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 ) – p.1/9

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

Periodic Measles Recruitment Incidence Spectral density Measles (New York City, USA) Power spectrum 30000 1.5 e a 0.02 20000 1 0.01 10000 0.5 0 0 0 1928 1928 1938 1938 1948 1948 1958 1958 0 0.5 1 1.5 Year Frequency (1/yr) Measles Incidence in Liverpool, England power spectrum 1.5 800 case reports 600 1 400 0.5 200 0 0 50 55 60 65 0 0.5 1 1.5 year frequency power spectrum Measles Incidence in Ontario, Canada 1.5 8000 case reports 6000 1 4000 0.5 2000 0 0 1900 1910 1920 1930 1940 0 1 1950 0.5 1.5 year frequency (source: Mathematical Epidemiology ; Brauer et al., 2008) – p.3/9

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 y y − ξyz dv dt = ky − uv − βφvx dw cφwv dt = λ z − − d w w C 1 φv + K 1 − ( ρ + d z ) z dz cφvw pφvz fc m φvm dt = + + C 1 φv + K 1 C 2 φv + K 2 C 3 φv + K 3 C 4 φv + K 4 dm ρz c m φvm dt = − d m m − C 3 φv + K 3 C 4 φv + K 4 – p.4/9

In-Host Model Initiate the adaptive immune Adaptive immune Immunological Establishment of Infection response response memory Level of virus in plasma Day 0 10-11 14 17-18 21 Pathogen Infectiousness Symptoms Infectiousness Pathogen enters plasma begins appear ends is cleared (source: Heffernan and Keeling, 2008) – p.4/9

In-Host Model ( a ) 8 ( b) 200 low m (0) 6 150 4 100 2 50 high m (0) 0 5 10 15 20 25 0 20 40 60 time (days) time (days) ( d ) 300 250 200 150 100 0 50 100 150 200 initial memory, (source: Heffernan and Keeling, 2010) – p.4/9

Between-Host Model No vaccine dS 0 dt = B + qR 0 + w 1 S 1 − λS 0 − dS 0 dS i dt = qR i + w i +1 S i +1 − λS i − dS i − w i S i dE i dt = λS i − a i E i − dE i dI i dt = a i E i − g i I i − dI i dR i � dt = w i +1 R i +1 + b i,j g j I j − w i R i − qR i − dR i j � λ = β i 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

Between-Host Model With vaccine dS 0 dt = B (1 − p ) + qR 0 + w 1 S 1 − λS 0 − dS 0 dS v dt = Bp + qR v + w v +1 S v +1 − λS v − dS v − w v S v dS i dt = qR i + w i +1 S i +1 − λS i − dS i − w i S i ∀ i � = 0 , v dE i dt = λS i − a i E i − dE i dI i dt = a i E i − g i I i − dI i dR i � dt = w i +1 R i +1 + b i,j g j I j − w i R i − qR i − dR i j � λ = β i I i i The value v = 90 is determined by the within-host model. Dimension = (# S ) + (# E ) + (# I ) + (# R ) = 200(300) + 15 + 15 + 200(300) = 430(630) – p.5/9

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

Numerics for Measles → Steady States � � � � Disease Free Equilibrium, ( E i = 0 , I j = 0 ) � � � � t =0 t =0 τ = 20 0 10 p = 0.10 p = 0.50 p = 0.90 −1 10 −2 10 S −3 10 −4 10 −5 10 0 10 20 30 40 50 60 70 80 90 100 i 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 ). – p.7/9

Numerics for Measles → Steady States � � � � Disease Free Equilibrium, ( E i = 0 , I j = 0 ) � � � � t =0 t =0 p = 0.50 p = 0.50 0.2 0.8 0 0.6 S eig E eig −0.2 0.4 τ = 12 −0.4 τ = 20 0.2 −0.6 τ = 80 −0.8 0 0 50 100 150 200 0 5 10 15 i i p = 0.50 p = 0.50 0.2 0.06 0.15 0.04 R eig I eig 0.1 0.02 0.05 0 0 −0.02 0 5 10 15 0 50 100 150 200 i i unstable direction – p.7/9

Numerics for Measles → Steady States Endemic Equilibrium τ = 20 τ = 20 −4 4 x 10 0.04 p = 0.10 p = 0.10 p = 0.50 p = 0.50 0.03 3 p = 0.90 p = 0.90 S E 0.02 2 0.01 1 0 0 0 50 100 150 200 0 5 10 15 i i τ = 20 τ = 20 −4 −3 4 x 10 6 x 10 p = 0.10 p = 0.10 p = 0.50 p = 0.50 3 4 p = 0.90 p = 0.90 R 2 I 2 1 0 0 0 5 10 15 0 50 100 150 200 i 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

Numerics for Measles → Bifurcations Continuation of Hopf bifurcation 1.2 1.1 1 p H 0.9 0.8 0.7 0 10 20 30 40 50 60 70 80 90 100 τ initial period (years) 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100 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). – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.43 τ = 20 , p = 0.43 1 0.8 total(S) 0.8 total(E) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.43 τ = 20 , p = 0.43 0.25 1 new infect = 10 −10 total(R) 0.2 total(I) new infect = 10 −8 0.5 new infect = 10 −5 0.15 End. Equ. 0.1 0 0.05 0 −0.5 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.53 τ = 20 , p = 0.53 1 0.8 new infect = 10 −10 total(S) 0.8 total(E) 0.6 new infect = 10 −8 new infect = 10 −5 0.6 0.4 End. Equ. 0.4 0.2 0.2 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.53 τ = 20 , p = 0.53 0.25 1 total(R) 0.2 total(I) 0.5 0.15 0.1 0 0.05 0 −0.5 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.63 τ = 20 , p = 0.63 1 0.8 new infect = 10 −10 total(S) 0.8 total(E) 0.6 new infect = 10 −8 new infect = 10 −5 0.6 0.4 End. Equ. 0.4 0.2 0.2 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.63 τ = 20 , p = 0.63 0.25 1 total(R) 0.2 total(I) 0.5 0.15 0.1 0 0.05 0 −0.5 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.73 τ = 20 , p = 0.73 1 0.8 new infect = 10 −10 total(S) 0.8 total(E) 0.6 new infect = 10 −8 new infect = 10 −5 0.6 0.4 End. Equ. 0.4 0.2 0.2 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.73 τ = 20 , p = 0.73 0.25 1 total(R) 0.2 total(I) 0.5 0.15 0.1 0 0.05 0 −0.5 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.83 τ = 20 , p = 0.83 1 0.5 new infect = 10 −10 total(S) 0.8 total(E) 0.4 new infect = 10 −8 new infect = 10 −5 0.6 0.3 End. Equ. 0.4 0.2 0.2 0.1 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.83 τ = 20 , p = 0.83 0.25 1 total(R) 0.2 0.8 total(I) 0.15 0.6 0.1 0.4 0.05 0.2 0 0 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.93 τ = 20 , p = 0.93 1 0.5 total(S) 0.8 total(E) 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.93 τ = 20 , p = 0.93 0.25 1 new infect = 10 −10 total(R) 0.2 total(I) new infect = 10 −8 0.5 new infect = 10 −5 0.15 End. Equ. 0.1 0 0.05 0 −0.5 0 50 100 150 0 50 100 150 time time introducing infection into Disease Free Equilibrium – p.8/9

Numerics for Measles → Bifurcations τ = 20 , p = 0.53 τ = 20 , p = 0.53 1 0.04 total(S) total(E) 0.8 0.03 0.6 0.02 0.4 0.01 0.2 0 0 50 100 150 0 50 100 150 time time τ = 20 , p = 0.53 τ = 20 , p = 0.53 0.04 0.8 random start total(R) End. Equ. total(I) 0.03 0.6 0.02 0.4 0.01 0.2 0 0 0 50 100 150 0 50 100 150 time time SIMULATING from a RANDOM state – p.8/9