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Since the immune function of AIDS patients infected with SARS-CoV-2 is impaired, it is easier for SARS-CoV-2 to reproduce, replicate, and even mutate in the host. Therefore, the co-infection of SARS-CoV-2 and HIV in vivo deserves our attention. In this paper, a series of co-infection dynamic models of HIV, wild-type, and variant SARS-CoV-2 are developed and studied for four co-infected patients in South Africa. Based on the clinical data, such as the number of CD4+ T cells, HIV viral loads, and SARS-CoV-2 Ct value in four co-infected patients, we estimate the unknown parameters in the model by the affine invariant ensemble Markov chain Monte Carlo (GWMCMC) algorithm and select the best model for virus transmission in South African co-infected patients via the Akaike Information Criterion (AIC). The structural identifiability of the best model is investigated, and the model is qualitatively analyzed. Finally, we run sensitivity tests on the model parameters. The results show that SARS-CoV-2 mutates in vivo and quickly becomes the dominant strain for all four South African co-infected patients. In addition, it is shown that drugs used to treat HIV patients have no significant effect on the inhibition of the SARS-CoV-2 variant. Based on the theoretical analysis of the model, we obtain the basic reproduction number of the model and the stability of the equilibrium state, which qualitatively investigates the nature of co-infection dynamics. Moreover, we find that the systemic inflammation triggered by COVID-19 can cause the latent HIV reservoir's reactivation and transiently increase viral loads. In terms of already-dead patient 1, if the drug that inhibits the replication of SARS-CoV-2 is used, then it can reduce the virus replication rate by 99%. Only in this way can the basic reproduction number of the SARS-CoV-2 infection be reduced to less than 1. That is, SARS-CoV-2 can be completely suppressed. This means that only by reducing the risk of COVID-19 infection in HIV patients as much as possible can we avoid the aggravation of the disease for such patients. Therefore, it is essential to increase vaccination coverage for COVID-19 in countries with a high prevalence of HIV. • The models take into account the coexistence of HIV and SARS-CoV-2 in vivo. • Validation of the model with the data on simultaneous infection with HIV and SARS-CoV-2. • Using advanced algorithms to estimate parameters. • Numerical simulation shows that the systemic inflammation triggered by SARS-CoV-2 reactivates the latent HIV reservoir. • Model selection result shows that HIV drugs may not inhibit the assignment of SARS-CoV-2. [ABSTRACT FROM AUTHOR]

• The epicenter of COVID-19 keeps shifting to the south. • Coexisting HIV & COVID-19 is a worry for southern hemisphere countries. • Stability analysis is provided for a HIV-COVID-19 simple model. • Existence of backward and forward bifurcations is shown. • Predicted prevalence from a generalized COVID-19 model is given. Were southern hemisphere countries right to undertake national lockdown during their summer time? Were they right to blindly follow the self-isolation wave that hit European countries in full winter? As a southern hemisphere country like South Africa stands now as the most COVID-19 and HIV affected country in Africa, we use in this paper, recent COVID-19 data to provide a statistical and comparative analysis that may alert southern hemisphere countries entering the winter season. After that, we use a generalized simple mathematical model of HIV-COVID-19 together with graphs, curves and tables to compare the pandemic situation in countries that were once the epicenter of the disease, such as China, Italy, Spain, United Kingdom (UK) and United States of America (USA). We perform stability and bifurcation analysis and show that the model contains a forward and a backward bifurcation under certain conditions. We also study different scenarios of stability/unstability equilibria for the model. The fractional (generalized) COVID-19 model is solved numerically and a predicted prevalence for the COVID-19 is provided. Recall that Brazil and South Africa share number of similar social features like Favellas (Brazil) and Townships (South Africa) with issues like promiscuity, poverty, and where social distanciation is almost impossible to observe. We can now ask the following question: Knowing its HIV situation, is South Africa the next epicenter in weeks to come when winter conditions, proven to be favorable to the spread of the new coronavirus are comfily installed? [ABSTRACT FROM AUTHOR]