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Mathematical models of infectious diseases

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Mathematical models of infectious diseases

D. Gurarie

- Goals
- Methodology
- Basic SIR and SEIR
- BRN: its meaning and implications
- Control strategies: treatment, vaccination/culling, quarantine
- Multiple-hosts: zoonotics and vector-born diseases

- 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

- 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

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

Vector –borne diseases

schisto

malaria

3. Epizootic: WNV, Marburg, …

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

Explain outbreak pattern ?

Predict (peak, duration, cumulative incidence) ?

Control (drug, vaccine, quarantine) ?

S – Susceptible E – Exposed

I – InfectiousR - Removed /immune

R

I

E

S

Latency Infective stageImmune stage…

- 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}

Questions

- Outbreak duration, peak -?
- Cumulative incidence (other health statistics)-?
- Dependence on c (contact), p (transmissibility), r (recovery) - ?
- Control, prevention ??
- Drug treatment
- Vaccine
- Quarantine

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:

- 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

“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!

Smallpox

SIR

Flu

SmallpoxFlu

SIR

SEIR

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 -?

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

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.

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 ?

r

r

l=bI

l=bI

a

S

E

S

I

I

R

R

r

r

Differential

equations

Parameters

Phase-plane

S

Time series

Cumulative incidence

I

1/R0

BRN:

Endemic Equilibrium

- BRN

r – recovery

d –disease mortality

r – immune loss

Prevalence DE

Analysis:

- (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)

- 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

Viral: RVF, Dengue, Yellow fever,

Plasmodia: Malaria, toxoplasma

Parasitic worms: schistosomiasis, Filariasis

X

X

Y

Z

Z

R

R

v

u

w

w

u

Host:

Vector:

This diagram is provided by Center for Disease Control and Prevention (CDC).

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 (??)

- 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.