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Mosquitos’ mobility effectively spatially homogenizes human population locally, resulting in an R0 approaching N2.

Densities of human populations, the character and quality of available water supply, food, and shelter together with the frequency and range of contacts among individuals...affect disease patterns significantly…

…Great cities were, until recently, always unhealthy…

…all such local disturbances of ecological relations have worked within a biological gradient characterized by an increase in the variety and frequency of infections as temperature and moisture increased.

[William H. McNeil, Plagues and People (1976), page 51]

SE Asia

encephalitis in India

In Southwest US

Americas

D = P E H

- Geographic expansion
- Increased epidemic activity
- New pathogen
- New presentation
- Antimicrobial resistance

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

- Increased density
- Increased dispersion
- Increased mobility

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

- Regional Environmental Change
- Global Climate Change

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

Public Health Infrastructure

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

- Increased density
- Increased dispersion
- Increased mobility

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

- Regional Environmental Change
- Global Climate Change

D = P E H

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

Public Health Infrastructure

What are the environmental influences on disease?

Infectious disease ecology attempts to address this question by:

- Focusing on zoonotic and vector borne diseases; ~80% of EIDs especially if diseases that persist in the environment are included (e.g., polio virus, MRSA)
- Considering ‘intrinsic’ (host-pathogen life history) and ‘extrinsic’ (environmental) factors – (biotic vs abiotic)
- Attempting to elucidate and develop predictive models of disease emergence (as a basis for control and prevention)
- Using a theoretical/analytical approach draws on ‘host-parasite biology’, functionally describes viral, bacterial, and protozoan pathogens as ‘microparasites’.

Note: ‘infection’ in disease ecology ~ innoculation of a host not necessarily accompanied by a clinically detectable immune response.

D = P E H

- Geographic expansion
- Increased epidemic activity
- New pathogen
- New presentation
- Antimicrobial resistance

Infectious Disease Emergence =

Population growth x Environmental Change x Hygiene

What are the environmental influences on disease?

Infectious disease ecology attempts to address this question by:

- Focusing on zoonotic and vector borne diseases; ~80% of EIDs especially if diseases that persist in the environment are included (e.g., polio virus, MRSA)
- Considering ‘intrinsic’ (host-pathogen life history) and ‘extrinsic’ (environmental) factors – (biotic vs abiotic)
- Attempting to elucidate and develop predictive models of disease emergence (as a basis for control and prevention)
- Using a theoretical/analytical approach draws on ‘host-parasite biology’, functionally describes viral, bacterial, and protozoan pathogens as ‘microparasites’.

Note: ‘infection’ in disease ecology ~ innoculation of a host not necessarily accompanied by a clinically detectable immune response.

Zoonotic and Vector Borne Disease Definitions

Zoonotic diseases are diseases caused by infectious agents that can be transmitted between (or are shared by) animals and humans

Vector-borne diseases are diseases in which the pathogenic microorganism is transmitted from an infected individual to another individual by an arthropod or other agent, sometimes with other animals serving as intermediary hosts.

Intermediary hosts such as domesticated and/or wild animals often serve as a reservoir for the pathogen until susceptible human populations are exposed.

Direct versus Indirect Transmission

Definitive Host

Host

T2

T1

T1

Intermediate

Host*

T1 T2

Transmission parameters for the flow of the pathogen from the Defintive Host to the Intermediate Host and from the Intermediate Host to the Definitive Host

*may be a vector or other reservoir

Pathogen Growth in a Host Population and Epidemiological Dynamics

The SIR “Compartmental Model”

births

infected

Susceptibles (uninfected)

Recovered (immune)

deaths

deaths

deaths

Exercise: Pathogen Growth in a Host Population: Introduction to the SIR Model

births

infecteds

Recovered (immune)

susceptibles

R0, The Basic Reproductive Rate of a Disease

This single most important parameter determining whether or not a disease can spread, cause and epidemic, and whether or not a disease will be epidemic or endemic - or become a pandemic!

The quantity R0 determines expresses a combination factors including providing insights into:

(1) How transmissable disease is

(2) How controllable an epidemic will be

(3) How the disease can be controlled

For many microparasites with direct transmission R0 and Re

(The “effective reproductive rate) increases with:

The period of time over which an infected host remains infectious.

When higher host densities offer more opportunities for transmission.

The transmission rate of the disease, which depends both on the intrinsic infectiousness of the disease as well as on patterns of host behavior that increase the likelihood of infectious and susceptible hosts coming together.

I. Discrete, compartmental Model of R0

As in Begon et al. (where R0 is in theory in not a fixed “intrinsic rate” of growth, R0 is allowed to be like Re):

- For microparasites with direct, density dependent transmission R0 can said to increase with:
- the average period of time a host remains infectious, L
- the number of susceptible individuals, S
- the transmission coefficient, β

- So, overall:
- R0 = SβL

Expressing the Transmission Threshold

A critical population size, St, can be expressed where R0 = 1

So, at that threshold:

St = 1/(βL)

- In populations with numbers of susceptibles less than this, the infection will die out (R0 < 1)

Consider the Different Kinds of Population

- Microparasites are highly infectious (large βs)
- Microparasites give rise to long periods of infection (large Ls)
- ….produce high R0s
- What kinds of parasite populations would behave like this?
- Also, differences in the parameters β and L can determine whether a disease becomes endemic or not…..
- What other factors or parameters (hint: mobility)

II. Continuous Model (calculus!!) for Direct Transmission and the Basic Reproductive Rate of a Disease, R0

(as used by Anderson and May in continuous model derivations)

R0, “the basic reproductive rate,” is the average number of successful offspring a microparasite is intrinsically capable of producing.

More precisely, in the case of the compartmental model, R0 is average number of secondary infections when one infected individual is introduced into a population where everyone is susceptible.

R0 is mainly a theoretical concept that is extremely important in mathematical ecological epidemiology for deriving numerous equations valuable in infectious disease research.

Re, The Effective Rate of Reproduction

R0 is extremely difficult to measure for ecologists, so Re, the “effective reproductive rate of a disease” can be a more useful practical tool.

The effective reproductive rate, Re, is the is equal the basic reproductive rate, R0, discounted by the fraction, x*, of the host population that is susceptible at equilibrium. x*can be estimated from serological data.

At equilibrium, Re =1, therefore, R0x* = 1

The density-depend process of holding Re below R0 is simply the removal of susceptible individuals from the population by immunity.

In reality, various factors intervene to prevent “runaway” exponential growth of an infection in a population besides the increasing density of immune individuals – microparasites are affected by all of the same kinds of abiotic and biotic regulatory factors as any population of organisms.

Considering Immunity alone, if a proportion p is becomes immune (by natural infection or a public health immunization program) the proportion remaining susceptible is at most x* =1 – p.

Therefore, ReR0 (1-p)

If the right hand side of the equation is less than 1, then Re < 1, and the infection will not be able to maintain itself in the host population.

Thus, the critical proportion, pc of a population to be immunized (holding population density constant along with other assumptions like complete mixing) for eradication of a disease is:

pc = 1 – (1/ R0)

*For many diseases Immunity is by far the most important factor, outweighing all other factors together

from Anderson and May (1991)

Immunization coverage in Africa of about 80% succeeded in eradicating the smallpox virus, but not in India where the lowest population densities (17/km2) where still greater than the highest densities in Africa (13/km2). “Re” was basically reduced toless than one by immunizing a smaller fraction of the population in Africa than in India.

Host Threshold Density - A key Concept in Eco-Epidemiology

pc can be restated, as the concept was originally developed by Kermack and McKendrick (1927) as:

Host Threshold Density (NT) – the population density below which Re< 1 for a given parasite, and an epidemic or epizootic, thus the establishment of a pathogen in a human or animal reservoir (including vector) host population, is not possible.

Model for the Basic Reproductive Rate (R0): Directly Transmitted Microparasite

R0= e( S/Γ)-1

This is satisfactory in explaning/predicting the dynamic behavior of directly transmitted diseases independent of social behavior (like measles, pertussis, etc.)

Basic Reproductive Rate (R0): Directly Transmitted Microparasite

R0= e( S/Γ)-1

density of susceptibles

Satisfactory in explaning/predicting the dynamic behavior of directly transmitted diseases independent of social behavior (like measles, pertussis, etc.)

Basic Reproductive Rate (R0): Directly Transmitted Microparasite

R0= e( S/Γ)-1

rate of depletion of infective pool though death or recovery

density of susceptibles

Satisfactory in explaning/predicting the dynamic behavior of directly transmitted diseases independent of social behavior (like measles, pertussis, etc.)

Basic Reproductive Rate (R0): Directly Transmitted Microparasite

R0= e( S/Γ)-1

rate of depletion of infective pool though death or recovery

density of susceptibles

transmission coefficient

Satisfactory in explaning/predicting the dynamic behavior of directly transmitted diseases independent of social behavior (like measles, pertussis, etc.)

Actual Number of measles cases reported in New-York, 1928-1971

Model Simulation

Number of new cases of measles in a population of 1M individuals (birth rate equal to death rate = 0.0000351; R=15)

1 2 3 4 5 6 7 8

Rt 1 1 2 2/2 3/2 4/3 6/4 7/6

Chains of Transmission in SARS

(discrete generation model from Anderson et al. 2005)

Chains of transmission between hosts

Generation =time

1 2 3 4 5 6 7 8

Rt 1 1 2 2/2 3/2 4/3 6/4 7/6

Chains of Transmission in SARS(discrete generation model from Anderson et al. 2005)

Effective reproductive number = number of new infections caused by each new case at time, t.

1 2 3 4 5 6 7 8

Chains of Transmission in a Malaria

Rt 1 3/1 8/3 14/8 ………………

Mosquitos’ mobility effectively spatially homogenizes human population locally, resulting in an R0 approaching N2.

1 2 3 4 5 6 7 8

Chains of Transmission in a Malaria

Rt 1 3/1 8/3 14/8 ………………

Mosquitos’ mobility effectively spatially homogenizes human population locally, resulting in an R0 approaching N2.

R0 for malaria reported to be as high as 80, and even in 1000 locally!

Malaria Model Parameters

Assume a single primary case with a recovery rate of y, where the average time spent in an infectious state is 1/y.

During this time, the average number of mosquito bites received fromm susceptible mosquitoes, each with a biting rate a is am/y.

Of these mosquitoes a proportion c is actually infectious, which gives a total of amc/y mosquitoes infected by the primary human case.

Each of these mosquitoes survives for an average time of 1/, where is the per capita mortality rate. Each makes a total of ab/ bites, where b is the proportion of infectious bites on humans that produces a patent infection.

The total number of secondary cases is thus (ab/)(amc/y). Note that a enters into the equation twice since the mosquito biting rate controls transmission from humans to mosquitoes and mosquitoes to humans.

Discrete Model for the Basic Reproductive Rate (R0): Based on Malaria

R0 = ma2bc/µγ

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0):

density of susceptible mosquitoes

R0 = ma2bc/µγ

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0):

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0):

per capita mosquito mortality rate

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0):

per capita mosquito mortality rate

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

proportion of infectious bites that produces a patent infection

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0): Indirectly Transmitted Microparasite

per capita mosquito mortality rate

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

chance a mosquito gets infected biting an infected human

proportion of infectious bites that produces a patent infection

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Basic Reproductive Rate (R0): Indirectly Transmitted Microparasite

per capita mosquito mortality rate

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

per capita human recovery rate

chance a mosquito gets infected biting an infected human

proportion of infectious bites that produces a patent infection

Anderson and May (1991), after Ross (1911) and MacDonald (1957)

Model Explanation

Assume a single primary case with a recovery rate of y, where the average time spent in an infectious state is 1/y.

During this time, the average number of mosquito bites received fromm susceptible mosquitoes, each with a biting rate a is am/y.

Of these mosquitoes a proportion c is actually infectious, which gives a total of amc/y mosquitoes infected by the primary human case.

Each of these mosquitoes survives for an average time of 1/m and makes a total of ab/m bites.

The total number of secondary cases is thus (ab/m)(amc/y). Note that a enters into the equation twice since the mosquito biting rate controls transmission from humans to mosquitoes and mosquitoes to humans.

Example of Model’s to Understanding Epidemic vs. Endemic Malaria

The Ro equation for mosquitoes is usually derived algebraically to examine the stability properties of the differential equations to understand the dynamical properties of the model using graphical analysis.

For our purposes, consider the term, ac/m, the average number of bites made on a human host during the lifetime of a mosquito that lead to infection.

If this number is large, relatively small changes in the mosquito density, m, or the biting rate, a, will have little effect on the equilibrium prevalence of malaria – the result is ‘stable endemic malaria’.

The term ac/m has been called an index of stability. In areas where mosquitoes bite humans relatively often and have relatively long life spans, the index is high.

Where mosquitoes bite humans less often and have shorter life spans, the index is low and in these areas human populations tend to be subject to epidemic outbreaks. This is where R0 tends to be < 1.

Intrinsic Versus Extrinsic Factor Affecting Epidemiology of a Disease

Ecologists find it useful to distinguish environmental factors such as climate variability (seasonality and changes in the weather) and predicted global climate change from factors intrinsic to the host-pathogen transmission dynamics.

We can modify the values of parameters in the compartmental model based on experimental data. For example, using laboratory data on temperature induced changes in biting rates or life span in mosquitoes could be used to parameterize the model for different climate change scenarios.

Intrinsic versus Extrinsic Factors

Terms influenced by extrinsic factors

per capita mosquito mortality rate

density of susceptible mosquitoes

Mosquito biting rate

R0 = ma2bc/µγ

per capita human recovery rate

chance a mosquito gets infected biting an infected human

proportion of infectious bites that produces a patent infection

1 2 3 4 5 6 7 8

Chains of Transmission in a Malaria

Rt 1 3/1 8/3 14/8 ………………

Mosquitos’ mobility effectively spatially homogenizes human population locally, resulting in an R0 approaching N2.

1 2 3 4 5 6 7 8

Chains of Transmission in a Malaria

Rt 1 3/1 8/3 14/8 ………………

R0 for malaria reported to be as high as 80, and even in 1000 locally!

Paradox of Directly vs Indirectly Transmitted Parasites

R0 can approach N2 for vector borne diseases, e.g. for malaria R0 > 100 or even 1000!

[100% children in endemic regions]

Summary of Disease Ecology Principles ( and mathematical epidemiology)

- Rate of growth of an epidemic (R0)– varies significantly as a function of intrinsic and extrinsic factors.

2. Host Threshold Density (NT) – An infectious disease (along with its environmental circumstances) has a unique host threshold density at which R0 equals or exceeds 1, the ‘critical population’ size or Proportion (Pc) of Immunes – determining endemic and epidemic activity.

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