Mathematical models of
infectious diseases
D. Gurarie
Lecture outline
• Goals
• Methodology
• Basic SIR and SEIR
– BRN: its meaning and implications
– Control strategies: treatment, vaccination/culling,
quarantine
– Multiple-hosts: zoonotics and vector-born diseases
1. Math modeling: issues, problems
Spread of diseases in populations
•Biological factors (host-parasite interactions)
•Environmental – behavioral factors (‘transmission environment’)
Public health assessment (morbidity, mortality)
Intervention and control
•Drug treatment (symptomatic, prophylactic)
•Vaccines
•Transmission prevention
Modeling Goals
Develop mathematical/computer techniques, tools,
methodology to
i) Predict outcomes
ii) Analyze, develop control strategies
2. Early history of math. modeling (XVIII
century smallpox)
• Known facts:
– Short duration (10 days), high mortality (up to
75%)
– Life-long immunity for survivors
– Possible prevention: inoculation by cow-pox
• Q: could life expectancy be increased by
preventive inoculation?
• Approach: age-structured model of
transmission + ‘analysis’ =>
• Answer: gain of 2.5 years
Daniel Bernoulli
1700-1782
3. Transmission patterns
1. Direct: host – to-host (flu, smallpox, STD,…)
2. Vector –borne diseases
Macro-parasites: schistosome life cycle
schisto
malaria
This diagram is provided by Center for Disease Control and Prevention (CDC).
3. Epizootic: WNV, Marburg, …
4. Infection patterns: typical flu outbreak
Data (British Medical Journal, March 4 1978, p. 587)
Influenza Epidemic
Infectives
300
250
200
150
100
50
0
Day
0
2
4
6
8
10
12
14
Explain outbreak pattern ?
Predict (peak, duration, cumulative incidence) ?
Control (drug, vaccine, quarantine) ?
5. SIR –methodology: host ‘disease states’
and history
S – Susceptible
E – Exposed
I – Infectious
R - Removed /immune
I
R
E
S
Latency
Infective stage
Immune stage
…
6. SIR transmission in randomly mixing
community
• Community of N hosts, meet in random groups of c (or less) = contact rate
• Host states and transitions: S I  R
• Probability of infection/infectious contact = 1-p
• Recovery rate = 1-r (=> mean duration of I-state T=1/(1-r))
• Life long immunity
Groups
Day
1
2
3
…
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
{1},{2,9,13},{4,5,10},{6,7,8}
Simulated pattern: infection outbreak
Questions
•Outbreak duration, peak -?
•Cumulative incidence (other health statistics)-?
• Dependence on c (contact), p (transmissibility), r (recovery) - ?
• Control, prevention ??
•Drug treatment
• Vaccine
• Quarantine
II. SIR methodology: diagrams
SI
S
SIR
SEIR
S
I
Birth
I
R
SEIR
S
Death
E
I
R
V
S
E
I
Loss
R
recruitment
V
Variables: S, E, I, R, V (vaccinated) – host states, or populations /fractions
Models:
•Continuous (DE) for {S(t),… }- functions of time t
•Discrete {S(t),… } (t=0,1,2,…)
•Community level (populations)
• Individual level (agent based)
Discrete SIR: Reed-Frost
S+I+R=N (or S+E+I+R=N) - populations, or prevalences: S+E+I+R=1
Parameters:
c - contact rate (‘average # contacts’/host/day)
p – probability to ‘survive infectious contact’ (1-p = susceptibility)
l(p,c) – force of infection
q – probability to stay latent => latency duration =1/(1-q) - ??
r –probability to stay infected => infectious period=1/(1-r)
s –probability to stay immune => immune duration=1/(1-s)
l
S
1-s
S
I
1-r
R
l
E
1-q
I
1-r
1-s
R
Reed-Frost map (discrete time step)
“current state”  “next state”
(S,I,R)  (S’,I’,R’)
(S,E,I,R)  (S’,E’,I’,R’)
(S=S(t),… )  (S’= S(t+1),…)
Equations:
 S   p c I / N  S  1  s  R

 E   1  p c I / N  S  qE

 I   (1  q) E  rI

 R  1  r  I  sR
 S   p c I / N  S  1  s  R


c I / N 
S  rI
I   1  p

 R  1  r  I  sR

Parameter


Probability

Time (duration)
p
survive infectious contact
q
stay latent (E-state)
1/(1-q)
r
stay infected (I-state)
1/(1-r)
s
stay immune (R-state)
1/(1-s)
c
contact rate/day
c(I/N)
number of ‘infectious contacts’
No analytic solution!
SIR
Smallpox
Duration (days)
Probability = 1-1/T
latent (E-state)
3
q=.66
infected (I-state)
7
r=.86
immune (R-state)
All life
s=1
10
r=.9
All life
s=1
Flu
Duration (days)
Probability = 1-1/T
latent (E-state)
2
q=.5
infected (I-state)
5
r=.8
immune (R-state)
150
s=.993
7
r=.86
150
s=.993
Numeric simulations
Smallpox
Flu
SIR
Infect. period Epidemic peak Day Cumulative incidence BRN
10.
0.495439
17
0.984751
5.26803
Infect. p eriod
7.
1.0
1.0
0.8
0.8
0.6
0.6
Ep idemic p eak
0.442523
Day
15
Endemic levels
0.23, 0.03, 0.74
BRN
4.42514
0.4
0.4
0.2
0.2
0
20
40
60
80
Infection period Epidemic peak Day Cumulative incidence BRN
10.
0.192664
34
0.931552
3.16082
SEIR
50
100
150
200
100
Infect. p eriod
7.
1.0
1.0
0.8
0.8
0.6
0.6
Ep idemic p eak
0.188307
Day
27
Endemic levels
0.38, 0.01, 0.02, 0.59
BRN
2.63401
0.4
0.4
0.2
0.2
0
20
40
60
80
100
50
100
150
200
250
300
Analysis of outbreaks and endemic
equilibria
3 basic parameters
i) Susceptibility :1-p (‘resistance to infection’ = p)
ii) Contact rate: c
iii) recovery rate: r
Questions:
1. How (p,c,r) would determine infection pattern: outbreak,
endemic equilibria levels et al?
1. Control intervention -?
Key index: BRN
1
c
R0 
ln  
1 r  p 
R0 > 1 – stable endemic infection (flu); outbreak of increased strength (smallpox)
R0 < 1 – stable eradication (flu); no outbreaks (smallpox)
Control, prevention
•Drug treatment  r (“prophylactic MDT“-> p)
• Vaccine  ‘S- fraction’, p
• Quarantine  c
1. Effect of vaccine
0<f<1 – cover fraction
e>1 – efficacy (enhanced resistance):
(normal) p(vaccinated) pV = p1/e
1. Perfect vaccine (1/e = 0 – full
resistance) Vaccination = Effective
reduction of contact rate: c (1-f)c
 Reduced BRN
V
0
R
S
E
f
I
V
c 1  f   1 

ln    1  f  R0  1(?)
1 r
 p
If R0 is known (?), cover fraction f=1-1/R0 needed
to eradicate infection.
R
2. Imperfect vaccine (1/e>0)
0<f<1 – cover fraction
e>1 - efficacy
Effects of vaccine:
• reduce risk of infection under identical
‘infected contacts’: p pV = p1/e > p
• enhance recovery: r rV = re <r
l  1  p  lV  1  pV
cI
1-f
S
l
E
I
R
f
cI
 Effect (f,e) - ?
 BRN: R0(f,e) -?
Can BRN be brought <1 ?
S’
lV
E’
I’
III. Continuous (DE) models
r
r
lbI
r
S
I
R
S
lbI
dS
dS
Differential
equations
 b
dt
dI
dt
dR
dt
b
I
N
I
N
S  r R  ...
S  r    I
 1  d  rI   r    R
dt
dE
dt
dI
dt
dR
dt
Parameters
b  b
r
a
 /d
r
a
E
b
I
I
 b
N
I
N
r
R
S  ...
S  a    E
 aE   r    I
 1  d  rI  ...
transmission coefficient = “contact rate” x “prob. infection/contact”
recovery rate = 1/”mean duration”)
1/”latency period”
Natural/disease mortality
immune loss rate
2. Smallpox SIR (immune loss r=0)
 S   b SI
 ... Solution ...

 I  b SI  rI
Phase-plane
S
Time series
1.0
1.0
0.8
0.8
0.6
Cumulative
incidence
0.6
I
0.4
0.4
0.2
0.2
0
0.0
0.0
0.2
0.4
0.6
0.8
2
4
6
8
1.0
1/R0
BRN:
R0 
b
r
or
b
r   
Transmission
=
Re cov ery (+ loss)
10
3. SIR with immune loss (flu)
dS
Prevalence
DE
dt
dI
dt
dR
dt
  b IS  r R  dI
 b IS   r  d  I
r – recovery
d –disease mortality
r – immune loss
R0 
S *  1/ R0 ;
b
I *  1  1/ R0 
rd
Jacobian matrix for (S,I)
R0  1
R0  1
Eradication: (1,0,0)
 r / b
 0

...

1  1/ R0 
S , I , R 
   I  r / b   1  r / b  


Endemic:
*
*
*
*

I*
Saddle/ unstable
Sink /stable
Sink /stable
Saddle/ unstable
r
;
rr
r
R*  1  1/ R0 
;
rr
 rI  r R
Equilibrium
Analysis:
Endemic Equilibrium
BRN
0

4. BRN: meaning, implications
• (SIR with life-long immunity): R0 determines whether
outbreak occurs (R0 >1), or infection dies out (R0 <1)
R 1 rt
• BRN is related to initial infection growth : I  t   I 0e 
As e R 1  R0 , R0 approximately measures “# secondary
cases/per single infected” over “time range ” r t=1
• BRN (R0 >1) determines infection peak and timing, depending
on initial state I0
• For SIR with immune loss sets apart: (i) endemic equilibrium
state (R0 >1), or waning of infection (R0 <1)
0
0
5. Control intervention
• Vaccination (herd immunity):
– vaccinating fraction f of susceptibles decreases R0  (1f)R0. So f>1-1/R0 prevents outbreak
– culling of infected animals has the same effect I(1-f)I
• Demographics:
– increased population density N drives R0 = bN/r up
(enhanced outbreaks, higher endemicity)
• Transmission prevention:
– Lower transmission rate b decreases R0
IV. Vector mediated transmission
Viral: RVF, Dengue, Yellow fever,
Plasmodia: Malaria, toxoplasma
Parasitic worms: schistosomiasis, Filariasis
2. Coupled SIR-SEIR diagrams
Host:
X
Y
Vector:
w
X
Z
v
u
Z
w
R
R
u
3. Macro-parasites: schistosome life cycle
This diagram is provided by Center for Disease Control and Prevention (CDC).
4. Macdonald model: mean intensity-burden
(host) + prevalence (vector)
Infection intensity (burden) is important for macro-parasites
w=mean worm burden of H population;
y=prevalence of shedding snail;
Premises:
•Steady snail population and
environment
•Homogeneous human population, and
transmission patterns (contact
/contamination rates, worm
establishment ets)
b 2 HN
BRN: R0    
AB
=> equilibria, analysis and control (??)
Summary (math modeling)
• Models either ‘physical’ (mice) or ‘virtual’ (math) allow one to recreate ‘reality’ (or
part of it) for analysis, prediction, control experiment s
• Methodology:
• Models need not reproduce a real system (particularly, complex biological
ones) in full detail.
• The ‘model system’ is made of ‘most essential’ (in our view) components and
processes
• For multi-component systems we start with diagrams, then produce more
detailed description (functions, equations, procedures)
• Math models have typically many unknown/uncertain parameters that need
to be calibrated (estimated) and validated with real data
•Simple math. models can be studied by analytic means (pen and paper) to draw
conclusions
•Any serious modeling nowadays involves computation.