Mathematical modelling and controlling the dynamics of … · 2017-09-12 · • # future infecs. ( Opt. dist. beds Note: Methods/models for prediction were not accurate Main steps

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


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


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


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 models:

• Statistical-Based Methods

• Dynamical systems (State-Space Models -

SI, SIR, SIRS, SEIR)

• Empirical/Machine Learning-Based Models


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


Figure 1:

SIR

S(t)→ 0, I(t)→ 0, R(t)→ S0


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


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)∗∗.


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


• Malaria: (ShEhIhRh) - for human, (SmEmIm) - for mosquito

Figure 2:


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)


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))


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)


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}).


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 ?


α = const, β = const, ∀t ∈ [0, T ];

τ(t) = const in each [0, t1], [t1, t2], [t2, T ]


Figure 3: τ(t) ∈ {3, 4, 5} for t ≥ 11/Nov/2014


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)


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


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


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





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.


Case 2: τ1 = τ2 = 4.


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.


Case 3: τ1 = τ2 = 5.


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.


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


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


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


τ1 > τ2 Region 1: 178 215 100 20


Recall: Initial data (at t = 100):

• Region 1: 126 beds, 1259 cumulative-infecs

• Region 2: 60 beds, 675 cumulative-infecs


THANK YOU

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Mathematical modelling and controlling the dynamics of … · 2017-09-12 · • # future infecs. ( Opt. dist. beds Note: Methods/models for prediction were not accurate Main steps