On how not to apply mathematical models

Over at Aetiology, there's a furious war going on in the comments between Tara and an AIDS-HIV denialist. One of the things that the denialist has referenced is an essay by Professor Rebecca Culshaw, a mathematician who has worked on mathematical models of AIDS and HIV transmission, and who has become an HIV denialist.

I'll leave the science debate over at Aetiology, where it belongs. But there's definitely a mathematical aspect to this. Professor Culshaw lends her authority as a mathematician to the HIV denialist folks. Does her math support what she's saying?

Alas, no.

Professor Culshaw is not a bad mathematician - quite the opposite. What I can read of her publications shows very solid mathematical work, done extremely well.

The problem is that when she tries to apply the mathematics to the science of epidemiology, she fails miserably, and the reason why is mathematical.

Let me first present an excerpt of her essay. (I suggest reading the thing in it's entirety; it's an interesting piece of writing, even if I think she's wrong.)

Over the past ten years, my attitude toward HIV and AIDS has undergone a dramatic shift. This shift was catalyzed by the work I did as a graduate student, analyzing mathematical models of HIV and the immune system. As a mathematician, I found virtually every model I studied to be unrealistic. The biological assumptions on which the models were based varied from author to author, and this made no sense to me. It was around this time, too, that I became increasingly perplexed by the stories I heard about long-term survivors. From my admittedly inexpert viewpoint, the major thing they all had in common – other than HIV – was that they lived extremely healthy lifestyles. Part of me was becoming suspicious that being HIV-positive didn’t necessarily mean you would ever get AIDS.

By a rather curious twist of fate, it was on my way to a conference to present the results of a model of HIV that I had proposed together with my advisor, that I came across an article by Dr. David Rasnick about AIDS and the corruption of modern science. As I sat on the airplane reading this story, in which he said "the more I examined HIV, the less it made sense that this largely inactive, barely detectable virus could cause such devastation," everything he wrote started making sense to me in a way that the currently accepted model did not. I didn’t have anywhere near all the information, but my instincts told me that what he said seemed to fit.

Over the past ten years, I nevertheless continued my research into mathematical models of HIV infection, all the while keeping an ear open for dissenting voices. By now, I have read hundreds of articles on HIV and AIDS, many from the dissident point of view but far, far more from that of the establishment, which unequivocally promotes the idea that HIV causes AIDS and that the case is closed. In that time, I even published four papers on HIV (from a modeling perspective). I justified my contributions to a theory I wasn’t convinced of by telling myself these were purely theoretical, mathematical constructs, never to be applied in the real world. I suppose, in some sense also, I wanted to keep an open mind.

Reading this, you would conclude that what she had studied in her research was mathematical models of HIV transmission between people. You'd be wrong. Her work is on the transmission of HIV between infected and non-infected cells in culture. For example, here's one abstract of a paper available on the web:

We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases.We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1)

And there lies the problem. Mathematical models are highly dependent on the details of their formulation and their parameters. Her work on modeling viral behavior is not modeling viral behavior of HIV in a human host; it's modeling viral behavior in a cell culture. That's a useful thing to understand: it helps us understand how HIV attacks and spreads between cells. But it doesn't really tell us anything directly about how HIV should spread between cells in a dramatically different environment. Viruses don't behave the same in culture and in a complete host. (For example, see this paper, describing how differently Hepatitis-A behaves in culture from in a host.)

She's trying to take a mathematical model created to specifically model one environment, and to apply that model to a different environment, without modifying the model for the new environment.

You can't do that. It doesn't work. A good mathematical model of HIV infection in a human host is not the same as a good mathematical model of HIV infection in a cell culture. You can't use the math that you derived for describing a cell culture, and apply it to the new environment without modifying the model to account for the differences between the environments. That's just bad math. Professor Culshaw appears to be a good enough mathematician that she should know better.