Numerical qualitative analysis of a large-scale model for measles - - PowerPoint PPT Presentation

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)

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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

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Periodic Measles

1928 1938 1948 1958 10000 20000 30000 Measles (New York City, USA) 0.5 1 1.5 Frequency (1/yr) 0.5 1 1.5 Power spectrum 1928 1938 1948 1958 Year 0.01 0.02 Incidence Recruitment density Spectral

a e

50 55 60 65 year case reports 200 400 600 800 Measles Incidence in Liverpool, England 1900 1910 1920 1930 1940 1950 year 2000 4000 6000 8000 case reports Measles Incidence in Ontario, Canada 0.5 1 1.5 frequency 0.5 1 1.5 power spectrum 0.5 1 1.5 frequency 0.5 1 1.5 power spectrum

(source: Mathematical Epidemiology; Brauer et al., 2008)

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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 = λx − dxx − βφxv dy dt = βφxv − dyy − ξyz dv dt = ky − uv − βφvx dw dt = λz − cφwv C1φv + K1 − dww dz dt = cφvw C1φv + K1 + pφvz C2φv + K2 − (ρ + dz)z C3φv + K3 + fcmφvm C4φv + K4 dm dt = ρz C3φv + K3 − dmm − cmφvm C4φv + K4

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In-Host Model

Establishment of Infection Initiate the adaptive immune response Day Level of virus in plasma Immunological memory Adaptive immune response 14 17-18 21 10-11 Pathogen enters plasma Infectiousness begins Symptoms appear Infectiousness ends Pathogen is cleared

(source: Heffernan and Keeling, 2008)

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In-Host Model

5 10 15 20 25 2 4 6 8 (a) (b) d) time (days) low m(0) high m(0) 20 40 60 50 100 150 200 time (days) 50 100 150 200 100 150 200 250 300 initial memory,

(

(source: Heffernan and Keeling, 2010)

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Between-Host Model

No vaccine dS0 dt = B + qR0 + w1S1 − λS0 − dS0 dSi dt = qRi + wi+1Si+1 − λSi − dSi − wiSi dEi dt = λSi − aiEi − dEi dIi dt = aiEi − giIi − dIi dRi dt = wi+1Ri+1 +

• j

bi,jgjIj − wiRi − qRi − dRi λ =

• i

βiIi

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

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Between-Host Model

With vaccine dS0 dt = B(1 − p) + qR0 + w1S1 − λS0 − dS0 dSv dt = Bp + qRv + wv+1Sv+1 − λSv − dSv − wvSv dSi dt = qRi + wi+1Si+1 − λSi − dSi − wiSi ∀i = 0, v dEi dt = λSi − aiEi − dEi dIi dt = aiEi − giIi − dIi dRi dt = wi+1Ri+1 +

• j

bi,jgjIj − wiRi − qRi − dRi λ =

• i

βiIi

The value v = 90 is determined by the within-host model. Dimension = (#S) + (#E) + (#I) + (#R) = 200(300) + 15 + 15 + 200(300) = 430(630)

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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.

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Numerics for Measles → Steady States

Disease Free Equilibrium, (Ei

• t=0

= 0, Ij

• t=0

= 0)

10 20 30 40 50 60 70 80 90 100 10

−5

10

−4

10

−3

10

−2

10

−1

10

i S τ = 20

p = 0.10 p = 0.50 p = 0.90

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 (0 ≤ P ≤ 1, 10 ≤ τ ≤ 100).

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Numerics for Measles → Steady States

Disease Free Equilibrium, (Ei

• t=0

= 0, Ij

• t=0

= 0)

50 100 150 200 −0.8 −0.6 −0.4 −0.2 0.2

i

Seig

p = 0.50

5 10 15 0.2 0.4 0.6 0.8

i

Eeig

p = 0.50

5 10 15 0.05 0.1 0.15 0.2

i

Ieig

p = 0.50

50 100 150 200 −0.02 0.02 0.04 0.06

i

Reig

p = 0.50

τ = 12 τ = 20 τ = 80

unstable direction

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Numerics for Measles → Steady States

Endemic Equilibrium

50 100 150 200 0.01 0.02 0.03 0.04

i S τ = 20

5 10 15 1 2 3 4 x 10

−4

i E τ = 20

5 10 15 1 2 3 4 x 10

−4

i I τ = 20

50 100 150 200 2 4 6 x 10

−3

i R τ = 20

p = 0.10 p = 0.50 p = 0.90 p = 0.10 p = 0.50 p = 0.90 p = 0.10 p = 0.50 p = 0.90 p = 0.10 p = 0.50 p = 0.90

This stable equilibrium goes under a Hopf bifurcation and looses it stability at p = pH. The Hopf bifurcation is supercritical.

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Numerics for Measles → Bifurcations

Continuation of Hopf bifurcation

10 20 30 40 50 60 70 80 90 100 0.7 0.8 0.9 1 1.1 1.2

τ pH

10 20 30 40 50 60 70 80 90 100 2 4 6 8

τ

initial period (years)

This was our first guess for oscillation mechanism. BUT, soon we observed that the amplitudes of

• scillations were very small (not surprising for Hopf bifurcation).

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.43

50 100 150 0.2 0.4 0.6 0.8

time

total(E) τ = 20 , p = 0.43

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.43

50 100 150 −0.5 0.5 1

time

total(R) τ = 20 , p = 0.43

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.53

50 100 150 0.2 0.4 0.6 0.8

time

total(E) τ = 20 , p = 0.53

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.53

50 100 150 −0.5 0.5 1

time

total(R) τ = 20 , p = 0.53

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.63

50 100 150 0.2 0.4 0.6 0.8

time

total(E) τ = 20 , p = 0.63

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.63

50 100 150 −0.5 0.5 1

time

total(R) τ = 20 , p = 0.63

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.73

50 100 150 0.2 0.4 0.6 0.8

time

total(E) τ = 20 , p = 0.73

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.73

50 100 150 −0.5 0.5 1

time

total(R) τ = 20 , p = 0.73

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.83

50 100 150 0.1 0.2 0.3 0.4 0.5

time

total(E) τ = 20 , p = 0.83

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.83

50 100 150 0.2 0.4 0.6 0.8 1

time

total(R) τ = 20 , p = 0.83

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.93

50 100 150 0.1 0.2 0.3 0.4 0.5

time

total(E) τ = 20 , p = 0.93

50 100 150 0.05 0.1 0.15 0.2 0.25

time

total(I) τ = 20 , p = 0.93

50 100 150 −0.5 0.5 1

time

total(R) τ = 20 , p = 0.93

new infect = 10−10 new infect = 10−8 new infect = 10−5

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.53

50 100 150 0.01 0.02 0.03 0.04

time

total(E) τ = 20 , p = 0.53

50 100 150 0.01 0.02 0.03 0.04

time

total(I) τ = 20 , p = 0.53

50 100 150 0.2 0.4 0.6 0.8

time

total(R) τ = 20 , p = 0.53

random start

• End. Equ.

SIMULATING from a RANDOM state

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.63

50 100 150 0.01 0.02 0.03

time

total(E) τ = 20 , p = 0.63

50 100 150 0.01 0.02 0.03 0.04 0.05

time

total(I) τ = 20 , p = 0.63

50 100 150 0.2 0.4 0.6 0.8

time

total(R) τ = 20 , p = 0.63

random start

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.73

50 100 150 0.01 0.02 0.03 0.04

time

total(E) τ = 20 , p = 0.73

50 100 150 0.01 0.02 0.03 0.04

time

total(I) τ = 20 , p = 0.73

50 100 150 0.2 0.4 0.6 0.8

time

total(R) τ = 20 , p = 0.73

random start

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.83

50 100 150 0.01 0.02 0.03 0.04

time

total(E) τ = 20 , p = 0.83

50 100 150 0.01 0.02 0.03

time

total(I) τ = 20 , p = 0.83

50 100 150 0.2 0.4 0.6 0.8

time

total(R) τ = 20 , p = 0.83

random start

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

50 100 150 0.2 0.4 0.6 0.8 1

time

total(S) τ = 20 , p = 0.93

50 100 150 0.01 0.02 0.03 0.04 0.05

time

total(E) τ = 20 , p = 0.93

50 100 150 0.01 0.02 0.03 0.04

time

total(I) τ = 20 , p = 0.93

50 100 150 0.2 0.4 0.6 0.8

time

total(R) τ = 20 , p = 0.93

random start

• End. Equ.

introducing infection into Disease Free Equilibrium

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Numerics for Measles → Bifurcations

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.

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Numerics for Measles → Bifurcations

Unstable Disease Free Equilibrium Stable Endemic 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.

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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.

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