Dynamical Models of Epidemics: from Black Death to SARS
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Transcript of Dynamical Models of Epidemics: from Black Death to SARS
Dynamical Models of Dynamical Models of Epidemics: from Black Death Epidemics: from Black Death to SARSto SARS
D. GurarieD. Gurarie
CWRUCWRU
Epidemics in History – Plague in 14th Century Europe killed 25 million – Aztecs lost half of 3.5 million to smallpox – 20 million people in influenza epidemic of 1919
Diseases at Present – 1 million deaths per year due to malaria – 1 million deaths per year due to measles – 2 million deaths per year due to tuberculosis– 3 million deaths per year due to HIV – Billions infected with these diseases
History of Epidemiology. Hippocrates's On the Epidemics (circa 400 BC). John Graunt's Natural and Political Observations made upon the Bills of Mortality (1662). Louis Pasteur and Robert Koch (middle 1800's)
History of Mathematical Epidemiology. Daniel Bernoulli studied the effect of vaccination with cow pox on life expectancy (1760). Ross's Simple Epidemic Model (1911). Kermack and McKendrick's General Epidemic Model (1927)
HistoryHistory
SchistosomiasisSchistosomiasis Chronic parasitic trematode infectionChronic parasitic trematode infection 200-300 million people worldwide200-300 million people worldwide Significant morbidity (esp. anemia) Significant morbidity (esp. anemia) Premature mortalityPremature mortality Life-cycle is complex, requiring species-specific Life-cycle is complex, requiring species-specific
intermediate snail host intermediate snail host Optimal control strategies have not been Optimal control strategies have not been
established.established.
Geographic Distribution -1990
Smallpox: XVIII centurySmallpox: XVIII century
Known facts:Known facts:– Short duration (10 days), high Short duration (10 days), high
mortality (75%)mortality (75%)– Life-long immunity for survivorsLife-long immunity for survivors– Prevention: immunity by inoculation Prevention: immunity by inoculation
(??)(??) Problem: could public health (life Problem: could public health (life
expectancy) be improved by expectancy) be improved by inoculation?inoculation? Daniel Bernoulli
1700-1782“I simply wish that, in a matter which so closely concerns the well-being of mankind, no decision shall be made without all the knowledge which a little analysis and calculation can provide.”
Daniel Bernoulli, on smallpox inoculation, 1766
Bernoulli smallpox model Bernoulli smallpox model (1766)(1766)
1) Population cohort of age a, n(a), mortality (a)
2 4 6 8 10a
0.2
0.4
0.6
0.8
1
adnda
n; n0 n0 na n0a=(a) natural mortality
(a) exp-0a a survival function
L0 0 a life expectacy
Special cases :
1. = 0; 0 a L0; a L0
= 1; 0 a L00; a L0
2. Const = e a; L0 1
X - susceptible pop. (infected pop. Z n , rapidly die
or recover with immunity)
- death incidence (1 - survival rate)
- force of infection/capitadnda
n X ; n0 n0
dXda
X ; X0 n0
Solution: na n0a1 ea 1a
Life spans:L0 0 a 1
;
L1 0 1 a 1
;
Bernoulli estimates: 1/8year; =1/8; L1 26.5 years
Life gained: L0 L1 2.54years
2) Small pox effect
Caveat: if inoculation mortality is includedone would need <.5% for success!
Modeling issues and Modeling issues and strategiesstrategies State variables for host/parasiteState variables for host/parasite
– ““mean” or “distributed” (deterministic/stochastic)mean” or “distributed” (deterministic/stochastic)– Prevalence or level/intensityPrevalence or level/intensity– Disease stages (latent,…)Disease stages (latent,…)– Susceptibility and infectiousnessSusceptibility and infectiousness
TransmissionTransmission– Homogeneous (uniformly mixed populations): “mass action”Homogeneous (uniformly mixed populations): “mass action”– Heterogeneous: age/gender/ behavioral strata, spatially Heterogeneous: age/gender/ behavioral strata, spatially
structured contactsstructured contacts– Environmental factorsEnvironmental factors
Multi-host systems, parasites with complex life cycles, …. Multi-host systems, parasites with complex life cycles, …. Goals of epidemic modelingGoals of epidemic modeling
– PredictionPrediction– Risk assessmentRisk assessment– Control (intervention, prevention)Control (intervention, prevention)
Box (compartment) Box (compartment) diagramsdiagrams
S – Susceptible E – Exposed
I – Infectious R - Removed
V – Vaccinated …
S ISI
S E I R
V
SEIR
S I RSIR S E I R
V
SEIR
Birth Death
recruitment
Total population: N = S+I+…
SIR-type SIR-type modelsmodelsRoss, Kermak-Ross, Kermak-McKendrickMcKendrick
•Population size is large and constant•No birth, death, immigration or emigration •No recovery •No latency•Homogeneous mixing
SI
1 2 3 4 5 6S
0.5
1
1.5
2
2.5
3
3.5
4I SI phaseplot
5 10 15 20t
0.2
0.4
0.6
0.8
1SIR
N
RtItSt
Residual S(∞)>0
S I
2 4 6 8t
0.2
0.4
0.6
0.8
1
S,I
S
S I
I
S I Logistic
I
N IIIt N I0N I0I0e N t
Transmission : S rate of new infection
per I
SIR with immunity
S I R
S
S I
I
S I I
R
I
Reductions :
d Id S
1 S
d Sd R
S
S transmission; recovery rate
Basic Reproductionnumber: R0=N/R– endemicR0<1 - eradication
S R
S
S I R
I
S I I
R
I R
ReductionS
S I N S II
S I :
S transmission; recovery rate; loss of immunity
Equilibria Jacobian
I , N
A B N B
0
II N, 0 B0 N
Saddle node bifurcation in N , or R0 N
:
R0 1 stable endemic equilibriumIR0 1 eradicationstable IIendemic epidemic
100 200 300 400
0.2
0.4
0.6
0.8
1
RIS
0.4 0.6 0.8 10
0.025
0.05
0.075
0.1
0.125
0.15
S
I
Control(i) R0=“transmissiom”x”pop. density”/”recovery”<1. Hence critical density N>/b
to sustain endemic level(ii) Vaccination removes a fraction of N from transmission cycle: so eradication
(equilibrium I<0) requires (1-1/R0) fraction of N vaccinated
SIR with loss of immunity
““Smallpox cohort” SIRSmallpox cohort” SIR
X Y
X
X
XZ
X ZY
1 Z Y
n n ZtotalHigh recovery rateshort illness duration: ,
implies near equilibrated Z Z
X,
Hence Bernoulli equation :X
Xn n X
Growth models: variable population Growth models: variable population N(t)N(t)
Const recruitment
.01
100 200 300 400
10
20
30
40 NIS
0 5 10 15 20 25 300
5
10
15
20
25
30
S
I
Linear growth due to S(Voltera-Lotka)
Linear growth rate due to S,I
S
S I a
I
S I I N
a I;SI
a ;SIREquilibrium :
,
a. Jacobian : a
a
0
sinkspiral sink.Nt, Itstabilize at N, IendemicS
S I a S
I
S I I N
a S I;SI
a S ;SIRa growth rate of S
Equilibrium :,
a. Jacobian :0
a 0 centercyclesSt, Itdo in cycles.
S
S I a S b I
I
S I I N
a S b I;SI
a S ;SIRa, b growth rates of S, contributed by S, Ib Equilibrium :
,
a b. Jacobian : ab
b b
ab
0
sinkspiral sinkSt, Itstabilize at endemic levels, Nt
HIV/AIDS and HIV/AIDS and STD STD
• Variable population N=S+I• Natural growth a for S• Mortality =10/year for I• Transmission: S I/(S+I)
= mean number of partners/per IS/(S+I) probability of infecting S (S-fraction of N)
10 20 30 40tyears0.2
0.4
0.6
0.8
1
s,i,n
isnm
Typical collapse
Conclusion: a Transm. treatment
Treatment w/o prevention of spread can only increase (collapse!)
SI wo recoveryAIDS: S
a S S ISI
I
S ISI I
Basicparameters: a or R0
a
transmission
birth recovery
Analytic solution based on function : mt IS m0e
t
Case : 0;mtm0
uncontroled epidemics: SS0
eat; II0
e tislow natural growth : a aiiincrease IN fraction S decays faster than IiiiCollapse of S, I, N
Case : 0;mtm0
, butSS0
ea't;II0
ea' t;igrow at rate rate : a' a m01 m0
aiimaintain constant IN fraction endemicCase : 0;
mtm0
0 :SS0
ea t;II0
eat,irestore natural growth rate aiidecrease IN fraction eradication
20 40 60 80 100 120 140
0.25
0.5
0.75
1
1.25
1.5
1.75
IfemIhetIhomSfemShetShom
bhH bhT btF bfT f H.1 .01 .4 .6 5 .01
Parameters: Initial stateShom Shet Sfem Ihom Ihet Ifem.1 1 1 .01 0 0
AIDS for behavioral groups: 6D model
Data (trends) of several African countries
Heterogeneous transmission Heterogeneous transmission for distributed populationsfor distributed populations
• SIR type are only conceptual models• Idealize transmissions and individual characteristics (susceptibilities)• Real epidemics requires heterogeneous models:
• age structure • spatial/behavioral heterogeneity, etc.
Age structured models Age structured models (smallpox)(smallpox)
t n a n n a 0, t 0;BCna0 bana, t a;ICnt0 n0a;
ba birth rate
Equilibriumsurvival f n: na n0exp0a a,provided :
0
baexp0
a
a a 1
Continuous population strata n(a,t), age “a”, time “t”
Discrete population bins: n=(na)
ODS t n An with Leslie matrix
A
b1 1 1b2 ... bn1 1 20 0
0 1 ... 0
1 1 n
Equilibrium: nk jk11 j, provided: k1
n bkj1k1 j 1
t n a n n X
t X a X X a 0, t 0
n Xa0 ba'na', t a'n Xt0 n0a all newborn susceptible
Force of infection : =a, a'na' Xa' a't n
A n .n X
Xt X
A X
.n X
X
11 12 ...21 22 ...
Structured transmission
"susceptibility" "contact rate"
Example :
" contact matrixij" "suscept. vector 1 1 ... P"
Normal growth Infection
Example: 15-bin system with linear Example: 15-bin system with linear growth and structured transmissiongrowth and structured transmission
2 4 6 8 10 12 14
0.045
0.05
0.055
0.06
0.065
Natural mortality
2 4 6 8 10 12 14
0.2
0.4
0.6
0.8
1Survival function : 1.13
Fatality rate 10%
10 20 30 40
10
20
30
40
Total
10 20 30 40
0.2
0.4
0.6
0.8
1
1.2
1.4Susceptible
Fatality rate 75%
10 20 30 40
1
2
3
4
Total
10 20 30 40
0.5
1
1.5
2
2.5Susceptible
Age bins: red (young) to blue (old)
High survival
Low survival
Fisher’s Equation (1937)
Infection: S(x,t), I(x,t) – (distributed) susceptibles and infectives
• Population density is constant N • No birth or death • No recovery or latent period • Only local infection • Infection rate is proportional to the number of infectives• Individuals disperse diffusively with constant D
1t S S I Dx2 S
tI S I Dx2 I t I IN I Dx
2 I
2t S IS Dx2 S
tI S I Dx2 I t I IN
I Dx
2 I ; N NxEquilibria solutions of BVP :
uxx fu 0; 0 x Lu0 uL 0
Elliptic function u0xor 0Linearized stability for S L operator : M = x2 f'u0
Original motivation: spread of a genetype in a population)
-4-2
02
4 0
5
10
15
00.250.5
0.751
-4-2
02
4-4 -2 2 4
0.2
0.4
0.6
0.8
1
-2
0
20
20
40
60
80
100
0
0.05
0.1
-2
0
2
-3 -2 -1 1 2 3
-0.1
-0.05
0.05
0.1
0.15
Spreading wave in uniformmedium with const pop. density
Spreading wave with variable pop. density (red)
Solutions: propagating density Solutions: propagating density waveswaves
Problems:•Equilibrium, Basic Reproduction Number?•Speed of propagation (traveling waves)?•Parameters for control, prevention?
Some current modeling Some current modeling issues and approachesissues and approaches
Spatial/temporal patterns of Spatial/temporal patterns of
outbreaks and spreadoutbreaks and spread
Stochastic modelingStochastic modeling
Cellular Automata and Agent-Cellular Automata and Agent-
Based ModelsBased Models
Network Models (STD)Network Models (STD)