Bifurcations in an SEIQR Model for Childhood Diseases

Transcription

1 Bifurcations in an SEIQR Model for Childhood Diseases David J. Gerberry Purdue University, West Lafayette, IN, USA, Conference on Computational and Mathematical Population Dynamics Campinas, Brazil

2 Outline Description of the model Summary of literature Preliminary findings Hopf bifurcations Homoclinic bifurcation Comparison to real data

3 SEIQR Model for Childhood Diseases d dt S=Λ µs σsi 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 dt R= µr+ξq S=Susceptible E=Exposed I=Infectious Q = Isolated(Quarantined) R=Recovered A= ActivePopulation (S+E+I+R) Λ = Recruitment Rate µ=deathrate σ=infectionrate 1=γ 1 =Ave. LengthofLatentPeriod 1=γ 2 =Ave. LengthofInfectiousPeriod 1=ξ = Ave. Length of Isolation Period

4 The Cause of Sustained Oscillations in Childhood Diseases? (1906) Hamer [7]: suggested autonomous internal forces (1950 s) Bartlett [8,9]: stochastic effects (1973) London and Yorke [10]: external periodic forcing (1975) Bailey [11, p. 12]: this failure of deterministic models led to their abandonment in many quarters (1984) Schenzle [12]: school year/age structure

5 The Cause of Sustained Oscillations in Childhood Diseases? (Continued) (1993) Feng and Thieme [1]: include an isolated class Assumption: per capita contact rate is independent of total population size Can create sustained oscillations

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

7 Initial Findings Basic Reproductive Number: R 0 = 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 uses approach to persistence of Thieme [5, 6] Endemic Equilibrium: Exists in a biologically feasible region when R 0 >1

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 2 1 +O(ν 1/2 ) and ζ 1 (ν)= (θ1 1 θ θ 1 (1 θ 2 )2 2 (θ 1 +θ 2 ) 2+ (1 θ 2 ) 2 ) 2 ( (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. Observethatν θ 1 ;θ 2 ;ζ: 3. Note that the coefficients of the characteristic polynomial ofthejacobianareanalyticinν forν> δ: 4. Compute the roots of the characteristic polynomial whenν=0. 5. Usethesetocomputethefirsttwotermsintheseries expansionsforeachrootinν. 6. Higherordertermscanbeomittedsinceν isverysmall. 7. w 1 = ζ+o(ν),w 2 = (θ 1 +θ 2 )+O(ν), w 3,4 =±ai+bν+o(ν 2 ). 8. bgoesfromnegativetopositiveasζ becomessmaller thanζ 0 (ν).

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

12 Intersection of ζ 0 (ν) and ζ 1 (ν) ζ 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 θ 1 +1 uv+o(3)

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 } Homoclinic bifurcation occurs along HC= { (µ 1 ;µ 2 ):µ 2 = 6 7 µ2 1+O(µ 3 1);µ 1 >0 }

15 Homoclinic Bifurcation of nearby system (Cont d)

16 Comparison to Scarlet Fever Data Scarlet fever data from England and Wales, [3] 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) Anderson and May [3]observe 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 Primary References [1] Feng, Z., Thieme, H.R. Recurrent outbreaks of childhood diseases revisited: the impact of isolation. Math. Biosci., 128: , [2] Wu, L., Feng, Z. Homoclinic bifurcation in an SIQR model for childhood diseases. J. Differential Equations, 168: , [3] Anderson, R.M., May, R.M. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York, [4] Kato, T. Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, [5] Thieme, H.R. Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations. Math. Biosci., (111), , [6] Thieme, H.R. Persistence under relaxed point-dissipativity (with applications to an endemic model). SIAM J. Math. Anal., (24), , 1993.

19 Historical References [7] Hamer, W.H. Epidemic disease in England. Lancet, (1), , 1996 [8] Bartlett, M.S. Measles periodicity and community size. J. Roy. Stat. Soc. Ser. A, (120), 48-70, [9] Bartlett, M.S. The critical community size for measles in the United States. J. Roy. Stat. Soc. Ser. A, (123), 37-44, [10] London, W.P., Yorke, J.A. Recurrent outbreaks of measles, chickenpox and mumps, I. Seasonal variation in contact rates. Am. J. Epidemiol., (98), , [11] Bailey, N.T.J. The Mathematical Theory of Infectious Diseases and Its Applications, Griffin, London and High Wycombe, [12] Schenzle, D. An age-structured model of pre- and postvaccination measles transmission, IMA J. Math. Appl. Biol. (1), , 1984.