Population dynamics of infectious diseases

Question 1

List the key pieces of information that are conveyed by the basic reproduction number (R0) of an infectious disease.

Answer 1

Average number of secondary cases generated by a primary case in a totally susceptible population.

• fundamental measure of the transmission potential of a pathogen in a given setting
• determines whether a pathogen can invade
• determines the carrying capacity (proportion immune at equilibrium)
• hence dictates the proportion necessary to vaccinate for eradication.

Question 2

How can you think of infectious diseases in terms of resource dynamics?

Answer 2

Infectious disease is the product of an ecological interaction between pathogen and host species – host is a resource for pathogen that is exploited by the pathogen

Question 3

Discuss how different life history strategies of pathogens are reflected in their population dynamics.

Answer 3

Colonisers (eg. HIV) are modelled as SI systems
• SI systems exhibit logistic growth with carrying capacity at 1-1/Ro
• SI systems do not have a tendency to oscillate
• unable to persist in in small,
low-density aggregations of
hosts
• temporary colonisers (eg. gonorhoea) are modelled as SIS systems
• SIS systems have similar behaviour to SI systems

Pathogens conferring lifelong immunity - Invaders - (eg. Measles) are modelled as SIR systems
• SIR dynamics comparable to predator-prey systems
• SIR systems have an inherent tendency to oscillate (regular cycles)
• SIR systems are also subject to fadeouts
• able to persist in in small,
low-density aggregations of
hosts
• antigenically variable pathogens (influenza, malaria) are essentially a set of sequential SIR models – difficult to control through vaccination

Question 4

Measles life history

Answer 4

Measles – INVADERS - hit&and run

• only lasts ~ 1 week
• short incubation period where viremia is high and individual is infectious but asymptomatic
• fever, rash & other symptoms (you can die during this stage)
• antibodies act against surface antigens, T cell responses, these act to reduce level either completely or to v low levels
• thereafter controlled by natural immune response, which confers LIFELONG IMMUNITY

Question 5

HIV life history

Answer 5

HIV – COLONISERS - keep host alive as long as possible to keep transmitting, eventual breakdown into (AIDS).

• starts with acute spike in viremia
• rash and flu a bit like measles, viral set point (maintained by combination of cellular and humoral response)
• you can have extended ‘clinical latency’ phase at ‘set point’
• If virus gets through CD4 T cells, drop of T cells causes lack of control of viremia, leading to full blown aids.

Question 6

Measles virus

Answer 6

The measles virus is a paramyxovirus, genus Morbillivirus.
It is 120–250 nm in diameter, with a core of single-stranded RNA

Two membrane envelope proteins are
important in pathogenesis. They are the F (fusion) protein, which is responsible for fusion of virus and host cell membranes, viral penetration, and hemolysis, and the H
(hemagglutinin) protein, which is responsible for adsorption of virus to cells.

Question 7

5 stages of an infectious disease

Answer 7

incubation, prodromal, illness, decline, and convalescence periods

Incubation - occurs in an acute disease after the initial entry of the pathogen into the host

• It is during this time the pathogen begins multiplying in the host.
• However, there are insufficient numbers of pathogen particles present to cause signs and symptoms of disease.

Question 8

How does HIV persist and evade the immune system & maintain chronicity?

Answer 8

• Escape from CD8+ T cell responses – peptides can acquire mutations so no longer recognised by CD8 T cells (??)
• Escape from antibody responses - HIV capable of varying its envelope proteins
• Switch in tropism - HIV able to switch receptor. Allows it to exploit different resource

Question 9

Malaria

• caused by ?
• symptoms
• what affects disease severity?

Answer 9

Plasmodium spp.
• Apicomplexan protozoa
• Of 156 named Plasmodium species, 4 infect humans (P. vivax, P. falciparum, P. malariae, P. ovale)
• Also sets up chronic infections
• Malaria restricted to areas where environmental conditions allows parasite multiplication in the mosquito vector

Symptoms
• Symptoms include fevers and chills, headache, vomiting, neurological symptoms

Disease severity
• differs acc. to species of Plasmodium, immune status, previous exposure

▪ They use a unique form of actin-based gliding motility to target and invade host cell - this uses highly dynamic actin filaments

Question 10

Plasmodium (malaria) life cycle

Answer 10

Life cycle
• General plasmodium life cycle: Mosquito infects sporozoites, then liver, merezoites, rbcs, replication, differentiation into gametocytes which are picked up by mosquitos

1. Mosquito injects sporozoites into bloodstream when it bites human
2. Sporozoites travel to liver and take residence in hepatocytes
3. Sporozoites multiply asexually into schizont, which ruptures, releasing merozoites back into the bloodstream
4. Merozoites invade erythrocytes, forming ring-like trophozoite.
5. The ring stage trophozoites multiplies asexually to form schizonts, which rupture releasing merozoites when rbcs burst
6. This cycle of rbc invasion, multiplying and bursting continues, causing malarial symptoms such as chills and sweating
7. After several asexual cycles, the merozoites will form gametocytes when it infects a rbc
8. Gametocytes sucked up by another mosquito, digesting the gametocytes and allowing them to mature into gametes, which fuse together to form oocyte
9. Oocyst develops sporozoites which are released when oocyst ruptures
10. Sporozoites migrate to mosquito’s salivary glands

Question 11

What’s common in Plasmodium and Trypanosoma strategies to evade the immune system?

Answer 11

Both employ antigenic variability strategy

• Both Trypanosoma and Plasmodium have evolved different mechanisms of antigenic variation to evade host immune systems by expressing VSGs (variant surface glycoproteins)
• Able to make new VSGs from library of genes they have which they can swap in and out – great way of achieving promiscuity
• Allows rapid adaptation to changing environments

Question 12

African trypanosomiasis (sleeping sickness)

• caused by ?
• how do they reproduce?
• symptoms

Answer 12

Trypanosoma brucei
• Free-living (extracellular) parasite carried around in bloodstream of infected hosts
• One of the few pathogens to cross the blood brain barrier
• Carried by Tsetse fly vector between mammal hosts
• Flagellated (9+2)
• Has an unusual organelle called a kinetoplast made up of many copies of the mitochondrial DNA
• Have glycosomes (membrane bound organelles containing the glycolytic enzymes) which are derived from peroxisomes

How do they reproduce?
• In mammal blood & lymph, grow long and thin and multiply by binary fission, penetrate blood vessel endothelium and invade tissue including CNS
• In fly, multiply and enter salivary glands

Symptoms:

• Haemo-lymphatic stage causes fever, headaches, joint pain
• Neurological stage when parasite infects CNS
• Changes of behaviour, confusion, poor coordination
• Usually fatal without treatment

Question 13

How does Trypanosoma evade the immune system & maintain chronicity?

Answer 13

Trypanosoma
• Have surface glycoprotein – express VSG (variant surface glycoproteins)
• Able to make new VSGs from library of genes they have – great way of achieving promiscuity

Question 14

How does Plasmodium evade the immune system & maintain chronicity?

Answer 14

Plasmodium
• P. vivax hides in liver
• P. falciparum decorates surface of rbc with antigens (eg. PfEMP1) that are the primary target of antibody mediation

o Allows infected rbc to attach to epithelium
o Stops rbc being trafficked into spleen (where it would be recognised as foreign and destroyed)
o 60 copies of viral gene with it sequentially expresses so that antibodies produced are no longer effective against second variant
o Clonal antigenic variation allows P. falciparum to avoid immune system until it runs out of viral genes or another arm of immune system destroys it

Question 15

How does TB maintain chronicity?

Answer 15

Hides out in wbcs!!

Mycobacteria are phagocytosed by alveolar macrophages in the lungs and cannot be digested as its cell wall prevents fusion of the phagosome with the lysosome

Able to subvert immune responses and maintain chronic for whole life time

Question 16

Name infections that can become chronic

Answer 16

AIDS - HIV

Malaria - Plasmodium sp.

Sleeping sickness - Trypanosoma brucei

TB - Mycobacterium tuberculosis

Glandular fever/ mono etc. Herpes virus (eg. EBV)

Chickenpox (varicella) and shingles (herpes zoster) - Varicella-zoster virus

Question 17

The SI model!!
creating a model to understand colonisation

• describe
• what used for
• equations

Answer 17

Colonisers (eg. HIV) are modelled using SI systems as there is no real recovery term (RIP)

• Split up host pop into 2 compartments. Only 2 possible states
• S = no. susceptible
• I = no. infected
λ = rate of moving S into I = per capita risk of infection

• Assuming S able to move into I at rate λ, this creates a flow from S to I
• Susceptible are being depleted, net loss from this compartment is -λS.
• In I, rate of change is λS
• You can simplify into one equation as S + I = 1 so can use one variable (y=proportion of population infected)

dS/dt = -λS
dI/dt = λS

dy/dt = λ(1-y) where λ = By

• SI systems exhibit logistic growth with carrying capacity at 1-1/Ro
• SI systems do not have a tendency to oscillate

Question 18

How do you understand λ?

Answer 18

λ = By
therefore dy/dt = By(1-y)

λ = rate of moving S into I = per capita risk of infection

The risk of infection depends on the proportion of pop infected

B is a combination of parameters that relate risk of infection to prevalence of infection

A big B can mean lots of things depending on type of pathogen and how it’s transmitted

Question 19

How do you understand λ?

Answer 19

λ = By
therefore dy/dt = By(1-y)

λ = rate of moving S into I = per capita risk of infection

The risk of infection depends on the proportion of pop infected

B is a combination of parameters that relate risk of infection to prevalence of infection

A big B can mean lots of things depending on type of pathogen and how it’s transmitted

Question 20

For directly transmitted pathogens

How do you define

B (a combination of parameters that relate risk of infection to prevalence of infection)

Answer 20

B = βN

β = probability of transmission on contact
N = population size

therefore λ = βNy
y = proportion of population infected

Question 21

For sexually transmitted pathogens

How do you define

B (a combination of parameters that relate risk of infection to prevalence of infection)

Answer 21

B = βc

Remember - only 2 people are involved - not a population!!

β = per partner probability of transmission
c = average no. partners you have over a period of time

therefore λ = βcy
y = proportion of population infected

Question 22

How does the SI model behave?

• in the absence of births and deaths
• in what scenarios can a pathogen invade?

Answer 22

dy/dt = By(1-y)

Logistic growth which plateaus where every host is infected - In a realistic scenario you must modify to include births and deaths!!

It’s reasonable to ignore the usual birth/death turnover of the host population if the timescales we’re looking at for the spread of the disease are much shorter than host generation time.

Pathogen cannot invade unless dy/dt>0 when y=0
ie. rate of change of population infected must be >0 when pop is fully susceptible

For this to be true, B/μ>1 because risk and prevalence of infection (B) has got to be bigger than death term (μ) for invasion of pathogen into population to happen (makes sense!!)

R0 = B/μ and is a measure of the average no. secondary cases generated by a primary case in a totally susceptible population eg. R0=3, each primary case individual will infect 3 others

SO in order for a pathogen to invade and cause an outbreak R0 >1

If R0<1, transmission chains are short so epidemic never takes off (either people die so don’t pass on or become immune so can’t be infected again??)

Question 23

How does the SI model behave?

• with births and deaths

Answer 23

dy/dt = By(1-y) - μy

This is the modified version that includes births and deaths (when pop is stable ie. births = deaths)

o When you start off, slow spread of infection
o Builds up until sufficient no. to keep spreading (exponential)
o Eventually saturates to equilibrium (for each person that dies, a new susceptible becomes infected)

To find out the proportion of people infected at which this equilibrium settles,
Set derivative at dy/dt=0, solve for y, get y=1 - μ/B

This logistic growth is seen in real life! Eg. South Africa HIV

Question 24

Rate of change in proportion of population infected, TAKING into account host death

Answer 24

dy/dt = By(1-y) - μy

This is the modified version that includes births and deaths

o When you start off, slow spread of infection
o Builds up until sufficient no. to keep spreading (exponential)
o Eventually saturates to equilibrium (for each person that dies, a new susceptible becomes infected)

To find out the proportion of people infected at which this equilibrium settles,
Set derivative at dy/dt=0, solve for y, get y=1 - μ/B

This logistic growth is seen in real life! Eg. South Africa HIV

Question 25

Herpes viruses (EBV, HSV, VZV)

• how do they evade the immune system & maintain chronicity?

Answer 25

In all VZV and EBV...
Primary infection generally occurs in a
subclinical fashion in early childhood,
with subsequent lifelong persistence of
infection. 

CHRONICITY

During primary infection

• Herpes Simplex Virus (HSV) enters peripheral sensory nerves and migrates along axons to sensory nerve GANGLIA in the CNS
• VZV also infects sensory neurones and becomes latent there

• The intrahost reservoir of EBV is B cells, almost always remains latent in these cells

Question 26

Varicella-zoster virus (herpes)

• what does it cause?
• how does it spread?
• how does it cause damage?

Answer 26

Causes:
Chickenpox (varicella) and shingles (herpes zoster)

How does it spread?
▪ Highly contagious
▪ VZV particles reach mucosal epithelial sites of entry by air and direct contact
▪ VZV spreads by local replication, lymphoid tissues and T cells

Causing damage 2nd time round - shingles!!
▪ For shingles, which is caused by recurrent infection, VZV is reactivated in the sensory neurones, producing new VZV particles that infect epithelial cells in the skin
▪ These epithelial cells release the virus extracellularly from infectious skin lesions

Question 27

Name 3 types of herpes virus and what they cause

Answer 27

They’re all chronic!!

Epstein-Barr Virus (EBV) - mono/ glandular fever (infects almost 90% of humans worldwide (normally harmless))

Herpes Simplex Virus (HSV) - cold sores and genital herpes

Varicella-zoster virus (VZV) - chicken pox/shingles

Question 28

How do you use equations to figure out if a pathogen can invade and whether you’ll get an outbreak?

Answer 28

Pathogen cannot invade unless dy/dt>0 when y just above 0
ie. rate of change of population infected must be >0 when proportion of population infected is v small

For this to be true, B/μ>1 because risk and prevalence of infection (B) has got to be bigger than death term (μ) for invasion of pathogen into population to happen (makes sense!!)

R0 = B/μ and is a measure of the average no. secondary cases generated by a primary case in a totally susceptible population eg. R0=3, each primary case individual will infect 3 others

SO in order for a pathogen to invade and cause an outbreak R0 >1

If R0<1, transmission chains are short so epidemic never takes off (either people die so don’t pass on or become immune so can’t be infected again??)

Question 29

Basic Reproduction Number

List the key pieces of information that are conveyed by the (R0) of an infectious disease.

Answer 29

• R0 is the average no. secondary cases generated by a primary case in a totally susceptible population
• Fundamental transmission potential of a pathogen in a given setting
• determines whether a pathogen can invade
• determines the carrying capacity (proportion immune at equilibrium)
• hence dictates the proportion necessary to vaccinate for eradication
• If R0<1, transmission chains are short so epidemic never takes off. Black blobs – people who have been infected and become immune
• R0 >1, (in pic R0=3 ie. Each individual affects 3 others)

Question 30

How can you rearrange equations to relate parameters in logistic growth curve to simple SI model

Answer 30

Carrying capacity of system is essentially equivalent to 1-1/R0 (ie this is why the graph stabilises there)

Intrinsic growth rate r=B-μ (this is why B/μ gives measure of fundamental potential of the systems to grow)

Question 31

The SIR model

• describe
• what used for
• equations

Answer 31

Pathogens with life history strategy conferring lifelong immunity (eg. Measles) are modelled as SIR systems

dS/dt = -λS
dI/dt = λS - σI
dR/dt = σI

dy/dt = By(1-z) - σy
dz/dt = By(1-z)

• Incorporates the recovered class (R) who remain immune for rest of life!!
• σ is the rate of recovery from I to R!!

• You can write down the equations in 2 ways

• the 3 SIR compartments OR
• lump I and R together into ‘z’. This is because both I and R are no longer a resource for the pathogen (from an ecological perspective) so all we need to know is what’s in z and what’s not in z.

Question 32

How does the SIR model behave?

• in the absence of births and deaths
• can we be confident in the model?

Answer 32

Same issue that pathogen cannot invade unless rate of change > 0 when z=0 (fully susceptible population).
At this level, same going on as SI model.

• No.s initially infected will grow (looks exponential)
• Available pool of those that can be infected (z) is reduced so the rate at which the infection can grow is reduced
• Growth rate becomes negative, density of susceptible population has fallen below a certain threshold. (steep decline)

Can we be confident in the SIR model?
- There is some correspondence between the output of the model and what’s going on in real life (Bristol data) – this gives confidence that the model is good

Question 33

How does the SIR model behave?

• with births and deaths

Answer 33

Long term behaviour of simple SIR model with birth (coming into susceptible) and death (all compartments)
• You see a rapid increase in proportions infected and very rapid increase in no. immune
• Epidemic starts to die out
• Never completely get rid of it – very low levels
• DAMPED OSCILLATIONS of proportion infected – eventually reach equilibrium
• Prop immune settles down to exactly what prop infectious settled to in SI model before, 1-1/R0!! IT’S THE OTHER WAY AROUND

Question 34

What features do predator-prey systems share with infectious diseases which

(i) lead to chronic infection (eg. HIV and other SI type pathogens)?
(ii) cause lifelong immunity after a short period of infection (eg. measles and other SIR type pathogens), or

Answer 34

pathogen=predator, host=prey, predation=infection,
predator reproduction=transmission
- In all cases prey population is diminished by predation

(i) In the SI case:
the predator continues to feast on the prey (as if put in refrigerator) and therefore doesn’t undergo a precipitous decline once a proportion of the prey has been consumed – leads to logistic growth

(ii) In the SIR case:
the predator stops reproducing once a proportion of prey has been consumed – this leads to classic predator prey cycling

• a good answer might mention that predator-prey is a better analogy for SIR than SI

Question 35

Population dynamics of SIR model over a longer time period

Answer 35

NOT logistic

Fadeouts - SIR systems are subject to fadeouts eg. Measles in Iceland
• Epidemics occur, die out, long periods with no measles
• Susceptible pop starts to grow until measles is reintroduced and takes off again
• We rely on
1. Build up of susceptible pop AND
2. Reintroduction of measles for new epidemic in small island populations

Regular cycles - eg. In large populations, Measles exhibits regular cycles (Denmark, UK, US)
• Doesn’t rely on external introduction of virus – no.s of infected never decline to the point where it has completely disappeared
• Births occur until threshold is crossed and measles reoccur

If you add births and deaths, inherent tendency to oscillate (damped oscillations)

These dynamics (cyclic and oscillations) are comparable to predator-prey systems
• You can rearrange equations, you get equivalence getting eaten = becoming immune etc etc see slide

Antigenically variable pathogens (influenza, malaria) are essentially a set of sequential SIR models – difficult to control through vaccination

Question 36

the advantages of chronicity

Answer 36

avoid the problem of too much host death and immunity meaning pathogen is subject to fade outs in small populations and needs reintroducing!!

chronic infections have an immortal virus cycle even in small populations

Question 37

Effect of vaccine on measles

Answer 37

• Before vaccination generally regular biennial epidemics (with small annual epidemics in between)
• After vaccine – much lower incidence and more irregular epidemics
Eg. In the early 90s, MMR saw serious declines in cases

Question 38

1 - 1/R0

what settles to this in SIR and SI models?

Answer 38

SI - proportion infected (plateau of the logistic model)

SIR - proportion immune (oscillations dampen and converge)

Question 39

eradication

• objective
• how achieved

Answer 39

Objective: Reduce the average number of secondary cases
generated by an index case to less than one (ie. R0<1)

To achieve this, the proportion (p) immunised must exceed the fraction immune at equilibrium in the absence of vaccination :

p > 1 - 1/R0
ie. p>

Question 40

If the pathogen has an R0 of 4, you need to vaccinate X% of the population for ERADICATION and Y% for STOPPING SPEAD

Answer 40

1 - 1/4 = 0.75

If the pathogen has an R0 of 4, you need to vaccinate >75% of the population for eradication and 75% to stop spread
• Most bugs we deal with don’t have massive R0 values, which has allowed us to bring many under control

Question 41

Why has smallpox been eradicated and measles hasn’t?

Answer 41

Differences in Ro may explain why we have eradicated smallpox (R0 = 6) while measles (R0 = 14) still ranks as one of the leading causes of childhood mortality worldwide

Question 42

name an example of a human disease where we

have no vaccine
have a partially effective vaccine
have a fully developed vaccine

Answer 42

no vaccine = HIV, malaria

partially effective = TB, influenza

fully developed = measles, polio

Question 43

Why do we have a vaccine against measles but not influenza?

Answer 43

Both have Haemagglutinin (HA) glycoproteins on surface of viral envelope that allow attachment to host receptor cells

Measles - The receptor binding site of HA is a conserved neutralising epitope. Our antibodies recognise epitope (region within receptor site) – this means that measles cannot mutate to avoid immunity – something is conserved! This leads to lifelong protection

Influenza - the loops of HA have potential for variation, and these are mutated in influenza.
• Subtypes die out and replace each other eg. swine 2009 – new lineage, but HA v similar 1918 virus (illustrates that virus changes, but within limits)
• sequential dominance

Question 44

Malaria antigenic variation

Answer 44

Able to sequentially express PfEMP1 gene – which is stuck on surface of rbc.

• Within a population, for malaria, you do not see sequential dominance but co-circulating strains
• Depending on strain, end up with different disease symptoms (cerebral, severe anemia)

Question 45

How do models of infectious disease link the disease status of individuals to the spread of the infection at a population level?

Answer 45

Models of infectious disease divide the host population into compartments.

What happens within the host determines:

(i) how the compartments are structured
(ii) the rates at which individuals flow from one compartment to the other

For a pathogen (like the measles virus) which sets up short infections followed by lifelong immunity within an individual, the appropriate compartments are Susceptible, Infected and Recovered, and the rate of loss of infectiousness is high (if the infection is 1 week long, the rate of loss is 1/52 years, for example).

For a pathogen (like HIV) which causes chronic infection, the Recovered compartment is not necessary.

The population dynamics of an SIR matches our observations of measles incidence in the prevaccine era (ie. it oscillates with a frequency determined by the length of the infection period and its basic reproduction number) while that of an SI matches the logistic growth exhibited by HIV prevalence. This demonstrates that the compartmental models are able to link within-host dynamics with the spread of infection at a population level.

A very good answer may contain details of the within-host dynamics (ie. graphs) – this was shown in the lectures

• discuss how the infectious period could be split up into more compartments to reflect nuances such as levels of infectiousness varying during this period
• mention virulence as a within-host factor which would influence death rates within the infected class
• mention that within host dynamics (such as viral titres) could influence force of infection

Question 46

frequency of oscillation of an SIR pathogen determined by

Answer 46

length of infection period

basic reproduction number R0