Why haven’t we found a cure for HIV yet?

It’s estimated that HIV infected humans as far back as the late 1800s or even earlier, but it took until 1983 for the virus itself to be identified. Even then, we’ve had over 30 years to create a vaccine or a cure, and have nothing to show for it. So why is HIV so dang hard to eradicate?
It comes down to the nature of the virus itself. Viruses—unlike bacteria and parasites—are not alive. What this means is viruses cannot grow or reproduce on their own, but instead need to attach themselves to a living cell to do these things. Viruses are often found within or on your own body cells—meaning your body often can’t find the infection in the first place.
Further, a virus can’t be killed with something like traditional antibiotics because there’s nothing alive to kill. If you destroy a part of a bacterium that allows it to make more bacteria, the bacterial infection can’t spread; with viruses, you would have to destroy actual human cells to do the same. Further, for an HIV vaccine to work, it would have to stop HIV in the few moments before it is allowed to enter human cells (and thus becomes unreachable)—a very tiny window of opportunity.
HIV is one evil virus
However, HIV is even more insidious than a regular virus. First, it’s actually a retrovirus, meaning it doesn’t just enter a cell to replicate; it enters itself into the cell’s very DNA in order to make more copies of the virus. HIV only infects the very cells that would normally kill it—immune system cells known as CD4 cells.
HIV hides inside these cells, waiting for them to get triggered by an infection. When this happens, HIV replicates inside the cells—and extremely quickly, meaning the virus mutates very quickly as well. After enough copies are made, the virus bursts out of the cells (which kills the cells) before finding another uninfected CD4 cell to enter.
The current medications we have can kill actively-working infected CD4 cells, but one single medication isn’t enough to kill all mutated forms of the virus—HIV can quickly change into a form the medication can’t kill. To combat this, HIV-positive patients usually take multiple antiretroviral drugs at the same time, in an effort to kill off all mutated versions. Eventually, though, the virus will find a form that works—meaning the drug combination fails after some time.
However, not every CD4 cell activates with an infection; often, many remain inactive. And sometimes, after HIV enters a CD4 cell, that CD4 stays inactive—meaning that as the medications only destroy the active infected cells, there are many that hide HIV undetected. After the medication clears, these cells are free to quickly activate and create more HIV in the body.
While this seems to paint a pretty dire image, there is hope. For example, one patient has been declared cured of HIV: a man from Berlin, who received treatment in 2007. The man was an HIV-positive leukemia patient who was treated for his cancer in a standard way: his blood cells were mostly killed off with chemotherapy, in an attempt to eradicate the cancer amongst them, and then he was given an infusion of blood-cell creating stem cells from a donor. The new blood cells created then attacked and killed off the remaining white blood cells in his body—a typical reaction between a graft and host.
By killing off these cells, the process also killed off hidden CD4 cells carrying HIV. However, there was no guarantee that this process would kill every single infected cell, so the doctor added in an extra measure: the blood cells donated to the patient were HIV-resistant, as they lacked the “door” by which HIV usually enters cells (CCR5).
He continues to show no signs of an HIV infection, causing doctors to label him as cured.
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