Mathematical models of infectious diseases
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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.

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Mathematical models of infectious diseases

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Mathematical models of infectious diseases

Mathematical models of infectious diseases

D. Gurarie


Lecture outline

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

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

    • Predict outcomes

    • Analyze, develop control strategies


2 early history of math modeling xviii century smallpox

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

3. Transmission patterns

Direct: host – to-host (flu, smallpox, STD,…)

Vector –borne diseases

schisto

malaria

3. Epizootic: WNV, Marburg, …


4 infection patterns typical flu outbreak

4. Infection patterns: typical flu outbreak

Data (British Medical Journal, March 4 1978, p. 587)

Explain outbreak pattern ?

Predict (peak, duration, cumulative incidence) ?

Control (drug, vaccine, quarantine) ?


5 sir methodology host disease states and history

5. SIR –methodology: host ‘disease states’ and history

S – Susceptible E – Exposed

I – InfectiousR - Removed /immune

R

I

E

S

Latency Infective stageImmune stage…


6 sir transmission in randomly mixing community

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

Day

1

2

3

Groups

{1},{2,9,13},{4,5,10},{6,7,8}


Simulated pattern infection outbreak

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

II. SIR methodology: diagrams

S

I

Birth

Death

S

I

R

SIR

recruitment

SEIR

SEIR

S

S

E

E

I

I

R

R

V

V

SI

Variables: S, E, I, R, V (vaccinated) – host states, or populations /fractions

Loss

  • Continuous (DE) for {S(t),… }- functions of time t

  • Discrete {S(t),… } (t=0,1,2,…)

  • Community level (populations)

  • Individual level (agent based)

Models:


Discrete sir reed frost

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

I

l

1-q

S

E

I

1-r

1-r

1-s

1-s

R

R


Reed frost map discrete time step

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:

No analytic solution!


Mathematical models of infectious diseases

Smallpox

SIR

Flu


Numeric simulations

Numeric simulations

SmallpoxFlu

SIR

SEIR


Analysis of outbreaks and endemic equilibria

Analysis of outbreaks and endemic equilibria

3 basic parameters

Susceptibility :1-p (‘resistance to infection’ = p)

Contact rate: c

recovery rate: r

Questions:

1. How (p,c,r) would determine infection pattern: outbreak, endemic equilibria levels et al?

Control intervention -?


Key index brn

Key index: BRN

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

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

f

S

E

I

R

V

If R0 is known (?), cover fraction f=1-1/R0 needed to eradicate infection.


2 imperfect vaccine 1 e 0

2. Imperfect vaccine (1/e>0)

0<f<1 – cover fraction

e>1 - efficacy

1-f

  • Effects of vaccine:

  • reduce risk of infection under identical ‘infected contacts’: p pV = p1/e > p

  • enhance recovery: r rV = re <r

S’

S

E’

E

I’

I

R

f

  • Effect (f,e) - ?

  • BRN: R0(f,e) -?

  • Can BRN be brought <1 ?


Iii c ontinuous de models

III. Continuous (DE) models

r

r

l=bI

l=bI

a

S

E

S

I

I

R

R

r

r

Differential

equations

Parameters


2 smallpox sir immune loss r 0

2. Smallpox SIR (immune loss r=0)

Phase-plane

S

Time series

Cumulative incidence

I

1/R0

BRN:


3 sir with immune loss flu

3. SIR with immune loss (flu)

Endemic Equilibrium

  • BRN

r – recovery

d –disease mortality

r – immune loss

Prevalence DE

Analysis:


4 brn meaning implications

4. BRN: meaning, implications

  • (SIR with life-long immunity): R0 determines whether outbreak occurs (R0 >1), or infection dies out (R0 <1)

  • BRN is related to initial infection growth :

    As , 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)


5 control intervention

5. Control intervention

  • Vaccination (herd immunity):

    • vaccinating fraction f of susceptibles decreases R0 (1-f)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

IV. Vector mediated transmission

Viral: RVF, Dengue, Yellow fever,

Plasmodia: Malaria, toxoplasma

Parasitic worms: schistosomiasis, Filariasis


2 coupled sir seir diagrams

X

X

Y

Z

Z

R

R

v

u

w

w

u

2. Coupled SIR-SEIR diagrams

Host:

Vector:


3 macro parasites schistosome life cycle

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

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)

BRN:

=> equilibria, analysis and control (??)


Summary math modeling

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.


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