Numerical qualitative analysis of a large-scale model formeasles 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
1928 1938 1948 19580
10000
20000
30000Measles (New York City, USA)
0 0.5 1 1.5
Frequency (1/yr)
0
0.5
1
1.5Power spectrum
1928 1938 1948 1958
Year
0
0.01
0.02
Incid
en
ce
Re
cru
itm
en
t
de
nsity
Sp
ectr
al
a e
50 55 60 65
year
case
rep
ort
s
0
200
400
600
800
Measles Incidence in Liverpool, England
1900 1910 1920 1930 1940 1950
year
2000
4000
6000
8000
case
rep
ort
s
0
Measles Incidence in Ontario, Canada
0 0.5 1 1.5
frequency
0
0.5
1
1.5power spectrum
0 0.5 1 1.5
frequency
0
0.5
1
1.5
power spectrum
(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= 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
– p.4/9
In-Host Model
Establishment of Infection
Initiate the
adaptive
immune
response
Day
Level of
virus in p
lasm
a
Immunological
memory
Adaptive immune
response
14 17-18 210 10-11
Pathogen
enters plasma
Infectiousness
begins
Symptoms
appear
Infectiousness
ends
Pathogen
is cleared
(source: Heffernan and Keeling, 2008)– p.4/9
In-Host Model
0 5 10 15 20 25
2
4
6
8(a) (b)
d )
time (days)
low m(0)
high m(0)
0 20 40 60
50
100
150
200
time (days)
0 50 100 150 200100
150
200
250
300
initial memory,
(
(source: Heffernan and Keeling, 2010)
– p.4/9
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 clearthe virus before T-cell activation preventing boosting. Class S refers to those individuals who have lostthis short-term protection
– p.5/9
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 6= 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)
– 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 partialdifferential 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, (Ei
∣
∣
∣
∣
t=0
= 0, Ij
∣
∣
∣
∣
t=0
= 0)
0 10 20 30 40 50 60 70 80 90 10010
−5
10−4
10−3
10−2
10−1
100
i
Sτ = 20
p = 0.10p = 0.50p = 0.90
Extensive numerical simulations show that the Jacobian at the disease free equilibrium always has oneand only one positive eigenvalue. Hence, this equilibrium is always unstable and there is no localbifurcation for our desired parameter range (0 ≤ P ≤ 1, 10 ≤ τ ≤ 100).
– p.7/9
Numerics for Measles → Steady States
Disease Free Equilibrium, (Ei
∣
∣
∣
∣
t=0
= 0, Ij
∣
∣
∣
∣
t=0
= 0)
0 50 100 150 200−0.8
−0.6
−0.4
−0.2
0
0.2
i
Sei
gp = 0.50
0 5 10 150
0.2
0.4
0.6
0.8
i
Eei
g
p = 0.50
0 5 10 150
0.05
0.1
0.15
0.2
i
I eig
p = 0.50
0 50 100 150 200−0.02
0
0.02
0.04
0.06
i
Rei
g
p = 0.50
τ = 12τ = 20τ = 80
unstable direction
– p.7/9
Numerics for Measles → Steady States
Endemic Equilibrium
0 50 100 150 2000
0.01
0.02
0.03
0.04
i
Sτ = 20
0 5 10 150
1
2
3
4x 10
−4
i
E
τ = 20
0 5 10 150
1
2
3
4x 10
−4
i
I
τ = 20
0 50 100 150 2000
2
4
6x 10
−3
i
R
τ = 20
p = 0.10p = 0.50p = 0.90
p = 0.10p = 0.50p = 0.90
p = 0.10p = 0.50p = 0.90
p = 0.10p = 0.50p = 0.90
This stable equilibrium goes under a Hopf bifurcation and looses it stability at p = pH . The Hopfbifurcation is supercritical.
– p.7/9
Numerics for Measles → Bifurcations
Continuation of Hopf bifurcation
0 10 20 30 40 50 60 70 80 90 100
0.7
0.8
0.9
1
1.1
1.2
τ
p H
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
τiniti
al p
erio
d (y
ears
)
This was our first guess for oscillation mechanism. BUT, soon we observed that the amplitudes ofoscillations were very small (not surprising for Hopf bifurcation).
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.43
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(E)
τ = 20 , p = 0.43
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.43
0 50 100 150−0.5
0
0.5
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.53
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(E)
τ = 20 , p = 0.53
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.53
0 50 100 150−0.5
0
0.5
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.63
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(E)
τ = 20 , p = 0.63
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.63
0 50 100 150−0.5
0
0.5
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.73
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(E)
τ = 20 , p = 0.73
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.73
0 50 100 150−0.5
0
0.5
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.83
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
time
tota
l(E)
τ = 20 , p = 0.83
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.83
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.93
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
time
tota
l(E)
τ = 20 , p = 0.93
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
time
tota
l(I)
τ = 20 , p = 0.93
0 50 100 150−0.5
0
0.5
1
time
tota
l(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
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.53
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(E)
τ = 20 , p = 0.53
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(I)
τ = 20 , p = 0.53
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(R)
τ = 20 , p = 0.53 random startEnd. Equ.
SIMULATING from a RANDOM state
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.63
0 50 100 1500
0.01
0.02
0.03
time
tota
l(E)
τ = 20 , p = 0.63
0 50 100 1500
0.01
0.02
0.03
0.04
0.05
time
tota
l(I)
τ = 20 , p = 0.63
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(R)
τ = 20 , p = 0.63 random startEnd. Equ.
introducing infection into Disease Free Equilibrium
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.73
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(E)
τ = 20 , p = 0.73
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(I)
τ = 20 , p = 0.73
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(R)
τ = 20 , p = 0.73 random startEnd. Equ.
introducing infection into Disease Free Equilibrium
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.83
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(E)
τ = 20 , p = 0.83
0 50 100 1500
0.01
0.02
0.03
time
tota
l(I)
τ = 20 , p = 0.83
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(R)
τ = 20 , p = 0.83 random startEnd. Equ.
introducing infection into Disease Free Equilibrium
– p.8/9
Numerics for Measles → Bifurcations
0 50 100 1500
0.2
0.4
0.6
0.8
1
time
tota
l(S)
τ = 20 , p = 0.93
0 50 100 1500
0.01
0.02
0.03
0.04
0.05
time
tota
l(E)
τ = 20 , p = 0.93
0 50 100 1500
0.01
0.02
0.03
0.04
time
tota
l(I)
τ = 20 , p = 0.93
0 50 100 1500
0.2
0.4
0.6
0.8
time
tota
l(R)
τ = 20 , p = 0.93 random startEnd. Equ.
introducing infection into Disease Free Equilibrium
– p.8/9
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 itsstrong absorbance in some directions.
� Therefore (almost) inaccessible from Disease Free Equilibrium by introducingnew infected individuals.
– p.8/9
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 cyclesfound. Currently there is no numerical method/software that can investigate homoclinic-like cycles forlarge-scale systems. A combination of analytic and numerical techniques should be developed and used.
– p.8/9
Ongoing and Future Work
� Confirm and find exact values for parameters at bifurcations using continuationmethods.
� Develop a framework for extraction of underlying low-dimensional dynamics.
– p.9/9
