June 14th, 2009 ecoli
Describing a model of infectious disease, as done in the field of mathematical epidemiology, helps scientists and doctors maximize the effectiveness while lowering cost of treating a population of individuals. To do this, however, one must first be able to describe how an infection moves through a population or, more accurately, how the population changes over time due to infectious agents.
This has been done using dynamic differential equation models for a long time now (and they can get quite complex). The most common of these models is called the SIR model (which stands for Susceptible -> Infected -> Recovered) representing the dynamic conditions of individuals within a general population as they “flow through” these different disease states at various rates.
The first ‘discovers’ of the simple SIR model was W. O. Kermack and A. G. McKendrick in 1927, in a paper published in the Proceedings of the Royal society of London. Their model is, ex post, appropriately titled The Kermack-McKendrick model.
The paper itself goes through some complex derivations but the results are suprisingly simple and intuitive. In a closed population, N, where there is no population growth, they assume instantaneous disease incubation, homogeneous population with no age, spatial, or social structure and fixed transmission and recovery rates. The result is this:
The susceptible population, which is the total population at t=0, decreases as an infection spreads and individuals become infected. This occurs at the constant rate of β as a function of the number of infected individuals (the source of new infections) times the number of susceptible individuals (where the infectious agents are spreading to). This means that to sustain an epidemic a disease requires a healthy reservoir of susceptibles or risk “burning” itself out. This equation looks like this:
The dynamics of the infected population depends on the growth of new infections minus the recovery rate (or even death rate) or infected individuals, in much the same way that was described for the change in susceptible population:
Finally, the infected population recovers at a constant rate, γ, at some fraction of the population.
They get a pretty simple model, which they solve mathematically, but, with all these assumptions, can they get any useful data?
Yes!
If you can’t read the chart’s description, this is an epidemic of the plauge on the “Island of Bombay” which fits the The Kermack-McKendrick model for epidemics quite precisely. When the infection gets introduced, although there are plenty of susceptible individuals around, the infection grows slowly (because the number of infected individuals starts out small). However, the exponential trend takes off as the disease creates many new infections (which in turns create more new infections) . But then the disease level crashes as it runs out of new individuals, as individuals recover or, in this case, they die. Whereupon the epidemic is over.
Can we describe or predict the time evolution of an epidemic in a more general way?
Yes (according to the Kermack-McKendrick model)!
The so-called time evolution of the model is described by the so-called epidemiological threshold:
Ro can be thought of as the ‘potential’ for new infections, or the number of secondary infections caused by a primary infection. When Ro < 1, the number of secondary infections is falling, and so the infection is dying out. When Ro > 1 each infected individual will infect more than 1 other person, thereby causing the infection to spread positively within a population.
This quantity Ro (the details of which differ depending on the specific model being used) has been described as one of the most important variables in epidemeology, with consideration for disease transmission potential and therefore disease evolution (in more complex considerations).
Despite the fact that the model’s assumptions may not accurately reflect most real human diseases, Kermack and McKendrick come up with genuinely interesting and relevant conclusions. They find that a population level requires some threshhold value in order for an epidemic to occur, a population size under this threshhold (which is disease specific) will not experience and epidemic. An infection in a population that exceeds this threshhold will reduce the populationas far below the threshhold as it was above it. Small increases in the infectivity rate can lead to dramatic changes in the dynamics of an epidemic, making them stronger. Epidemics generally end well before the entire population of susceptibles has been exhausted. Similar results are seen for diseases transmitted through an intermediate host (like vector bourne diseases).
Although this differential equation model lacks the accuracy of more complex models which relax assumptions and new types of stochastic models, the field of epidemeology has a lot to thank Kermack and McKendrick, for basically creating a new branch, bridging biology, mathematics and medicine.
Posted in classic science, mathematics, medicine, microbiology | 31 Comments »
May 21st, 2009 ecoli
This is going to be a multi-part series. As usual, my source.
The way we model the spread of infectious diseases may seem counter intuitive at first. We’re not looking at the movements of the bug, itself but rather populations of individuals infected with a pathogens. We split a general population into different types and describe their infection status, as a group. We then express the frequency of infection/recovery as rates, proportional to that population.
This will become clear in a moment.
Imagine we have a population of individuals who are susceptible to some disease. The infection will be transmitted to some subset of those susceptible individuals at a certain rate, and they become infected. Now, for this model, we assume individuals do not recover, the infection is non-lethal and we will ignore natural birth and death rates of the population.
The schematic for this simple model follows:
where β is the rate of infection (transmission). Since infections spread directly from person to person, the effective population of infected individuals and susceptibles vary directly with the rate. The infection spreads faster if there are more susceptible and infected individuals. We can see the dynamics more effectively with a math model, however. We can describe this schematic with a differential equation:
since the population is closed, S + I = N => S = N – I so,
Lets take a look at this equation: we have linear function of the infected population, varying with βNI. The the stuff in the parenthesis looks like a “saturation” limiter. When the number of total individuals is much higher than the infected, I/N is a small number. 1 minus a small number is close to 1, and you get a linear picture. However, as I approaches N (it can never exceed in this case) I/N = 1. Since 1-1 = 0, growth becomes saturated and, eventually, becomes zero. This looks very much like the logistics equation of population growth (in quadrant 1):
This result is logical: No matter how fast the infection rate is, eventually you’ll run out of new individuals to infect.
This is the last mathbio post I’m going to make in a while, probably.
Posted in mathematics, microbiology | No Comments »
August 7th, 2008 ecoli
This story has been big in the news lately. As you probably have already heard, the person who sent anthrax spores (Bacillus anthracis) through the mail has commited suicide. The culprit is army scientist Bruce Ivins.
Ivins worked with Anthrax in order to develop a vaccine, so one might think this is hypocritical to his work. However, current investigations lead us to beleive that Ivins was psychologically disturbed. Prosecuters speculate that Ivins was hoping to incense the public awareness of anthrax and hopefully get more funding. I guess scientists will go a long way to get research grants these days, but this is a new low.
My concern is how this event will ultimately change things for microbiologists and infectious disease researchers (a community I consider myself a part of). My lab doesn’t have anthrax, but we do have plenty of other deadly infectious diseases like Francisella tulerensis (tularemia), Yersinia pestis (bubonic plauge) and Borellia (Lyme’s disease).
Will researchers now have to get psych evals before we’re allowed to work with pathogens? Are intitutions going to tighten security, make us take even more safety classes, hire guards to check bags and not let us work alone? I hate to be the one to point this out, but it wouldn’t be exactly difficult to take pathogens out of the lab and even undergrads are given keys to the labs.
I have to sacrifice ease of access for security, but maybe the risks aren’t worth it? It just takes one incident to ruin it for the rest of us.
Perhaps (and hopefully) I’m wrong, and this will blow over as a unique incident, but it wouldn’t surprise me if universities, national and armed forces labs will at least start to discuss greater security measures.
Posted in medicine, microbiology, musings, news | 2 Comments »
August 6th, 2008 ecoli
This one is from Sci Am.
via ERV
Posted in medicine, microbiology | 2 Comments »
June 25th, 2008 ecoli
A while ago I posted a bit about my research woes. Well, I’m happy to report that my efforts since then have been fruitful. Creating the new constructs seem to have done the trick. There must have been something wrong with the original recombinant plasmid I was using (one that I didn’t actually create myself, by the way).
I’d never thought I would be so happy about seeing a little purple band appear on a little peice of paper before, but that band indicates the right protein is getting expressed and I can see it via western blot.
Posted in microbiology, my research | 10 Comments »
June 25th, 2008 ecoli
DaveScot, an IDiologist who writes for the uncommondescent blog has claimed to have found a mistake in Richard Lenski’s paper. (which I first talked about here)
He points to a statement made by Lenski: (DaveScot’s emphasis)
However, selection requires heritable variation
generated by random mutation, and even beneficial mutations
may be lost by random drift.
And then points to a study done by the Scripps institute that would seem to contradict this statement:
The bold portion is patently wrong. Selection operates on any heritable variation whether random or not… The Scripps researchers, in a nutshell, discovered that E. coli, when stressed (such as running out of food as in Lenski’s experiment or in the presence of antibiotics in the Scripps experiment) selectively increases the mutation rate on certain genes.
What DaveScot has wrong (and I believe he was refering to this experiment) is that the study says that mutations are purposefully induced on specific genes. However, the doesn’t say that only beneficial mutations are induced or that mutations were localized only to specific genes. This is a case of an organism increasing the rate of random mutations, which is a good survival strategy for a population to increase its genetic diversity. However, it does not appear to be the case, that the bacteria is select for their own survival.
So DaveScot is wrong in saying:
Thus the mutations in this case are not random but rather directed at a certain area in an attempt to solve a certain problem.
There is no basis to this teleological statement. Bacteria aren’t attempting to solve a problem, not in the same humans do when we create a new medicine or drug. They are simply increasing their genetic diversity (and probably not on purpose either, more likely in response to selection pressures), so that when antibiotics are around, the probability of a random mutation conferring antibiotic resistance is increased.
DaveScot FAIL
Posted in creationism, evolution, genetics, link out, microbiology | 9 Comments »
June 25th, 2008 ecoli
Thanks to bascule at SFN for this link.
Richard Lenski, whose groundbreaking work witnessed the evolution of citrate-using E. coli in a mere 44,000 generations over 20 years of work.
He responds directly to conservapedia’s ignorant creationism populace published by conservapedia (amazingly enough) here:
Lenski’s second letter is particularly good, filled with lots of good biochemistry, scientific philosophy and general good sense. A highly recommended read.
Discuss the exchange here.
Posted in biochemistry, creationism, evolution, link out, microbiology | 3 Comments »
June 9th, 2008 ecoli
It’s been a while since I’ve blogged about my own work. I thought I’ve give an update.
I have the chaperone of the CS1 pathway cloned onto a vector and in a host strain. But for some reason, I haven’t been able to induce expression to high enough levels to be useful for an overlay assay, in the periplasm prep.
In english, Gram-negative Enterotoxigenic E. coli (ETEC) have two membranes. I strip the outher membrane off and collect the proteins in the periplasm space. But, for some reason, the protein I’m trying to collect, which chaperones other CS1 pilus proteins between the two membranes, is not to be found in the periplasm.
Therefore, I’m going to take the gene out of the plasmid vector its currently in and try putting it into a different one. Wish me luck.
Posted in microbiology, musings, my research | 1 Comment »
June 4th, 2008 ecoli
No, it’s not a sandworm from Dune… (though the resemblance is uncanny):
It’s actually a Hookworm, a multicellular parasite that infects the intestine of humans and other mammals.
A study reported on in Microbiology Bytes explains a finding in which polyparasitic infections are implicated in the worsening of conditions of anemia in children. When hookworm and either Schistosoma japonicum or Trichuris co-infect a host higher levels of anemia are detected. This is despite the fact that these two parasites do not infect the same part of the host.
Interestingly, there is some synergistic affect that is occurring indirectly in the host. This provides some compelling evidence to conduct further epidemiological studies,to help control anemia by its corresponding parasitic pathogens. Further physiological studies are also needed, it appears to determine the synergistic role that these parasites play.
Posted in link out, medicine, microbiology, news | 2 Comments »
June 3rd, 2008 ecoli
For those of you who aren’t in the know, right now the American Society for Microbiology (ASM) conference is being held in Boston, Massachusetts. I’m not there, unfortunately, but it sounds like an amazing place to be if you’re a microbiologist. This enormous conference is being held at the Boston Convention Center, has drawn 12,000 people, taking up 23,100 hotel rooms. According to this report from Boston.com, the conference is estimated to bring in $15.6 million dollars in “economic activity” into the Boston area.
I look forward to hearing stories about the conference.
post script: Also, there seems to be some Science Social Media breakfast, that a couple of bloggers have been talking about. Apparantly, its open to the public, but not directly associated with ASM. That’s about as much detail as I can find. Does anyone want to explain, in further detail, what exactly this thing is?
Posted in microbiology, news | No Comments »
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