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30 Cards in this Set
- Front
- Back
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When to use mathematical modeling? |
A newly deadly infection has emerged and the question of: How can the infection be controlled?
Trials could take too long, so an alternate is to have a model of various populations and the contact patterns between individuals in the population - aka simulate reality |
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What are recent examples of emerging infectious diseases? |
HIV, SARS, Swine Flu, seasonal influenza, Ebola, |
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Stochastic (variation has evolved) model of a full country, all inhabitants, and their contacts |
Ex. Thailand: Over time several curves of how this epidemic may look (every line is a different stimulation model) and gray area is where all the curves may be found. The thick line is the average course of said epidemic. Put into the computer about everything about Thailand and hypothetical avian influenza outbreak. Graph D, the proportion of infected people is somewhat higher for children b/c
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Main conclusions of Thailand example of avian influenza. |
If you want to eliminate is often feasible using "geographic targeting" and stockpile of 3 million courses of antiviral drugs is sufficient |
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What information do we need to use the model for other epidemics? |
-How the disease is spread? such as in animal vectors (zoonosis)
Spread to human via respiratory contact... -behavioral change and immunity built in over time -incubation time (infectious via early or late) -combination of contacts and how infectious a person can be -probability of developing immunity/death
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Compare scenarios with or without an intervention to suspend 99.9% of air travel from affected countries. Ex. Hong Kong is country of origin, starting by June. (Probability city has experienced major epidemic) |
Blue lines show flights going on, and then eventually air restriction. Air travel shows no control of transmission. Only control of local transmission most effective, air travel restrictions give small delay at most so only some time to perhaps stockpile medicines and better treatment protocols. |
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Ex. 2 Promising new intervention:
UNAIDS is interested in the impact of a roll-out of interventions to reduce HIV infection: - Male circumcision, PrEP, etc. - Promising vaccine candidate from the RV144 trail (Thailand)
S |
Modeling could be the solution: Using 3 parameters:No vaccincation, One-off vaccination 30% coverage and then 60% covereage and turns out it is not that effective.
Conclusions: the new vaccine holds promise for controlling HIV in Thailand; Regular boosting of immunity and avoidance of risk compensation is essential, and targeting sex workers would achieve the greatest reduction in incidence per vaccination. |
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Risk compensation |
Behavioral change due to vaccine so people tend to be less safe b/c they assume people have had the vaccine etc so less likely to use condoms etc. which could negate the use since the vaccine is only partially effective. |
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Ex. 3 Bioterrorism |
Small pox is a potential bioterrorism weapon. Countries need to be prepared and since experiments are obviously not possible modeling can give crucial insights. |
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5 reason to model infectious disease |
1) Explore the course of new infections 2) Predict the impact (cost-effectiveness) of interventions 3) Gain insight into mechanisms that influence disease spread 4) Structure our thinking about transmission 5) With a prefect model, then can explore management options; decision-support |
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History of modeling infectious diseases |
Daniel Bernoulli (1766)
Studied the effect of variolation on smallpox mortality
The main objective was to calculate the gain in life expectancy if small pox were to be eliminated as a cause of death |
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Ronald Ross |
Book: The Prevention of MAlaria (1911)
Introuction of the idea of a "critical" threshold (R0)
If mosquito density could be brought down sufficiently, malaria would disappear even if transmission would not be interuppted completely. |
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Kermack & McKendrick (Deterministic model aka large populations) |
Studied SIR models (1927 - 1933) (Susceptible, Infected, Recovered)
Delineate course of S, I, and R over time
Be able to make the basic SIR model!!
Susceptible begins at 100 and drops down to 0; Infected starts at 0, makes a bell curve going back down to 0 Recovered starts a 0 and slopes up to 100 |
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What does Stochastic mean: |
"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics. They are used when these fluctuations are important, as in small populations. |
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Ro |
mean number of secondary cases generated by one primary case in a fully susceptible population
Ro needs to be greater than 1 for an epidemic to occur. Less than 1 the infection will die out. |
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Ro estimates of Measles, Rubella, Chicken pox, |
M: 16-16, CP:10-12, |
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SIR model: rates and formulas |
Contact rate: b(beta) (how often people are in contact with each other)
Recovery rate: v(nu)
Note: 1/v (nu) = average duration in I( staying in the infected zone)
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dS (change in susceptibility)/dt(change in time) =
dI (change in infected persons) /dt =
dR (change in recovered population) /dt =
dN(total population)/dt = |
dS (change in suceptibility)/dt(change in time) = - (beta) x(S)x(I)
dI/dt = (+(beta)x S x I ) - (v (nu)) x I (infected))
dR/dt = + v(nu) x I
dN(total population)/dt = 0 (aka population stays stable) |
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dI/dt should always be greater than 0 so ((Beta) x S x I) minus (v(nu) x I) > 0
Cross out I on both sides and S=N ; ((beta) x N) minus v(nu) > 0
(beta) x N > v(nu)
(N)(beta) / (nu) > 1 ===== Ro
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(N)(beta) / (nu) > 1 ===== Ro = N / (nu / beta)
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Limitations of SIR model / Ro |
-simply a threshold, not the average number of secondary infections. -heterogenity (no age in this model) -No birth and death -Life-long immunity -No extra death of infected people -No seasonality -problematic if there are intermediate vectors between hosts, such as malaria |
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Populations and Endemic vs Epidemic |
Endemic : measles in England and Wales
Epidemic : measles in Iceland (not enough of newborns who are susceptible to
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Macdonald : Ro for malaria |
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Critical Vaccination level
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Ex. For SARS the Ro was 3 so 1- 1/3 (67 % of the population must be vaccination in order for SARS to not spread
Condition for epdemic : Ro > 1
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Critical Vaccination level also referred to as herd immunity threshold |
The indirect population experienced by unvaccinated individuals resulting from the presence of |
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Ro and Rt |
Ro is the basic reproduction number Rt is the effective reproduction number = mean number of secondary cases generated by one primary case over time
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Delineate the course of an epidemic in terms of Rt |
y axis is in |
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Discuss the usefulness of the different interventions in the case of SARS? |
Mandatory reporting was clear to be useful
Perhaps the usefulness of the stringent control was not necessary to stop the epidemic however slower than without the stringent controls.
Answer strong political commitment and a centrally coordinated response was the most important factor int he control of SARS
The stringent control measures - including canceling May Day - were introduced after the decline of the epidemic was already underway
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Summart of Rt |
.. |
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Compartmental models |
-Divide the population according to infection status -assusme that all individuals ina compartment behave equally -SIR model -SIS, SIRS (malaria), SEIR, etc -sometimes arrive analytical solution |
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Individual-based models |
-collection of heterogeneous individuals -rules specifying individual behaviors -stochastic micro-stimulation: STDSIM (everything that happens to these individuals is a roll of the dice to memic reality) -flexible in design and assumptions |
