Epidemic Curve

S

I

R

I transmits to S

I recovers to R

I dies of disease

Exponential growth

Decay due to Susc. decline

DO NOT run out of S's

Time (days)

Compartment size

Susceptible

Infected

Recovered

Foot and Mouth Disease (UK, 2001)

Ebolavirus (2014)

Five determinants of curve structure

1. Communicable period

2. Latent period

3. Recovery rate

4. Degree of immunization

5. Between-host transmission rate

R_{0}

R_{0}

and invasion

R_{0}

R_{0}

: # of secondary infections produced by a single infectious individual in an otherwise fully susceptible population

(Management goal)

R_{0} \leq 1

R_{0} \leq 1

=> pathogen cannot invade

\frac{dS}{dt} = -\beta SI

\frac{dS}{dt} = -\beta SI

\frac{dI}{dt} = \beta SI - (\gamma + \alpha) I

\frac{dI}{dt} = \beta SI - (\gamma + \alpha) I

\frac{dR}{dt} = \gamma I

\frac{dR}{dt} = \gamma I

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

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

Disease	Mode of transmission
Measles	Airborne	12-18
Diptheria	Saliva	6-7
Smallpox	Airborne droplets	5-7
Polio	Fecal-Oral	5-7
Rubella	Airborne droplets	5-7
Mumps	Airborne droplets	4-7
HIV	Sexual	2-5
Pertussis	Airborne droplets	5.5
SARS	Airborne droplets	2-5
Influenza (1918)	Airborne droplets	2-3
Ebola (2014	Bodily fluids	1.5-2.5

R_{0}

R_{0}

High R0 = Good invader

Estimating

R_{0}

R_{0}

R_{0}

R_{0}

: # of secondary infections produced by a single infectious individual in an otherwise fully susceptible population

=> look at growth during BEGINNING of epidemic

Max. # cases

2014 Ebola dynamics in Sierra Leone

Population growth

is multiplicative, not additive

=> LOG!!

Max. # cases

New Cases!!

(NOT cumulative)

"Exponential" growth

(\text{``}R_{0}\text{"})

(\text{``}R_{0}\text{"})

Coefficients:
                Estimate Std. Error t value Pr(>|t|)   
(Intercept)    -185.1590    79.8263  -2.320   0.0274 * 
sl.early$Date3    0.9013     0.3068   2.938   0.0063 **

\hat{R_{0}} = \frac{\text{growth rate}}{\text{recovery rate}} + 1

\hat{R_{0}} = \frac{\text{growth rate}}{\text{recovery rate}} + 1

\text{set }\gamma = 5.6 \text{ from similar strain in the Congo, 1995}

\text{set }\gamma = 5.6 \text{ from similar strain in the Congo, 1995}

\hat{R_{0}} = \frac{0.90}{5.6} + 1

\hat{R_{0}} = \frac{0.90}{5.6} + 1

= 1.16

= 1.16

Growth rate: slope of linear model of log(new cases) ~ day

Goal of management is to drive R below 1

= \frac{\text{slope}}{\gamma} + 1

= \frac{\text{slope}}{\gamma} + 1

Persistence or Fade-out

Model equilibria

Option 1: Pathogen goes extinct (I* = 0)

Option 2: Static persistence (I* = k)

Option 3: Stable cycles

Pathogens fade-out when they run out of Susceptible hosts.

1. Transmission is much faster than births

2. Hosts artificially move from S to R

Critical Community Size

Minimum population size in which a host-specific agent can persist indefinitely

Does not exist for frequency-dependent diseases

For measles, ~300,000 people

Depends on population turnover and transmission rate

Herd immunity

Calculation

Pop is p% Immune and (1-p)% Susceptible

When a high-enough proportion of population is immune so that R0 <= 1 and transmission stops

R_{0} \times (1-p) = 1

R_{0} \times (1-p) = 1

For an epidemic to take off,

R_{0}\times S = 1

R_{0}\times S = 1

\rightarrow (1-p) = \frac{1}{R_{0}}

\rightarrow (1-p) = \frac{1}{R_{0}}

\rightarrow p_{c} = 1-\frac{1}{R_{0}}

\rightarrow p_{c} = 1-\frac{1}{R_{0}}

Goal for vaccination.

Measles herd immunity

p_{c} = 1-\frac{1}{18} = 1-0.055 = .945

p_{c} = 1-\frac{1}{18} = 1-0.055 = .945

Lower vaccination rates required for less-infectious agents

Strategies for persistence

Transmission

Virulence

Multi-host persistence

Canine Distemper Virus

(Morbillivirus)

Canine Distemper Virus Persistence??

1994

1996-2002

2003-2012

Rabies

Age-Seroprevalence

"Ways" to be seropositive

1. Acquired immune response after infection

Antibodies can't have "waned" yet

2. Antibodies acquired through passive transfer

Antibody titers wane exponentially following birth

Acquired through infection or vaccination

Passive immunity

Density- vs. Frequency-dependence

How does transmission rate scale with population density?

Frequency-dependent

Density-dependent

Transmission term =

\beta SI

\beta SI

Transmission term =

\frac{\beta SI}{N}

\frac{\beta SI}{N}

Take S, I, R to be Proportions

(S + I + R = 1)

Brucellosis

Brucella abortus

Elk

Bison

Cattle

=$$

Brucellosis

Transmission is 1) through abortion events

2) sexual

Brucellosis

Movement

Brucellosis

Movement

FD vs. DD transmission

Mode of transmission

Movement patterns

Spatial scale

Transmission often DD at fine-enough scale

elk on a feedground
bighorn sheep in a ewe-group

At a coarser scale, transmission "looks" more FD

Disease and Food

Supplemental Feeding

Elk on Wyoming feedgrounds

Flying fox camp in urban Australia

House finch with conjunctivitis

Bioenergetic diseases

Caloric costs of heat loss due to mange

Caloric costs of arousal bouts for bats with White Nose Syndrome

Predators and "Healthy Herds"

Predators consume sickest prey

Reduces overall transmission rate

=> Prey pop size INCREASES with predator introduction

Two Evolutionary Hypotheses

What sort of relationship should pathogens seek with their hosts?

H1: Evolution toward avirulence

Epidemic number

Rabbits and Myxoma

Worst strains decline through time

H2: Virulence-Transmission Trade-off

Pathogen needs large-enough population to transmit...

... but too high a pathogen load can kill the host

Pathogen population size

P(Transmits to a new host)

Not enough pathogen to produce infectious dose

So much pathogen host dies before transmitting

Optimal load

Virulence-Transmission in a protozoan parasite of the Monarch

Dilution or Amplification

How does biodiversity relate to disease risk?

Lyme disease (Borrelia Burgdorferi)

Ixodes scapularis (deer tick)

Different hosts for different instars

Host spp. vary in "competence"

When biodiversity is high, fewer

I. scapularis feed on "competent" hosts

parasiteecology.wordpress.com

Vacated Niches and Pathogen Emergence

Smallpox and Monkeypox

Orthopox viruses of primates

Some cross-immunity

Smallpox declared eradicated 1979

Variola case-fatality rate = 30%

Monkeypox case-fatality rate = 1-10%

Monkeypox case, DRC, 1986

Smallpox and Monkeypox

Lloyd-Smith, Phil. Trans., 2013

Did smallpox eradication leave a "vacated" niche for monkeypox to invade?

Epidemic Curve

S

I

R

Five determinants of curve structure

and invasion

High R0 = Good invader

Estimating

2014 Ebola dynamics in Sierra Leone

Goal of management is to drive R below 1

Persistence or Fade-out

Model equilibria

Pathogens fade-out when they run out of Susceptible hosts.

Critical Community Size

Herd immunity

Goal for vaccination.

Measles herd immunity

Lower vaccination rates required for less-infectious agents

Strategies for persistence

Transmission

Virulence

Multi-host persistence

Canine Distemper Virus

Canine Distemper Virus Persistence??

Age-Seroprevalence

"Ways" to be seropositive

Density- vs. Frequency-dependence

How does transmission rate scale with population density?

Brucellosis

Brucellosis

Brucellosis

Brucellosis

FD vs. DD transmission

Disease and Food

Supplemental Feeding

Bioenergetic diseases

Predators and "Healthy Herds"

Two Evolutionary Hypotheses

What sort of relationship should pathogens seek with their hosts?

H1: Evolution toward avirulence

H2: Virulence-Transmission Trade-off

Dilution or Amplification

How does biodiversity relate to disease risk?

Lyme disease (Borrelia Burgdorferi)

Cross-scale questions

Vacated Niches and Pathogen Emergence