Gabriela Gomes

IGC FIOCRUZ

Epidemiological Thresholds and Control Strategies I: field

Lecture 1

2000 BC

1000 BC

2000 AD

1000 AD

Arithmetics

4000 years of mathematics

Europe

Ronald Ross 1857-1932

Arabia

Pure mathematics

Greece Babylon Egipt India China

2000 AD

1000 AD

2000 BC

1000 BC

China

4000 years of malaria

Discovery mosquitoes transmit malaria (Ross – Nobel Prize 1902)

First mathematical model for malaria transmission (Ross)

Rome

First reports of malaria symptoms

Association between malaria and swamps

Medicinal plants

China

Discovery of the parasite in blood of patient (Laveran – Nobel Prize 1907)

Renaissance

DDT Global Eradication Campaign 1955-1978

Susceptible Infectious

Host classification

Host population

Sir Ronald Ross demonstrated that the parasite of malaria is transmitted by mosquitoes and, in 1902, he received the Nobel Prize of Medicine. He developed mathematical models for malaria transmission, and was a pioneer in mathematical epidemiology.

infection

recovery

SIS model

€

dSdt = − β I S + γ I

dIdt = β I S − γ I

S I β Ι

γ

β – transmission coefficient

γ – recovery rate

Appropriate for infections that induce no effective immunity (e.g. malaria).

SIS model

€

dIdt = β I(1−I) − γ I

As S + I = 1, the model is one dimensional and represented by the equation

This is a type of growth law known as logistic growth, and it appears commonly in population dynamics models in the form

€

dIdt = (β−γ) 1− I

1−γ β

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

I

The solution can be obtained analytically

€

I(t) =I(0) 1−γ β⎛

⎝ ⎜

⎞

⎠ ⎟

I(0) + 1−γ β−I 0( )⎛

⎝ ⎜

⎞

⎠ ⎟ e− β−γ( ) t

SIS model

Steady states:

1.  Disease free equilibrium

2.  Endemic equilibrium

Given an initial condition, I(0) = 10-6, the proportion of infected individuals grows as

Parameters: β = 3 , γ = 1

€

I = 0, S = 1

€

I = 1− γβ, S = γ

β

SIS model

The basic reproduction number is defined as the

average number of secondary infections produced by an average infectious individual, during its entire infectious period,

in a totally susceptible population

The basic reproduction number is calculated as

R0 = (rate of transmission from an infectious individual) x (infectious period)

= β x (1 / γ) = β / γ

Disease R0 Smallpox 4

Measles 17

Rubella (England and Wales) 6

Rubella (Gambia) 15

R0 is a nondimensional number, and depends on both the environment (physical and social) and the disease.

Basic reproduction number, R0

€

dIdt = R0 (1−I) I − I

= R0 1−1R0−I

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

I

If time is measured in units of infectious period, D = 1 / γ, then the SIS model becomes

The endemic equilibrium is rewritten as

€

I = 1− 1R0, S = 1

R0

Epidemic threshold: Infection can invade a totally susceptible population if and only if

€

R0 > 1

Nondimensional SIS model

Field data: clinical malaria by age Bakau, Gambia

Foni Kansala, Gambia

Sukuta, Gambia

Mponda, Malawi

Kilifi, Kenya

Chonyi, Kenya

Ifakara, Tanzania

Siaya, Kenya

S

I1

R

I2

λ

γ

λ γ / σ

α

susceptible

clinical malaria

partially immune

uncomplicated malaria

€

λ = β I1 + I2( )Force of infection:

Boundary conditions:

S(t,0) = µ R(t,0) = I1(t,0) = I2(t,0) = 0

System of partial differential equations (SIRI-like for malaria)

€

∂S∂t +

∂S∂a = α R − S λ + µ( )

∂I1∂t +

∂I1∂a = λ S − I1 γ + µ( )

∂R∂t +

∂R∂a = γ I1 +

1σI2

⎛

⎝ ⎜

⎞

⎠ ⎟ − R λ +α + µ( )

∂I2∂t +

∂I2∂a = λ R − I2

γσ

+ µ⎛

⎝ ⎜

⎞

⎠ ⎟

Global parameters:

average duration of clinical malaria ~ 1 month [γ = 14.12 yr-1]

average duration of uncomplicated malaria ~ 6 month [σ = 6.33]

average duration of partial immunity ~ 12 month [α = 1.07 yr-1]

Model fitting and parameter estimation for malaria

Local parameters:

€

R0 = β γ

Multi-population malaria trends

Decreasing disease with increasing transmission

Sustainable transmission for R0 < 1 (hysteresis)

epidemic threshold

reinfection threshold

Bistable regime with implications for malaria elimination and resurgence

Epidemic threshold: Infection can invade a population where every individual is “totally susceptible” if and only if

€

R0 > 1

Epidemiological thresholds of the SIRI model

Reinfection threshold: Infection can invade a population where every individual is “partially immune” if and only if

€

R0 > 1σ

€

dSdt = − β I S

dIdt = β I S − γ I

dRdt = γ I

S I R β Ι γ

β – transmission coefficient

γ – recovery rate

Appropriate for infections that induce highly effective immunity (e.g. measles, mumps, rubella).

SIR model

In 1927, Kermack e McKendrick fitted the model to various epidemic curves. They established the notion of the epidemic threshold, under a slightly different formulation: the growth of an epidemic requires that, on average, an infected individual infects at least one susceptible. An epidemic falls when the density of susceptibles is below a threshold: S < 1 / R0.

€

S> γβ

€

S< γβ β = 3

γ = 1

SIR model fitted to a plague epidemic in Bombay 1906

If time is measured in units of infectious period, D = 1 / γ, then the SIR model becomes

As individuals become immune, this system always approaches the disease free steady state, I = 0. As the epidemic progresses, the level of susceptibles decreases, and the level of recovered individuals increases. The important question is the final balance between these two classes – does the disease die out before all the susceptibles are exhausted?

€

dSdt = − R0 I S and dR

dt = IdIdt = R0 I S − I

€

dSdR = −R0S ⇒ S = exp−R0 R

⎛

⎝ ⎜

⎞

⎠ ⎟

Using the fact that R = 1 – I – S, and at equilibrium I = 0, we get

€

R* = 1 − exp−R0 R*⎛

⎝ ⎜

⎞

⎠ ⎟

Nondimensional SIR model

Final epidemic size

In the long term, the susceptible pool is replenished by births generating the conditions for new epidemics to occur.

e e e e

S I R R0 I

SIR with host demography

Steady states:

1.  Disease free equilibrium:

2.  Endemic equilibrium:

The new steady states are obtained from the model

€

dSdt = e − R0 I S − eSdIdt = R0 I S − I

The new parameter, e, represents the birth and death rate in units of infectious period. This is equivalent to D / L, where D is the average duration of infection and L is the life expectancy at birth. Assuming that D = 1 month, and L = 70 years, we get e = 0.0012.

SIR with host demography

Partial immunity SIRI model

S

I

R

R0

susceptible

infectious

partially immune

σ R0

A common scenario is that immunity is not fully protective, but reduces the risk of further infections by some factor (e.g. tuberculosis).

€

dSdt = e − R0 I S − eSdIdt = R0 I S+σ (1−S−I)⎛

⎝ ⎜

⎞

⎠ ⎟ − I

Partial immunity SIRI model

reinfection threshold (R0 = 1/σ)

The collective effect of a vaccination programme is to reduce the pool of susceptible individuals.

S I R R0 I

Vaccination in SIR model

The collective effect of a vaccination programme is to reduce the pool of susceptible individuals.

S I R

(1-v) e ve

v: vaccination coverage

R0 I

Vaccination in SIR model

Partial immunity induces a reinfection threshold, R0 = 1 / σ, above which the prevalence of infection is high and insensitive to vaccination.

S I R

σ : susceptibility reduction factor

R0 I

σ R0 I v e (1-v) e

Vaccination in SIRI model

uncontrollable controllable

Reinfection threshold and effectiveness of vaccination programs

Typically vaccines do not confer a level protection superior to that induced by natural infection. However, this is a prime goal in vaccine research. In order to predict the epidemiological impact of such vaccine we need an extension to the partial immunity model.

The vaccine is more potent than natural infection if and only of σV < σ.

Defining effications vaccines

Defining effications vaccines

Cape Town

China

Taiwan India

Malawi

Rwanda Madras

Bangladesh

Spain Houston Denmark

Australia

Lombardy

US & Canada

Field data: tuberculosis reinfection

€

dSidt = µ pi − λi+µ⎛

⎝ ⎜

⎞

⎠ ⎟ Si

dPidt = λi Si + σ λi Li+Ri

⎛

⎝ ⎜

⎞

⎠ ⎟ − δ+µ⎛

⎝ ⎜

⎞

⎠ ⎟ Pi

dIidt = φδPi + ω Li+Ri

⎛

⎝ ⎜

⎞

⎠ ⎟ − γ+µ( ) Ii

dLidt = 1−φ⎛

⎝ ⎜

⎞

⎠ ⎟ δPi − σλi+ω+µ⎛

⎝ ⎜

⎞

⎠ ⎟ Li

dRidt = γ Ii − σ λi+ω+µ⎛

⎝ ⎜

⎞

⎠ ⎟ Ri

λi = αiλ

System of ordinary differential equations (SIRI-like for tuberculosis)

Model fitting and parameter estimation for tuberculosis

p

Global parameters

Multi-population tuberculosis trends

epidemic threshold

reinfection threshold

Estimates of R0 may be very sensitive to heterogeneity in host susceptibility

Bibliography

Gomes MGM, White LJ, Medley GF (2004) Infection, reinfection, and vaccination under suboptimal immune protection: Epidemiological perspectives. J Theor Biol 228: 539-549.

Gomes MGM, Franco AO, Gomes MC, Medley GF (2004) The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc R Soc Lond B 271: 617-623.

Gomes MGM, White LJ, Medley GF (2005) The Reinfection Threshold. J Theor Biol 236: 111-113.

Águas R, White LJ, Snow RW, Gomes MGM (2008) Prospects for malaria eradication in sub-Saharan Africa. PLoS ONE 3(3): e1767.

Gomes MGM, Águas R, Lopes JS, Nunes MC, Rebelo C, Rodrigues P, Struchiner CJ (2012) How host selection governs tuberculosis reinfection. Proc R Soc Lond B 279: 2473-2478.