free equilibrium. (The second is after an epidemic, when R = N.)

9 Linearize equations First, we ll shift the variables so that the origin is at the equilibrium (N, 0) (0, 0): S = N S I = I 0 = I Then ds = dn dt dt ds dt = β I N S = β I N (N S ) di dt = di dt = β I I S γi = β N N (N S ) γi

10 Linearize equations We are only going to consider small deviations from the equilibrium, so that S and I are small. That means that any terms with higher powers of S and I are very small, so we neglect them. This is linearization. ds = β I dt N (N S ) = βi β I S N βi di dt = β I N (N S ) γi = βi β I S N γi βi γi

11 Our linearized equations are Linearize equations ds = βi dt di dt = βi γi (Autonomous) linear differential equations have solutions ve rt. Using that form of solution, without yet knowing r, we get ds = ds 0e rt = S 0 re rt = βi 0 e rt dt dt di dt = di 0e rt = I 0 re rt = (β γ)i 0 e rt dt

12 Linearize equations Dividing by e rt, which is never 0, gives the linear algebra problem I 0 rs 0 = βi 0 ri 0 = (β γ)i 0 Transforming that into a standard vector matrix problem gives the standard eigenvalue problem ( ) [ ] ( ) [ ] ( ) S0 0 β S0 r β S0 r = = = 0 0 β γ 0 r β + γ I 0 I 0

13 Linearize equations This is solved by setting the determinant of the matrix to 0: ([ ]) r β det = r(r β + γ) ( β)0 = r(r β + γ) = 0 0 r β + γ This gives the two eigenvalues r = 0 and r = β γ. (In general, for an N N matrix, we get N eigenvalues.) Here, the eigenvalue of interest is r = β γ.

14 Linearize equations With the eigenvalue r = β γ, we have solutions with e rt. If r = β γ > 0, the solutions grow away from the equilibrium. The equilibrium is unstable. For our model, this is an epidemic. (R 0 = β γ > 1.) If r = β γ < 0, the solutions contract back towards the equilibrium. The equilibrium is stable. For our model, this is no epidemic. (R 0 = β γ < 1.)

15 Linearize equations The quick way If we first write the model as a vector differential equation d dt Generically, this is in the form ( ) ( S β I(t) = I β I(t) N d dt v = f(v) N S(t) ) S(t) γi(t)

16 Linearize equations The quick way Taylor s theorem for a scalar-valued function of a scalar (f : R R) with enough smoothness f (x) = f (x 0 ) + (x x 0 )f (x 0 ) + (x x 0) 2 f (x 0 ) + 2 This is also true for a vector-valued function of a vector (f : R n R n ) f(v) = f(v 0 ) + J(f, v 0 )(v v 0 ) + where J(f, v 0 ) is the Jacobian of f, the generalization of the first derivative f (x 0 ). The Taylor expansion explicitly separates the function into its constant part, its linear part, etc.

17 Linearize equations The quick way Then, letting v 0 = (N, 0), the equilibrium d dt ( ) ( S β I = N S I β I N S γi + = 0 + = d ds d ds ) = ( β I N S ) (β I N S γi ) [ β I N β I N ] [ 0 β 0 β γ ( β I N S ) β I N S γi ) ( β I N S β S N β S N γ d di d di ] S=N,I=0 ) (β I N S γi S=N,I=0 (v v 0 ) + S=N,I=0 (v v 0 ) + (v v 0 ) +

18 Linearize equations The quick way First, note that the constant part of the Taylor expansion, f(v 0 ) = 0 when v 0 is an equilibrium: this is how we found the equilibria! Following the same steps as before, we see that ultimately, all we need the eigenvalues of the Jacobian matrix [ ] 0 β 0 β γ As before, the eigenvalues are 0 and β γ.

19 Vector Fields Non-Linear Model ds dt = β I N S di dt = β I N S γi Model Linearized at S = N, I = 0 ds dt = βi di = (β γ)i dt

20 Vector Fields Non-Linear & Linear Zoomed In

21 Hartman Grobman Theorem All of this is backed up formally by the Hartman Grobman Theorem, which roughly says that near an equilibrium point, the dynamics of the original (nonlinear) system are the same as those for the linearized system. (This theorem requires that none of the eigenvalues have 0 real part. The model we looked at did not satisfy this condition because one of the eigenvalues was 0, but we could show that this theorem is true nonetheless.)

22 Endemic Model Constant population size: ds I(t) = µn β S(t) µs(t) dt N di dt = β I(t) S(t) γi(t) µi(t) N dr = γi(t) µr(t) dt dn dt = ds dt + di dt + dr dt = 0.

23 So divide by population size Endemic Model s = S N, i = I N ds dt = d ( ) S = 1 dt N N di = βis γi µi dt dr = γi µr dt ds dt = µn N β I S N N µ S N s + i + r = 1 = µ βis µs

24 Find equilibria ds dt = di dt = 0 ds = 0 = µ βis µs dt di = 0 = βis γi µi dt Two equilibria: Disease-free equilibrium: Endemic equilibrium: ( E e = E 0 = (s = 1, i = 0) s = γ + µ β ) µ(β γ µ), i = β(γ + µ)

25 Linearize equations Write as vector differential equation d dt By Taylor s theorem ( ) s = i f(s, i) = f(s 0, i 0 ) + J(s 0, i 0 ) ( ) µ βis µs = f(s, i) βis γi µi [( ) s i ( )] s0 i 0 + At equilibrium, f(s 0, i 0 ) = 0, so the dynamics near (s 0, i 0 ) are governed by the linear part J(s 0, i 0 )

26 Jacobian derivative of f J(s, i) = [ f1 s f 2 s ] f 1 i f 2 i Disease-free equilibrium J(1, 0) = Analysis = Eigenvalues { µ, β µ γ} λ 1 = µ < 0 λ 2 = β µ γ β µ γ < 0 β µ γ > 0 [ ] βi µ βs βi βs γ µ [ ] µ β 0 β γ µ β γ+µ β γ+µ R 0 = < 1, stable, No epidemic > 1, unstable, Epidemic β γ + µ

27 Analysis Endemic equilibrium ( γ + µ J β ) [ µ(β γ µ) µβ, = β(γ + µ) γ+µ γ µ µ(β γ µ) γ+µ 0 { } Eigenvalues µβ µ+γ ± µ 2 β 2 4µ(β γ µ) (µ+γ) 2 R 0 = R 0 = β γ+µ β γ+µ > 1, stable < 1, unstable ]

28 Summary R 0 = β γ + µ E 0 = (s = 1, i = 0) E e = ( s = γ + µ = 1 µ(β γ µ), i = = µ ) β R 0 β(γ + µ) β (1 R 0) R 0 < 1 Disease-free equilibrium is stable Endemic equilibrium is unstable (and nonsense!) R 0 > 1 Disease-free equilibrium is unstable Endemic equilibrium is stable

29 Epidemic Curves N Susceptible Infective Recovered 0 t

30 Vaccination model ds = (1 p)µ βis µs dt di = βis γi µi dt dr = γi µr dt dv = pµ µv dt s + i + r + v = 1

31 Disease-free equilibrium: Analysis E 0 = (s = 1 p, i = 0, v = p) Jacobian: βi µ βs 0 J(s, i, v) = βi βs γ µ µ µ β(1 p) 0 J(E 0 ) = 0 β(1 p) γ µ µ

32 Analysis µ β(1 p) 0 J(E 0 ) = 0 β(1 p) γ µ µ λ 3 > 0 R v = Stability determined by R v λ 1,2 = µ < 0 λ 3 = β(1 p) γ µ β γ + µ (1 p) = R 0(1 p) > 1 λ 3 < 0 R v < 1

33 Critical vaccination level R v = R 0 (1 p ) = 1 = p = 1 1 R 0 p > p = R v < 1 No epidemic!