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Mathematical modelling and controlling the dynamics of infectious diseases
Mathematical modelling and
controlling the dynamics of
infectious diseases
Musa Mammadov
Centre for Informatics and Applied OptimisationFederation University Australia
25 August 2017, School of Science, RMIT
Mathematical modelling and controlling the dynamics of infectious diseases
Joint work Rob Evans (University of Melbourne):
• R.J. Evans and M. Mammadov, Dynamics of Ebola epidemics in
West Africa 2014, F1000Research 2015, 3:319 (doi:
10.12688/f1000research.5941.2)
(This article is included in the Ebola collection, Peter Piot)
• R.J. Evans and M. Mammadov, Predicting and controlling the
dynamics of infectious diseases, CDC-2015: 54th IEEE
Conference on Decision and Control (CDC), 5378–5383
Mathematical modelling and controlling the dynamics of infectious diseases
Introduction
• Mathematical models
– SI, SIR, SEIR, SEIRS, · · ·
– time-delay
• Possible optimal control models
New results
• Ebola epidemics in West Africa 2013-2016
• Optimal control problems
– capacity of beds in hospitals
– time to isolation (hospitalization)
– numerical example
Mathematical modelling and controlling the dynamics of infectious diseases
Introduction
The West African Ebola virus epidemic of 2013 to 2016
was the most widespread epidemic of Ebola virus disease in
history.
Country Cases Deaths
Liberia 10,675 4,809
Sierra Leone 14,122 3,955
Guinea 3,814 2,543
Total 28,616 11,310
Mathematical modelling and controlling the dynamics of infectious diseases
Mathematical models:
• Statistical-Based Methods
• Dynamical systems (State-Space Models -
SI, SIR, SIRS, SEIR)
• Empirical/Machine Learning-Based Models
Mathematical modelling and controlling the dynamics of infectious diseases
Model SIR :
N(t) = S(t) + I(t) +R(t)
• N(t) - total population
• S(t) - susceptible population
• I(t) - infectious population
• R(t) - recovered population
dS
dt= −β S I
dI
dt= β S I − γ I
dR
dt= γ I
• F (S, I) = β S I - force of infection
• N(t) = constant : dNdt
= 0
Mathematical modelling and controlling the dynamics of infectious diseases
Figure 1:
SIR
S(t)→ 0, I(t)→ 0, R(t)→ S0
Mathematical modelling and controlling the dynamics of infectious diseases
More Realistic Model SIR :
• λ - birth rate
• µ - natural death rate
• γ - recovery rate
dS
dt= λ− µS − β S I
dI
dt= β S I − (γ + µ) I
dR
dt= γ I − µR
Mathematical modelling and controlling the dynamics of infectious diseases
Analysis/Investigations on SIR :
• Stability of solutions: (S(t), I(t), R(t)) as t→∞
• Reproduction Number: R0 = βλµ(µ+γ)
• Equilibriums:λ− µS − β S I = 0
β S I − (γ + µ) I = 0
γ I − µR = 0
– R0 ≤ 1 : (S, I,R)∗ = (λµ, 0, 0) - disease-free equilibrium
– R0 > 1 : (S, I,R)∗∗ = ( γ+µβ, µβ
(R0 − 1), γβ
(R0 − 1))
• Theorem:
If R0 ≤ 1 then limt→∞(S(t), I(t), R(t)) = (S, I,R)∗;
If R0 > 1 then limt→∞(S(t), I(t), R(t)) = (S, I,R)∗∗.
Mathematical modelling and controlling the dynamics of infectious diseases
Generalizations
• SIRS : part of “Recovered” population become “susceptible”
(R(t)→ S(t))
• Take into account death from disease:
I = β S I − (γ + µ+ α) I
• Adding new compartments; e.g. “exposed” population SEIR :
S = λ− µS − β S I + ξ R
E = β S I − σ E
I = σ E − (γ + µ+ α) I
R = γ I − µR− ξ R
E(exposed) becomes I(infectious) after some time (2-21 days in
Ebola)
• Vaccination, age
Mathematical modelling and controlling the dynamics of infectious diseases
• Malaria: (ShEhIhRh) - for human, (SmEmIm) - for mosquito
Figure 2:
Mathematical modelling and controlling the dynamics of infectious diseases
Generalizations
• Pros: Good for the study particular effects (e.g.
Vaccination - newborns or non-newborns)
• Cons: Not suitable for prediction (overfitting)!
– 50-70 data points (weekly for I and Deaths)
– Ebola: many papers (Mid-2014) predicted 100,000s or
millions of cases for Dec 2014.
Reality: around 30,000 (50-100 times less!)
Infection control: it is crucial to have
• Accurate/predictive models
• Control parameter(s) ⇔ Practical measures
(realistic control parameters)
Mathematical modelling and controlling the dynamics of infectious diseases
Time delay: a good way to simplify the model
Assumption: E(exposed) becomes I(infectious) after τ1 (days)
S(t) = λ− µS(t)− β S(t) I(t) + ξ R(t)
E(t) = β S(t) I(t)− e−µτ1 β S(t− τ1) I(t− τ1)
I(t) = e−µτ1 β S(t− τ1) I(t− τ1)− (γ + µ+ α) I(t)
R(t) = γ I(t)− µR(t)− ξ R(t)
If ξ = 0 (no R→ S):
S(t) = λ− µS(t)− β S(t) I(t)
I(t) = e−µτ1 β S(t− τ1) I(t− τ1)− (γ + µ+ α) I(t)
Do we really need S(t) ? E.g. in Sierra Leone:
S(t) ≈ 7, 000, 000 I(t) ≤ 14, 122 (0.2% of S(t))
Mathematical modelling and controlling the dynamics of infectious diseases
Control models:
• Possible measures/interventations:
– distribution strategies for vaccination
– antibiotic programs, safe burials and community engagement
– travel restrictions
• Many papers consider β as a control parameter (→ 0)
Our approach
Key point: it uses a second time delay to take into account
“isolation” (hospitalization, beds)
Mathematical modelling and controlling the dynamics of infectious diseases
I(t+ 1) = βτ−1∑i=0
(1− αω(i)) I(t− τ1 − i).
Here ω - gamma distribution, R0 = β[τ − α∑τ−1
i=0 ω(i)]
There are 2 time-delays:
• τ1 is the latent period (infected becomes infectious);
• τ the average time of isolation (i.e. hospitalization)
There are 3 parameters:
• α (death rate) and β (transmission coeff.) ⇒ constants;
• τ ⇒ time dependent (τ(t) ∈ {3, 4, 5}).
Mathematical modelling and controlling the dynamics of infectious diseases
The main features of the model:
• Simple (only 3 parameters) ⇒ predictions more reliable;
• Since τ(t) ∈ {3, 4, 5} it is easy to create future scenarios;
• τ the average time of isolation/hospitalization:
– it can be well “connected” to preventive measures to
control the spread of infection
How it works in data fitting and prediction ?
Mathematical modelling and controlling the dynamics of infectious diseases
α = const, β = const, ∀t ∈ [0, T ];
τ(t) = const in each [0, t1], [t1, t2], [t2, T ]
Mathematical modelling and controlling the dynamics of infectious diseases
Figure 3: τ(t) ∈ {3, 4, 5} for t ≥ 11/Nov/2014
Mathematical modelling and controlling the dynamics of infectious diseases
After rapidly building new infrastructure and increasing the
capacity of beds the outbreak slowed down significantly. Starting
from January 2015, the epidemic has moved to the ending phase
that involves ensuring “capacity for case finding, case
management, safe burials and community engagement”
(from WHO, Ebola Situation Report, 14 Jan 2015)
“Each of the intense-transmission countries has sufficient
capacity to isolate and treat patients, with more than 2
treatment beds per reported confirmed and probable case.
However, the uneven geographical distribution of beds and cases,
and the under-reporting of cases, means that not all EVD cases
are isolated in several areas”
(from WHO, Ebola Situation Report, 28 Jan 2015)
Mathematical modelling and controlling the dynamics of infectious diseases
Optimal control Problem:
Optimal distribution of bed capacities
Difficulties:
• Opt. dist. beds ⇐ # future infecs.
• # future infecs. ⇐ Opt. dist. beds
Note: Methods/models for prediction were not accurate
Main steps in our approach:
• Predict future infections (a few scenarios)
– by setting τ(t) constant over 2-3 months periods
• Find optimal distribution of new beds according to
each scenario
Mathematical modelling and controlling the dynamics of infectious diseases
Optimal distribution of beds (Example)
Minimize(λ1,λ2,λ3,λ4)
150∑t=101
[h1(t : τ1,x1)
b1(t)+h2(t : τ2,x2)
b2(t)
]
subject to : λk ∈ [0, 1], k = 1, 2, 3, 4; t ∈ [101, 150];
xr(t+ 1) = βrr
τr(t)−1∑i=0
(1− αω(i))xr(t− d− i), r = 1, 2;
b1(t) = b1(100) +
|{T bj≤t; j=1,2,3,4}|∑
i=1
λi ·∆bi;
b2(t) = b2(100) +
|{T bj≤t; j=1,2,3,4}|∑
i=1
(1− λi) ·∆bi
Mathematical modelling and controlling the dynamics of infectious diseases
Here xr(t) is the # of infectious individuals and
hr(t : τ,x) =
σ∑i=τ(t)
(1− αω(i))xr(t− d− i) (1)
where σ = 11 (ave.stay in hosp) and d = 6.
• # of beds at t = 100: b1(t) = 126, b2(t) = 60
• Cumulative infs. at t = 100: 1259 and 675
• New beds are introduced weekly:
– 350 new beds at t = 101
– 300 new beds at t = 108
– 100 new beds at t = 115
– 20 new beds at t = 122
Mathematical modelling and controlling the dynamics of infectious diseases
Case 1: τ1 = τ2 = 3.
The optimal distribution of additional beds:
Region1 : 192.3 183.4 72.9 20
Region2 : 157.7 116.6 27.1 0
Total : 350 300 100 20
• The average number of bed occupancy over the time
interval [100, 150] is 0.45 for Region 1 and 0.29 for Region 2.
• The maximum occupancy rates are: 0.83 (that is, average
0.83 patient per bed) in Region 1 and 0.55 in Region 2; that
is, the demand for hospital beds is met at every point
t ∈ [100, 150].
Thus the solution obtained is feasible.
Mathematical modelling and controlling the dynamics of infectious diseases
Case 2: τ1 = τ2 = 4.
The optimal distribution of additional beds:
Region1 : 192.0 183.4 74.5 20
Region2 : 158.0 116.6 25.5 0
Total : 350 300 100 20
Again, the demand for hospital beds is met at every point
t ∈ [100, 150] and accordingly the optimal solution obtained
is feasible.
Mathematical modelling and controlling the dynamics of infectious diseases
Case 3: τ1 = τ2 = 5.
The optimal distribution of additional beds:
Region1 : 191.6 183.3 75 20
Region2 : 158.4 116.7 25 0
Total : 350 300 100 20
• The average number of bed occupancy over the time
interval [100, 150] is 0.83 for Region 1 and 0.51 for Region 2.
• The maximum occupancy rates are: 1.91 (that is, 1.91
patient per bed) in Region 1 and 1.05 in Region 2.
Mathematical modelling and controlling the dynamics of infectious diseases
Table 1: Optimal distr. add. beds (Not trivial !)
τ1 τ2 time = week1 week2 week3 week4
τ1 = τ2 Region 1: 192 183 75 20
Region 2: 158 117 25 0
τ1 < τ2 3 4 Region 1: 204.7 183.9 9.1 0
Region 2: 145.3 116.1 90.9 20
4 5 Region 1: 206.2 188.2 31.6 0
Region 2: 143.8 111.8 68.4 20
τ1 > τ2 4 3 Region 1: 179.3 223.3 100 20
Region 2: 170.7 76.7 0 0
5 4 Region 1: 177.1 208.5 100 20
Region 2: 172.9 91.5 0 0
Mathematical modelling and controlling the dynamics of infectious diseases
Summary
• Consider future scenarios; in the example:
– τ1 = 3, 4 or 5 and τ2 = 3, 4 or 5
– in total: 9 scenarios
– in fact (τ1, τ2) = (5, 5) - most likely (!?)
• Find optimal distribution of new beds according toeach scenario; in the example:
– in total: 9 optimal bed distributions
• Analyze all optimal bed distributions/patterns; in theexample:
– only 3 different distributions
Mathematical modelling and controlling the dynamics of infectious diseases
Table 2: Optimal bed distributions: 3 different patterns
τ1, τ2 time = week1 week2 week3 week4
τ1 ≈ τ2 Region 1: 192 183 75 20
(scenario1) Region 2: 158 117 25 0
τ1 < τ2 Region 1: 205 185 20 0
(scenario2) Region 2: 145 115 80 20
τ1 > τ2 Region 1: 178 215 100 20
(scenario3) Region 2: 172 85 0 0
Recall: Initial data (at t = 100):
• Region 1: 126 beds, 1259 cumulative-infecs
• Region 2: 60 beds, 675 cumulative-infecs
Mathematical modelling and controlling the dynamics of infectious diseases
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