On the rate of clinical AIDS on diagnosis: The mathematical interpretation and goal for the successful control of HIV/AIDS.

The most widely used measurement of transmission dynamics in real time is the effective reproduction number $ R\left(t\right) $. However, in the context of human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS), $ R\left(t\right) $ has not been used frequently, possibly because of the slowly progressing nature of HIV infection that limits the knowledge of recent infection events. Gaining deeper insights into the practically used epidemiological metrics of HIV/AIDS is therefore vital. Notably, in many high-income countries, including Japan, the rate of clinical AIDS on diagnosis, $ Q\left(t\right) $, has been routinely measured by calculating the proportion of newly diagnosed AIDS cases out of all new HIV infections that are diagnosed at a given calendar time. However, there has been no clear indication of whether the control of HIV/AIDS is effective in relation to this metric in Japan. In this study, we formulated the rate of clinical AIDS on diagnosis using a mathematical model and offered interpretations of it using the hazard rate of diagnosis among previously undiagnosed HIV-infected individuals. We showed that by taking the inverse of the odds of $ Q\left(t\right) $ and multiplying it by the inverse of the mean incubation period, we obtained $ \alpha \left(t\right) $, which is the hazard rate of diagnosis among undiagnosed HIV-infected individuals. We also showed that $ \alpha \left(t\right) $ can be related to the goal of the diagnosed proportion $ {P}_{0} $ among all people living with HIV. In addition to the rate of clinical AIDS on diagnosis $ Q\left(t\right) $, $ \alpha \left(t\right) $ can be calculated using a simplistic equation and can potentially act as a practical epidemiological metric for monitoring during surveillance.