Numerical qualitative analysis of a large-scale model for measles spread

Transcription

1 Numerical qualitative analysis of a large-scale model for measles spread Hossein Zivari-Piran Department of Mathematics and Statistics York University (joint work with Jane Heffernan) p./9

2 Outline Periodic Measles From In-Host Model to Between-Host Model Numerical Bifurcation Analysis of Large-Scale Systems Numerics for Measles Ongoing and Future Work p.2/9

3 Periodic Measles 3 2 Incidence a Measles (New York City, USA) Year Recruitment.2. Power spectrum e Spectral Frequency (/yr) density case reports M easles Incidence in Liverpool, England year.5.5 power spectrum.5.5 frequency case reports M easles Incidence in Ontario, Canada year.5.5 power spectrum.5.5 frequency (source: Mathematical Epidemiology; Brauer et al., 28) p.3/9

4 In-Host Model The within-host model consists of uninfected peripheral blood mononuclear cells (PBMCs, the main target of measles infection) (x), infected PBMCs (y) and virus (v), as well as naive (w), activated (z) and memory (m) CD8 T-cells: dx dt = λ x d x x βφxv dy dt = βφxv d yy ξyz dv = ky uv βφvx dt dw dt = λ z dz dt = dm dt = cφwv C φv +K d w w cφvw C φv +K + pφvz (ρ+d z)z + fc mφvm C 2 φv +K 2 C 3 φv +K 3 C 4 φv +K 4 ρz C 3 φv +K 3 d m m c mφvm C 4 φv +K 4 p.4/9

5 In-Host Model Establishment of Infection Initiate the adaptive immune response Adaptive immune response Immunological memory Level of virus in plasma Day Pathogen enters plasma Infectiousness begins Symptoms appear Infectiousness ends Pathogen is cleared (source: Heffernan and Keeling, 28) p.4/9

6 In-Host Model (a) 8 (b) 2 6 low m() high m() (days) (days) ( d) initial memory, (source: Heffernan and Keeling, 2) p.4/9

7 Between-Host Model No vaccine ds = B + qr + w S λs ds dt ds i dt = qr i + w i+ S i+ λs i ds i w i S i de i = λs i a i E i de i dt di i dt = a ie i g i I i di i dr i = w i+ R i+ + b i,j g j I j w i R i qr i dr i dt j λ = i β i I i Class R refers to individuals protected by short-term immune memory (or humoral responses), who clear the virus before T-cell activation preventing boosting. Class S refers to those individuals who have lost this short-term protection p.5/9

8 Between-Host Model With vaccine ds = B( p) + qr + w S λs ds dt ds v = Bp + qr v + w v+ S v+ λs v ds v w v S v dt ds i dt = qr i + w i+ S i+ λs i ds i w i S i i,v de i = λs i a i E i de i dt di i dt = a ie i g i I i di i dr i = w i+ R i+ + b i,j g j I j w i R i qr i dr i dt j λ = i β i I i The value v = 9 is determined by the within-host model. Dimension = (#S) + (#E) + (#I) + (#R) = 2(3) (3) = 43(63) p.5/9

9 Numerical Bifurcation Analysis of Large-Scale Systems Commonly Used Bifurcation Software AUTO (Doedel & Oldeman), XPPAUT (B. Ermentrout) [C, Fortran, Python] BIFPACK (R. Seydel)[Fortran] MATCONT(Dhooge & Govaerts & Kuznetsov)[Matlab] CONTENT(Kuznetsov & Levitin & Skovoroda) [C++] Methods Adapted for Large-Scale Problems (discretizations of partial differential equations) CL MATCONTL (Bindel & Friedmany & Govaertsz & Hughesx & Kuznetsov): Steady-State (Find-Continue), Hopf (Find-Continue), Fold (Find-Continue) [Matlab] PDECONT (K. Lust): Steady-State (Find-Continue), Periodic Solutions (Find-Continue) [C] LOCA (A. G. Salinger, et al.): Steady-State (Find-Continue), Hopf (Find-Continue), Fold (Find-Continue), Phase Transition (Find-Continue) [C] These methods are based on (some kind of) subspace continuation. p.6/9

10 Numerics for Measles Steady States Disease Free Equilibrium, (E i t= =, I j t= = ) τ = 2 p =. p =.5 p =.9 2 S Extensive numerical simulations show that the Jacobian at the disease free equilibrium always has one and only one positive eigenvalue. Hence, this equilibrium is always unstable and there is no local bifurcation for our desired parameter range ( P, τ ). i p.7/9

11 Numerics for Measles Steady States Disease Free Equilibrium, (E i t= =, I j t= = ).2 p =.5.8 p =.5 S eig.2.4 τ = 2.6 τ = 2 τ = i p =.5 E eig i p =.5 I eig. R eig i i unstable direction p.7/9

12 Numerics for Measles Steady States Endemic Equilibrium S τ = 2 p =. p =.5 p =.9 E 4 x τ = 2 p =. p =.5 p =.9. I x i τ = i p =. p =.5 p =.9 R x i τ = 2 p =. p =.5 p = i This stable equilibrium goes under a Hopf bifurcation and looses it stability at p = p H. The Hopf bifurcation is supercritical. p.7/9

13 Continuation of Hopf bifurcation.2. p H initial period (years) τ This was our first guess for oscillation mechanism. BUT, soon we observed that the amplitudes of oscillations were very small (not surprising for Hopf bifurcation). τ

14 τ = 2, p =.43 τ = 2, p =.43.8 total(s) total(e) total(i) τ = 2, p =.43 new infect = new infect = 8 new infect = 5 End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

15 total(s) τ = 2, p =.53 total(e) τ = 2, p =.53 new infect = new infect = 8 new infect = 5 End. Equ τ = 2, p = τ = 2, p =.53 total(i) total(r) introducing infection into Disease Free Equilibrium

16 total(s) τ = 2, p =.63 total(e) τ = 2, p =.63 new infect = new infect = 8 new infect = 5 End. Equ τ = 2, p = τ = 2, p =.63 total(i) total(r) introducing infection into Disease Free Equilibrium

17 total(s) τ = 2, p =.73 total(e) τ = 2, p =.73 new infect = new infect = 8 new infect = 5 End. Equ τ = 2, p = τ = 2, p =.73 total(i) total(r) introducing infection into Disease Free Equilibrium

18 total(s) τ = 2, p =.83 total(e) τ = 2, p =.83 new infect = new infect = 8 new infect = 5 End. Equ τ = 2, p = τ = 2, p =.83 total(i) total(r) introducing infection into Disease Free Equilibrium

19 τ = 2, p =.93 τ = 2, p =.93.5 total(s) total(e) total(i) τ = 2, p =.93 new infect = new infect = 8 new infect = 5 End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

20 τ = 2, p =.53 τ = 2, p =.53.4 total(s) total(e).3.2. total(i) τ = 2, p =.53 random start End. Equ. total(r) τ = 2, p = SIMULATING from a RANDOM state

21 τ = 2, p =.63 τ = 2, p =.63.3 total(s) total(e).2. total(i) τ = 2, p =.63 random start End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

22 τ = 2, p =.73 τ = 2, p =.73.4 total(s) total(e).3.2. total(i) τ = 2, p =.73 random start End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

23 τ = 2, p =.83 τ = 2, p =.83.4 total(s) total(e).3.2. total(i) τ = 2, p =.83 random start End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

24 τ = 2, p =.93 τ = 2, p =.93.5 total(s) total(e) total(i) τ = 2, p =.93 random start End. Equ. total(r) τ = 2, p = introducing infection into Disease Free Equilibrium

25 Question: WHAT HAPPENES to the medium-sized cycle? SHORT Answer: Neimark-Sacker bifurcation happens in the Poincare map of the cycle The resulting invariant two-dimensional torus is still stable; however, it loses its strong absorbance in some directions. Therefore (almost) inaccessible from Disease Free Equilibrium by introducing new infected individuals.

26 Stable Endemic Equilibrium Unstable Disease Free Equilibrium This is based on strong evidence from numerical simulations and eigenvalue investigation. The middle cycles should be continued, and stable/unstable pair is verified if fold bifurcation of cycles found. Currently there is no numerical method/software that can investigate homoclinic-like cycles for large-scale systems. A combination of analytic and numerical techniques should be developed and used.

27 Ongoing and Future Work Confirm and find exact values for parameters at bifurcations using continuation methods. Develop a framework for extraction of underlying low-dimensional dynamics. p.9/9