MIDTERM 2 STUDY GUIDE

R0 is Fundamental, it gives you a sense of the maximum capacity of a disease to transmit in a  population.  

Also, it is important that you can estimate it from epidemic data.  

Damped oscillations: - through time what happens to the proportion of infections, this is an  intrinsic property of infectious disease that you see in these simple models.  

Transmission varies through the year: This can be input into the model. Simplest is to make  the transmission term sinusoidal.  

If you do that and know R0 and infectious periods, the simple SIR model will predict real multi annual outbreaks of disease.  

Why is the oscillation an intrinsic property of infectious disease? There are delays (time it  takes to become immune, how long you are infectious/rate of recovery) that are specific to  certain diseases. Some diseases are highly infectious, so it comes into the population really fast  and hard, with fewer delays in those systems. Other diseases are slower and take longer to ramp up. 

When you have multiple of these factors, and a nonlinear system with delays, it inherently leads  to oscillation and periodicity. 

THRESHOLDS AND SPACE 

THRESHOLDS 

- We have an expression for R0 – transmission rate * susceptible population size /  virulence + natural death + recovery 

- R0>1 for invasion – it depends on the susceptible population size 

- Threshold population size is the one for R0=1 – known as the critical community size - See it with measles – smaller places less likely to see measles 

- Critical community size: smallest pop of S hosts below which a density dependent transmitted  disease goes extinct, and above which disease persists.  

SPACE

• See that there are patterns of spatial spread in disease – often mostly local with some long distance

• Leptokurtic dispersal leads to faster spread – see this.. many examples of accelerated spread

• Control of epidemic spread: 

  • Isolation: Identify infected individuals to limit their dispersal OR prevent 

whole population dispersal

• Quarantine: prevent dispersal during disease incubation period

• Ring vaccination/culling: pre-emptive to prevent infections in an area

  • Vaccination or culling local individuals 

VACCINATION 

Smallpox is an incredibly major reason for the history of the Americas. Europeans coming with diseases that the locals never had any exposure to and caused widespread epidemics.

Edward Jenner: the person who noticed the milkmaid connection often milkmaids didn’t get smallpox - They would get cowpox on their hands.

First vaccination: paper where he vaccinates a child with cowpox and then gives them smallpox and sees what happens. The thing that started and led to modern vaccination in the west. But vaccination had been going on around the world before that.  Lady Mary Wortley Montagu: she spent time in Turkey and knew about Turkish customs where people would do vaccinations for smallpox in that part of the world. People in the upper classes were regularly doing vaccinations. 

Smallpox was a major disease in the 1920s with a large peak in the 1950s. See it start being eliminated in different parts of the world through time. By 1977 got rid of it.

How many people should we vaccinate?

To eradicate smallpox: need to reduce Susceptible so that R0 is a lot less than 1. 

Vaccination: vaccination takes you right from S to R without ever going through I – it increases the recovered pool and decreases the S pool. 

Take the equation for R0 and try and get a threshold for how much we will have to reduce our susceptible class for R0 to get to 1. Vaccinate someone from S -> R and they have lifetime immunity (something like smallpox). Can calculate what that proportion is just from the equation. 

Greatest equation: proportion needed is 1 – 1/R0. 

Vaccination goals: protect others, reduce disease in the population. Herd immunity. Prevent an epidemic in a whole population. The greatest equation tells you what you need to do to actually eliminate it in a population. 

plot R0 and the p (critical vaccination threshold). Look at the shape of the curve. 

• With smallpox you need to get towards 80% of the population vaccinated. A large chunk but we managed to do it. Did it through hitting the critical vaccination threshold which allowed us to eliminate the disease

• Measles is 10-15 and puts it towards 95% needed for herd immunity. The US used to have that in 2003. WHO declared measles eliminated from the US because most parents vaccinated their children to go to school. Recently in Washington state ~100 cases of measles. Outbreak of measles in Madagascar ~9000 cases, 900 deaths. Vaccination coverage in Madagascar is about 60%. CA had an outbreak in Disney world, 2015. Most public institutions in CA require vaccination but not private schools or in other states.

Vaccination conflicts: people not vaccinating for different reasons. Conflicts that come into play. 

The concept of herd immunity where you need to vaccinate a large proportion of the population to get the benefit. One of the things that can happen is that once you have herd immunity (measles early 2000s) and people stop worrying about it. No individual motivation if you don’t see the disease. But we have not eradicated the disease from the entire world. Populations that did have herd immunity become vulnerable again when control is not maintained, and new infections enter the population. 

Fraction to be vaccinated is high, but not 100%: the “greatest equation” gives you the critical proportion of the population that needs to be immunized. 

• Smallpox was quite low. Measles is high 90-95% due to high R0

• Depends on R0 which also depends on population size

• Less dense population lower population that needs to be vaccinated

• Typically, the vaccinated proportion is high but not 100% of the population except for something like malaria in a hyperendemic region (99.99%). But that’s if you’re only vaccinating to try and eradicate. If you have multiple control measures you can still get eradication locally (in the US), make vaccination compulsory for the child to go to kindergarten then you can get up to the 90%s (at least in the US).

Factors important in vaccination: - why could we eliminate smallpox?

• R0 is low. Transmission isn’t that great

• High efficiency of the vaccine. Critically there is a lifetime of immunity through the vaccination

• Convincing case: very easy to get everyone on board with something like smallpox when there is a high mortality rate (good chance you’ll die from it and even if you recover then there is disfigurement). 

• High vaccine coverage: 80-90%, can get higher than this easily with US and UK with childhood diseases since you can have a semblance of gatekeeping (can’t go to school if not vaccinated). 

• You can tell what you are dealing with. One of the big issues with diseases like malaria is that you don’t know the incidence of malaria because there isn’t perfect reporting and no visible obvious symptoms. 

• No evolution of immune-evasion/vaccine resistance. Smallpox virus has never been able to evolve to avoid the vaccine. 

Host Population Regulation

How much are parasites responsible for regulating their hosts populations?

Black plague

• Estimates of the world population took a huge hit with plague. At least in the past diseases have had some big impacts on human population

Wild 

• If you look at populations in the wild, why are there only so many wildebeest and elephants 

  • Resources and predators can stop it

  • Disease 

Serengeti

• Ultimate predator prey system. Are wildebeest being controlled by lions primarily? 

• Rinderpest was removed by vaccinating cattle on the edge of the park. Cattle all around the park and they started vaccinating cattle, big program, and then eliminated rinderpest. See what happens to the wildebeest numbers: 200,000 to 1.2 million even though they didn’t vaccinate any wildebeest

• Numbers increased by 8 times, and buffalo also increased

• Can see striking evidence of how much a pathogen was controlling their population numbers when that pathogen is removed and the population numbers skyrocket.

• Large predators also increased in numbers 

  • Lion populations 

  • See that the drop offs in population are all linked with canine distemper virus 

  • Why did we not see these crashes before in the lion population? What has changed in the lion population? 

    • Lion population has gone up, denser population of lions, has it gone over a critical population size? 

Predators vs pathogens 

• Predators: slower birth rate, gets full for some time – therefore there is a handling time 

• Pathogens: much faster rate of population increase, insatiable. If it’s density dependent its populations just keep going. 

Graphs

• Predators you see them tend to be satiating

• Vectored and STD tend to plateau off. OID they might keep going up and up for a long while. Much less propensity for them to saturate. 

Important insights

Population regulation

• Deaths=Births; this is occurring when the population is otherwise at equilibrium 

• It requires this density dependence. As density goes up, more deaths due to disease, everything levels off at some point. 

Population effects

• Births come from susceptible and infectious individuals. But often you get fewer births from infected(I) individuals than S. 

• SI model showing that the infected class contributes slightly less to the birth of the new susceptible class

Disease virulence

• Graph showing the relationship between increased mortality with the equilibrium population size. If mortality is very high, very high virulence (bottom of the R0 equation, infectious period goes down, R0 goes down) at some point R0 goes below 1 and the disease dies out. Very virulent diseases don’t persist in populations. When mortality is very low there are not enough deaths happening in the population, so it doesn’t regulate the population either. 

• Disease regulates the population at intermediate levels of mortality.

Sterility

• As it goes higher then it sterilizes the Infected class completely. This doesn’t matter to the spread of the disease (in contrast to mortality that shortens the infectious period). Higher sterility will regulate the population more and more. 

Summary

• Sterilizing diseases would have bigger impacts on populations than deadly ones

HUDSON ET AL – GROUSE

Grouse – a bird valued for shooting - where the environment is managed on moorlands

• Famously shows cycles – high some years and too low to shoot others – this is a problem as you want to have regular shooting. 

• Infected by worm – with classic life cycle – notice that it has a free living stage in the environment

• Large funded research program to understand impact of the disease – 

• Show less chicks when individuals have more worms

• Higher breeding mortality when you have more worms

Model

• Similar to SIR model but now we have a macroparasite (the worm) where individual hosts have different worm burdens and we know this is important – need to model it

• Modeled with a distribution – again k tells you how random (everyone the same) or clumped (some with a lot – most with few) – this distribution is.  

• Notice that the infection reduces reproduction as well as increases death (this is important in a wildlife disease)

• Model predicts which factors lead to destability – i.e. might cause cycles.

Test to see if the destabilizing processes predicted by the model are true

Random distribution of worms – kind of.. 

Fecundity - yes

Time delays – yes

Population level experiments

Remove worms in the field and see if populations stop cycling

Very important and rare to do such an experiment

Worked… 

Take away: we currently in this field have far more questions than answers. Huge space for new researchers and diverse ideas and thoughts to help us figure it out.