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 28th, 2009 ecoli
Dr. James Holland Jones has an (older) post on scientific uncertainty and fat tails on his monkey’s uncle blog. Its one of the better explanations I’ve read, so do your self a favor and get educated. He also asks:
As scientists with an interest in policy, how do we communicate this type of uncertainty?
Before you can even ask this question, you have to ask if scientists, biologists in particular, are even interested in communicating uncertainty. Statistics and math are just not stressed in undergrad bio courses, and not at all in high school, so students are coming up with very little conceptions about uncertainty in data. I admit to being guitly of this but, unlike myself, how many scientists are willing to learn? Do their egos permit discussing statistical uncertainty?
This reminds me of one of Jorge Chan’s PhD comics on the science news cycle:
The cycle is exacerbated even in the first step, when scientists are not even well trained to understand or compute uncertainty themselves.
Posted in education, link out, mathematics, musings | No 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 »
March 16th, 2009 ecoli
In my last post I introduced Mathematical biology by explaining simple, 1st order, linear, differential systems.
Now, I’d like to get into non-linear systems, which better represent biological phenomenon, because the solutions output is rarely in direct proportion to the input.
Consider, for example, an enzymatic reaction. At low levels of substrate concentration, adding more substrate has a direct correlation with the amount of product form. However, the concentration of enzyme in this case limits the maximum velocity of the reaction, because the enzyme gets saturated with substrate. Therefore consider the changing population of Substrate and product like this:
-dS/dt = dP/dt = Vmax x [S] / (Km + S)
This is the Michaelis-Menten equation. Plotted it looks like this:
On the y axis is reaction velocity (dS/dt) and the y axis is substrate concentration. While this equation is specific for enzymatic reactions, a similar model for logistic growth can be applied to natural populations, growing in a limiting environment.
Consider first a continously growing population of a single species. As we went over last time, the size of population N at time n determines the population at Nn+1. Although the population can only be assesed at censuses, let us consider a continuosly reproducing species.
If a population doubles during every time period of τ then we get population growth that looks like this:
Time Number
0 N0
τ 2N0
2τ 4N0
3τ 8N0
This is a clear example of exponential growth – the coefficient of xN0 doubles for every linear increase of τ. The general equation which fits this exponential growth is N(t) = N0 2^(t/τ) - In the above chart, t is always given as a function of the doubling time.
Let us now consider natural death in a population (assume no competition). In this case, contrary to the growth rate, the death rate is directly proportional to the size of the population.
dN/dt = -γN where γ is the per capita death rate which, when solved, gives N(t) = N0 e^(-γt) an exponentially decaying function.
To combine terms and get a useful model for a growing species, lets first put the birth equation into a more usable form:
N(t) = N0 2^(t/τ) = N0 e^ln 2 * (t/τ) and let β = (ln2)/τ so that N(t) = N0 e^(βt) and dN/dt = βN(t)
Combining the growth and death terms, we get:
dN/dt = βN(t) – γN(t) = (β – γ) N(t) => N(t) = N0 e^(β – γ)t
for all β > γ you get an exponentially growing population.
But of course, exponential growth is never seen in nature (with people being the exception). This is because finite resources limit growth. Therefore competition ensures that the death rate will exceed the birth rate under certain conditions. Rather than defining these rates as complex functions, we can substitute the logistics equation of growth (which pretty accurately describes population growth).
dN/dt = (β – γ) N(t) [1-(N/k)]; where the new term (k) is the carrying capacity of the population (which can usually be determined experimentally and depends on the most limiting resource).
solving the logistics equation gives:
N(t) = kNoe^(Rot) / [K - No+ No^(Rot)] where Ro = (β – γ).
The logistics curve, N(t), looks something like the above (ignore the negative portion). Here growth is initially exponential but flattens out to K. (K equals Nmax).
In our next episode, we will consider growth on non-continuously reproducing species and intraspecific competition.
Posted in ecology, mathematics, musings | 19 Comments »
March 12th, 2009 ecoli
I’m taking my first mathbio class this semester and I would like to share with you all the insights I’ve learned so far.
Most biological processes are 1st order dynamic systems. Meaning that, unlike many physical phenomena, the ‘momentum’ of a system has no affect on the trajectory. The only state that plays a role on the current state is the one directly preceding it.
Consider 2 changing populations with the same birth rate and no natural deaths. Lets say one starts at a 10 individuals and the other starts at 15. In the next generation, both populations double; 20 and 30. In population two, I now remove 10 individuals and allow the populations to double again. Both populations will end up with 40 individuals. Even though the second population started out with more, the only state that matters is the one directly preceding it.
Lets consider a linear, 1st order, system with two variables;
dx/dt = ax + by and dy/dt = cx/dy
What are some properties of this system (that we can extend to linear dynamic systems in general)?
1) at x = y = 0, the velocities are zero
2) If velocity at (x,y) is (Vx,Vy)
AND velocity at (x2,y2) is (Vx2,Vy2)
Then V(x1+x2,y1+y2) = (Vx1+Vx2,Vy1+Vy2)
3) If V @ (x,y) is (Vx,Vy)
then V @ (cx,cy) is (cVx,cVy)
Here is an example of the above. Consider:
dx/dt = -x & dy/dt = -y
In the graph we see a description of the differential equation system. Every arrow represents the velocity; direction and magnitude. All paths point towards the origin, and spot at the stationary point at (0,0).
Lets know consider a more complicated system with two variables.
dx/dt = x + y & dy/dt = -x + y
Again, at (x,y), V = 0. When y = 0, dx/dt = x and dy/dt = -x
In this scenario, we have a unique case at x = y; dx/dt = 2x and dy/dt = 0
Graphing this system of equations, we get something like:
In this system, we get what looks like an destabilizing stationary point at the origin, with a spiral shape pushing outwards.
If you were to imagine x and y as 2 populations over time, it would look as if x and y alternating maximums, but both maximums getting higher as time progresses:
While its doubtful that a population of two species look like this in nature, it helps demonstrate how changing populations over time can be modeled with differential equations. For example, if species y is a predator of species x and species x has ample resources to grow, a growth pattern like this could occur, as blooms in one species causes a bloom in the other, and species x is able to take advantage of crashing y populations, and ample other resources to rebuild its own numbers even greater than before.
I continue this series next time with more modeling of populations.
Posted in ecology, mathematics, medicine, musings | No Comments »
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