Hello All,
I have posted the next version of the growth plots. These are done a little differently this time. I have plotted both the total number of cases and also the total number of deaths. The separation between these two are the number of active cases. I have also plotted the best fit exponential to the existing, complete data set, rather than keeping the old curve.
Some comments:
1. The data appear somewhat consistent. The cumulative cases and cumulative deaths are growing at the same exponential rate.
2. The number of active cases (averaged!!!) is growing at the same rate as well (as it should).
3. The value of D_c/I_c has remained roughly constant at 90%, and is consistent with the growth rate. (Although for this most recent data set it ominously dropped to 83%, which suggests an increased growth rate of 40 days. However, as this is a single data point, we should wait before concluding anything).
4. The exponential e-folding time (exponential time constant) is now 93 days.
5. The usual caveats and conditions apply. This analysis applies only to the reported data and does not consider those cases potentially uncounted.
6. The actual numbers are fluctuating, but the general, underlying growth is clear. The fluctuations are to be expected for such small numbers and given the unusual reporting conditions in Angola.
7. I have plotted two curves. The first one (rhs) is just the data. The second one (lhs) is the data and the fit. The fit is extended out to 200 days into CY 2005.
8. The legend on the curve fit is incorrect. It reads "days vs. predicted." It should read: "Predicted Cases". (sorry)
my take on your data:
1. we do not have this bastard contained
2. it is more robust than Ebola
3. that is one lethal bug.
correct?
There have been several issues discussed, but one of the central issues was whether this disease would "burn itself out." So, I wanted to investigate this and model it, if possible. Although I am not a virulogist, and this is certainly not my expertise, I will draw on the brain trust from this thread for my base assumptions, and also state those assumptions explicitly here. It was explained on this thread that the essential process for "burning out" is that the the genetic material of the virus mutates. There are several processes for this (evidently), but the one most likely in this instance is random changes in base pairs. Now, RNA is (evidently, oh hell, assume "evidently" whenever I state an assumption), not as robust as DNA; it mutates more easily. However, it still has 4 possible base pairs and is in a strand. During any reproductive cycle, there is a finite chance that any of the base pairs will mutate. So during any reproductive cycle, none, one, or many base pairs may change, and these changes will be random. (Random is defined by the fact that there is no mechanism to predict which base pairs will change. However, it one previous post the refered medical article stated that the RNA of the filoviruses did not mutate as fast as expected. (To be precise, they don't know that because all they measure is the distribution). Now, it is reasonable to assume that there is an "optimum" genetic makeup that has the highest growth rate of all of the genetic variants. Moreover, we are not talking about a single virus particle, but a whole populaton of virus particles. As the disease grows in an individual, two processes ocurr: 1) the population of virus particles grow, and 2) the genetic variation of that population increases in diversity. Note also, that all of the scientific and medical tests for Marburg, including the genetic makeup of this Marburg variant, can only be done on the population. Finally, the general pathology of Marburg / Ebola is due to explosive growth of the virus particles. Other viruses have a much lower growth, and induce damage by other means. However, the principal effect of Marburg is explosive growth so that the cells a filled up with virus particles until they litterally burst open. No cell wall, no tissue structure. The body liquifies. Hence, hemorrhagic fever. Richard Preston (c.f. The Hot Zone) provides a very graphic description. There are ancillary pathological effects but they are not principal and I am ignoring them.
These are the only assumptions I am making. These assumptions define the underlying dynamics of the virus population growth in an organism, and are sufficiently well-defined that it can be modelled mathematically.
So, I modelled it. This basic process defines something in math called a "Markov Process". Indeed, it is essentially an ideal Markov Process. A Markov Process is colloquially known as the "drunken walk problem." I model the genetic variation as a parameter "x", which is the total number of genetic variations from the nominal, ideal, case. Now, a Markov process is easily modelled with a type of differential equation (2nd order partial differential equation in time and distance). In some limiting cases it is called the Diffusion Equation. I have discussed this already in posts 570, 590, and 844, and the mathematics is somewhat described in post 620, all in this thread. I am not going to go into all of the mathematical detail here.
The diffusion equation is straight forward to understand but difficult to solve. It is generally not solvable analytically (it is not here, for example), except in a few special cases. It describes a broad range of phenomenon, from heat transfer to impurity deposition in semiconductor manufacturing.
Now, we modify this equation to include other effects (which is why it has to be solved numerically). First, we consider the effect of immune response on the virus population. We assume that the immune response acts over the entire distibution, but builds up in time to a maximum level. Second, we assume that the virus population grows (that seems to be the basic issue here). Moreover, the "optimum" genetic makeup grows the fastest. As you diverge from the genetic optimum, the growth rate slows (that is why the optimum configuration is, well, optimum. I use the guassian distribution to model the relative growth rate as a function of genetic distance, "x". (There are lots of good math and physics reasons for this that I won't go into here, but trust me, it is a reasonable choice). We can vary the width and strenght of the growth function, the strength and time response of the the immune response, and the diffusion rate.
So we have a simple equation that we solve numerically. This equation models the three principal process that define the dynamics:
1. The virus population grows proportional to the growth function, a Gaussian centered around an optimal case and falling off with genetic distance "x".
2. The virus population decreases proportional to the immune response, which is isotropic across genetic variation but increases with time.
3. The virus diffuses in genetic makeup. With each growth cycle some genetic changes ocurr in the population.
Now, lets look at the numerical results. These plots are the solution to the equations.
Figure A is the base case. The 3-D graph plots number of virus particles on the vertical axis, time on the axis running from the origin to lower right, and genetic variation "x" on the axis running from the origin to the lower left. The time duration is 100 time units. The genetic variation is from +50 to -50 total base pair changes. Figure A is the simple, basic diffusion case. There is NO growth and NO immune response. An initial, relatively narrow distribution of virus particles are introduced. They go through genetic mutations in reproductive cycles. (Total numbers remain constant). During each cycle there is a finite probability that some of the base pairs will change. (Note that no units are used for time. The reason is that all units can be expressed as dimensionless ratios with the coefficients in the equations.) The behavior of the population is classic diffusion (right out of a textbook). The distrubution spreads out in time to become more, well, diffuse. The peak value drops and the over all population is more spread out. Presumably, because these genetic variants are less "optimum", they would grow less. This process indicates the basic method that a virus "burns out." "Burning out" is a bad metaphor. The virus dilutes itself to death. As it spreads out, it becomes less "optimal".
Figure B is similar to the results of Figure A in that there is an initial population distribution and the diffusion mechanism is still in effect. However, in this case we have added the immune response (which grows in time to reach a steady-state) and also the virus growth. The growth rate is modest and the width of the growth function is quite narrow (i.e. only those variants close to the ideal grow fast, others grow slow). The results are pretty much what you would expect. After initial introduction the virus population begins to grow. However, the immune response increases and soon the growth is reversed.
However, the growth function is narrow and therefore, the population distribution becomes more narrow. The population distribution is following the growth function. The faster growing variants last the longest because they grow the most. Therefore, the distribution becomes narrow because the growth function is narrow.
This narrowing effect is evolution in action. Although the natural, stochastic base-pair mutations are still ocurring, and the diffusion process is still in effect, those slower growing variants are killed quickly. As we will soon see, this isn't just evolution; it is evolution on steroids.
For this particular case, the immune response is sufficiently strong to kill the virus. The general dynamics modelled here is essentially the process for most of us getting a cold. We catch a cold. It gets worse for a while. Our immune system kicks in and we get better.
Note also that the diffusion effect and the immune response work in concert. Both are processes that remove viruses from the central peak of the distribution.
Figure C is identical to Figure B except that I doubled the growth rate. Now the growth overwhelms the immune response. The individual eventually dies. In real life, different people have different immune strengths. A strong person undergoes the process of figure B. A person with a weak immune system suffers response C.
Note also the peaking of the distribution. Again, we have evolution on steroids. Those genetic variants that are slower-growing are killed by the combined diffusion and immune system response. The faster growing variants outgrow the ability of the immune system to kill them. The distribution narrows.
Moreover, this process shows why the genetic makeup of the virus "appears" stable. Although the individual virus particles are mutating like mad, and trying to diffuse away from the "optimum" case, the optimum cases grow faster and survive. The distribution peaks toward the high growth cases. The population is stable. This is an important point. Even though individual viruses are genetically stable, the population is genetically stable. It's evolution, folks.
The implications are that this virus will never "burn out" or mutate itself to dilution. The very same process that makes it a hemorrhagic fever, i.e. ultra fast growth, is the same process the makes the virus population genetically stable.
Figure D is identical to Figure C in all regards except that I doubled the width of the growth function. That is, the growth rate is the same as figure C, it is just over a larger number of genetic variants. Note the scale size on the 4 plots: figure A: .1, figure B: .25, figure C: 3.0, figure D: 10^12. That's right: 10 billion. Big change. That's why all you see is the end state.
Conclusion: a wide genetic optimum in the growth function makes for very fast growth. Growth rate and growth width are both important. You get super growth and super stability.
Now, what can we conclude from these simulations.
A Marburg variant may exist that has a wide optimum (for growth) in its genetic makeup. I am postulating that the Angola Marburg (as Judith Anne has named it), is a variant of Marburg with a fast growth rate and a wide optimum. That postulate will explain the following observations we have made here:
1. Since the process for lethality is raw growth in Marburg / Ebola, a wide optimum would explain the near 100% lethality.
2. The fast growth and wide optimum explain the genetic stability (over the population) of this variant. Individual virus particles may change. The population remains constant. This will not burn out.
3. A wide genetic distribution would explain the many negative test results for Marburg (false negatives we believe). Genetic variation may elude the genetic spectrum of the test.
4. A wide genetic distribution may explain why other animals (dogs, pigs, chimps, gorillas) may contract the disease (as has been reported on this thread).
Note, this is only a postulate. I do not have the CDC genetic typing to test this. However, it does fit the data. Moreover, it provides a testable model. If the genetic type of this Marburg is ever released, we can compare the genetic variation with earlier Marburg and Ebola outbreaks.
My speculation is that it is; Judith Anne is right. However, my reasons are different than what has been mostly discussed here.
Let me first state what my reasons are not.
I do not believe that the issue is that this is an engineered weapon. There is little evidence to support this and it really doesn't matter one way or the other.
I do not believe that this is presently a huge epidemic, and that the reported numbers are grossly in error. They are clearly in error somewhat, but I do not believe that there are, say, 10,000 infected people at present. The issue is not the present size of the epidemic.
I do not believe that there is any form of official coverup, nor do I particularly care.
At issue is four observations that we have jointly made on this thread. Taken together, they have, IMHO, considerable implications. These are:
1. This is (effectively) 100% fatal. You get it, you die.
2. There is no vaccine, and I believe little prospect for one. The growth rate is too explosive. (See my previous post or c.f. Richard Preston's The Hot Zone)
3. The virus is not going to die out or burn out. (See my previous post).
4. The epidemiological growth has been slow, but inexorable, and it is manifestly exponential.
The inescapable logical conclusion of the first three points is that the only mechanism for stopping this from becoming a "slate-wiper" as Richard Preston predicted, is isolation (c.f. The Hot Zone). And point 4 indicates that isolation isn't happening.
Consider the logical implications compared to existing viruses. All other viruses have an "escape mechanism." To wit:
1. You could protect yourself from infection by refraining from dangerous behaviors. (AIDS comes to mind). Doesn't work here; this is generally infectious. The only risky behavior to avoid here is leaving your home.
2. You could likely survive the sickness, particularly with hospical support (e.g. the flu). Nope, doesn't work here.
3. You could get vaccinated. (e.g. smallpox). Nope, doesn't work here.
4. You could hope to wait it out until it burns itself out (what I call genetic dilution). This has worked for previous outbreaks and other diseases such as the flu. Nope, see my post above, it doesn't work here.
The only survival mechanism for the species is isolation. And, at present, isolation is not working. Now, there are two counter arguments: 1) it is presently small in numbers, and 2) as a result of its small size there are exogenous variables that might result in its containment. This latter point was made by EternalHope. He is correct. What he is referring to is that the underlying process is stochastic (read basically random). My modelling averages over those random variables and ignores the inherent fluctuations (the mathematical term is called taking the expectation value). However, the counter-counter argument is that the measured fluctuation level (see data above) is just under the growth rate. This thing is growing, and the growth rate is beginning to increase beyond the fluctuation level.
Moreover, the larger the epidemic gets, the less likely that containment is even possible. The probability is straight forward to calculate.
The first issue, the small size, is deceptive. It is small now, but if it remains uncontained, it will not remain small. Moreover, the growth is exponential.
There is a mathematics parable (yes, Dorothy, there are math parables), about a wise man who did some great favor for the King of a small country. Saved the King's daughter or something. The King offered the wise man any wish. The wise man asked the King to place 1 grain of wheat on a chessboard on the first square, 2 grains on the second, 4 grains on the third, and so on. I.e. an exponential growth. It started real small: one grain of wheat. The King was surprised at the very modest request, and asked whether the wise man would want something else. The wise man said no, this was what he wanted. So the King agreed and had the royal pursar calculate the total amount of wheat required to fill the chess board (64 squares). The amount was greater than all of the wheat in the kingdom, indeed, it was greater than all of the wheat in the world.
Moral of the story: Watch out for the exponential function. If the present, now recalculated growth rate of 93 days is correct (see curve above), then making all of the assumptions I made before, including ignoring the nonlinear saturation effects (i.e. you run out of people), then we reach 7 billion people in 4.3 years. If the earlier calculated rate of 40 days is correct (from the cumulative death data alone or based on the most recent data point), then we reach 7 billion in 2 years. To me, the exact determination of the exponential time constant does not make a whole lot of difference.
So, it is easy to ignore this at present. But it is growing exponentially. The only method for stopping it is containment. Containment is not working. We are at the 3rd or 4th chessboard square.