Mathematical models of disease transmission

www.slides.com/keziamanlove/epimath

Infected/

Infectious (I)

Susceptible (S)

Recovered (R)

States

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\frac{dS}{dt} = -\beta SI - \mu S + \mu N

\frac{dS}{dt} = -\beta SI - \mu S + \mu N

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\frac{dR}{dt} = \gamma I - \mu R

\frac{dR}{dt} = \gamma I - \mu R

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

Framework 1: Differential equations

Framework 2: Stochastic (Markov) process

	S	I	R	(D)
S
I
R
(D)

1 - \beta SI - \mu

1 - \beta SI - \mu

\beta SI

\beta SI

0

State at t + 1

State at t

0

1 - \gamma - \mu

1 - \gamma - \mu

\gamma

\gamma

0

0

1 - \mu

1 - \mu

\mu

\mu

\mu

\mu

\mu

\mu

0

0

0

1

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

\text{Transmission} = \text{(host contact rate)} \times \text{P(contact with infected)}

\text{Transmission} = \text{(host contact rate)} \times \text{P(contact with infected)}

\times \text{P(Infection given contact with I)}

\times \text{P(Infection given contact with I)}

Transmission term,

\beta SI

\beta SI

\kappa

\kappa

\propto \frac{I}{N} \text{ or } I

\propto \frac{I}{N} \text{ or } I

1 - c

1 - c

\beta = -\kappa\log(1 - c)

\beta = -\kappa\log(1 - c)

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

R_{0}

R_{0}

Basic reproductive ratio,

Expected number of additional "I"s created by a single "I" in an otherwise entirely "S" population

R_{0} = \frac{\beta}{\gamma + \mu}

R_{0} = \frac{\beta}{\gamma + \mu}

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

R_{0}

R_{0}

Basic reproductive ratio,

Disease	Mode of transmission
Measles	airborne	12-18
Pertussis	airborne / droplet	12-17
Smallpox	Airborne droplet	5-7
HIV	Sexual contact	2-5
SARS	Airborne droplet	2-5
Influenza (1918)	Airborne droplet	2-3
Ebola (2014)	Bodily fluids	1.5-2.5

R_{0}

R_{0}

R_{0} \leq 1 \Rightarrow \text{Limited transmission and eventual fade-out}

R_{0} \leq 1 \Rightarrow \text{Limited transmission and eventual fade-out}

R_{0} > 1 \Rightarrow \text{Pathogen persistence}

R_{0} > 1 \Rightarrow \text{Pathogen persistence}

Equilibria

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

\text{Equilibria at } I\left(\beta S - (\gamma + \mu)\right) = 0

\text{Equilibria at } I\left(\beta S - (\gamma + \mu)\right) = 0

I^{*} = 0

I^{*} = 0

S^{*} = \frac{\gamma + \mu}{\beta} = \frac{1}{R_{0}}

S^{*} = \frac{\gamma + \mu}{\beta} = \frac{1}{R_{0}}

Disease-Free

Endemic

I^{*} = \frac{\mu}{\gamma}\left(1 - \frac{1}{R_{0}}\right) = \frac{\mu}{\beta}\left(R_{0} - 1\right)

I^{*} = \frac{\mu}{\gamma}\left(1 - \frac{1}{R_{0}}\right) = \frac{\mu}{\beta}\left(R_{0} - 1\right)

Age-Seroprevalence and

R_{0}

R_{0}

Seroprevalence: Proportion of population that has antibodies

to pathogen X (cumulative exposure; R + part of I)

For fully immunizing pathogens, Recovery is an absorbing state

A \approx \frac{1}{\mu(R_{0} - 1)}

A \approx \frac{1}{\mu(R_{0} - 1)}

Age at first infection (A) depends on infectiousness of pathogen ( ), and host life-span ( )

R_0

R_0

\mu

\mu

Assume constant pop size

Start with 3-D system

R(t) = N - S(t) - I(t)

R(t) = N - S(t) - I(t)

So, system reduces to 2-D

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\lambda = \text{Force of infection}

\lambda = \text{Force of infection}

\nu = \text{Recovery rate}

\nu = \text{Recovery rate}

\frac{dS}{dt} = -\beta SI - \mu S + \mu N

\frac{dS}{dt} = -\beta SI - \mu S + \mu N

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\frac{dI}{dt} = \beta SI - \left(\gamma + \mu\right) I

\frac{dR}{dt} = \gamma I - \mu R

\frac{dR}{dt} = \gamma I - \mu R

\frac{dS}{dt} = \mu - (\mu + \lambda(t))S(t)

\frac{dS}{dt} = \mu - (\mu + \lambda(t))S(t)

\frac{d\lambda}{dt} = (\nu + \mu)\lambda(t)(R_{0}S(t) - 1)

\frac{d\lambda}{dt} = (\nu + \mu)\lambda(t)(R_{0}S(t) - 1)

Damped Oscillator

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\lambda = \text{Force of infection}

\lambda = \text{Force of infection}

\nu = \text{Recovery rate}

\nu = \text{Recovery rate}

T \approx 2\pi\frac{L(\frac{1}{\nu} + \frac{1}{\sigma})}{R_{0} - 1}^{1/2}

T \approx 2\pi\frac{L(\frac{1}{\nu} + \frac{1}{\sigma})}{R_{0} - 1}^{1/2}

L = \text{Host life expectancy}

L = \text{Host life expectancy}

\sigma = \text{Incubation rate}

\sigma = \text{Incubation rate}

\beta SI

\beta SI

\gamma

\gamma

\mu

\mu

\mu

\mu

\mu

\mu

\mu = \text{Natural mortality}

\mu = \text{Natural mortality}

\beta = \text{Transmission rate}

\beta = \text{Transmission rate}

\gamma = \text{Recovery rate}

\gamma = \text{Recovery rate}

(R is no longer an absorbing state)

\kappa \times \beta SI

\kappa \times \beta SI

\kappa = \text{Ease of boosting}

\kappa = \text{Ease of boosting}

Why is whooping cough such a mess?

Hypothesis: Boosting through re-exposure

Lavine, King, Bjornstad (2011) PNAS

Lloyd-Smith et al., Nature, 2005

Focus: dispersion in # cases generated

Z = \text{RV representing number of } 2^{0} \text{ cases produced by a particular I}

Z = \text{RV representing number of } 2^{0} \text{ cases produced by a particular I}

\nu = R_{0} \Rightarrow Z \sim Pois(R_{0})

\nu = R_{0} \Rightarrow Z \sim Pois(R_{0})

\nu \sim Exp() \Rightarrow Z \sim Geometric(R_{0})

\nu \sim Exp() \Rightarrow Z \sim Geometric(R_{0})

\nu \sim Gamma\left(R_{0}, \kappa \right) \Rightarrow Z \sim \text{Neg Bin}(R_{0}, \kappa)

\nu \sim Gamma\left(R_{0}, \kappa \right) \Rightarrow Z \sim \text{Neg Bin}(R_{0}, \kappa)

\nu = \text{"individual reproductive number"} \sim \left(\mu = R_{0}\right)

\nu = \text{"individual reproductive number"} \sim \left(\mu = R_{0}\right)

\text{Var(Z)} = \frac{R_{0}}{1 + \frac{R_{0}}{\kappa}}

\text{Var(Z)} = \frac{R_{0}}{1 + \frac{R_{0}}{\kappa}}

Two (non-mutually-exclusive) within-host models

Antigen-driven dynamics

Predator-prey between immune system (D) and pathogen(L)

Network regulation model

Internal feedbacks control host's immune response

\frac{dL(\tau)}{d\tau} = \rho L(\tau) - \frac{\nu L(\tau)^{2}D(\tau)}{\psi^{2} + L(\tau)^{2}}

\frac{dL(\tau)}{d\tau} = \rho L(\tau) - \frac{\nu L(\tau)^{2}D(\tau)}{\psi^{2} + L(\tau)^{2}}

\frac{dD(\tau)}{d\tau} = D(\tau)\frac{\phi\nu L(\tau)^{2}}{\psi^{2} + L(\tau)^{2}} - \epsilon D(\tau)

\frac{dD(\tau)}{d\tau} = D(\tau)\frac{\phi\nu L(\tau)^{2}}{\psi^{2} + L(\tau)^{2}} - \epsilon D(\tau)

Holling Type III Functional response

Cross-Scale Considerations

Have to model age-of-infection for all I's

\Rightarrow{PDEs}

\Rightarrow{PDEs}

\frac{dS(t)}{dt} = \theta - \mu S(t) \int_{0}^{\infty}\beta(a)I(a, t)da

\frac{dS(t)}{dt} = \theta - \mu S(t) \int_{0}^{\infty}\beta(a)I(a, t)da

\frac{\partial I(a, t)}{\partial t} = -\frac{\partial I(a, t)}{\partial t} - \alpha(a)I(a, t)

\frac{\partial I(a, t)}{\partial t} = -\frac{\partial I(a, t)}{\partial t} - \alpha(a)I(a, t)

I(0, t) = S(t) \int_{0}^{\infty}\beta(a)I(a, t)da

I(0, t) = S(t) \int_{0}^{\infty}\beta(a)I(a, t)da

\theta = \text{birth rate}

\theta = \text{birth rate}

\mu = \text{host mortality rate}

\mu = \text{host mortality rate}

\alpha = \text{loss rate of individuals from I}

\alpha = \text{loss rate of individuals from I}

Mathematical models of disease transmission www.slides.com/keziamanlove/epimath