In the most interactive and open world than ever of this generation, both HIV and zika virus (ZIKV) infections are found to be a threat not only to adults but even more serious to newborns.In this thesis we have modelled HIV and zika virus (ZIKV) spread each at a time.That is, this thesis has at least two mathematical models. Recent researches have found that HIV and ZIKV spread are sharing one mode of transmission, that is sexual transmission.On the other hand the mode of transmission is quite different, in this case ZIKV spread has same mode of transmission with malaria, mosquito transmission.Accordingly, these modes of transmissions have been considered appropriately.　　We have modelled HIV spread for the sake of mathematical application to control HIV spread, one of a specific reasons is to support UNAIDS goal to end AIDS by 2030. Botswana from Africa has been demonstrated as a case study in this thesis.UNAIDS goal is linked to the basic reproduction number (R0).In the analysis of the model, no backward bifurcation has been found, so when R0＜1 the number of new HIV infections decreases.Hence, HIV spread is demonstrated as controllable under some conditions, Botswana has been simulated as an example.Records for PLHIV in Botswana have R0≤0.5051 which suggests her potential to achieve UNAIDS goal.By 2020, 92％ of PLHIV are expected to be under ART.Interestingly, in Botswana new HIV infections are mostly due to people who are not under ART.The value of R0 =0.6000 has been demonstrated as a threshold value below which UNAIDS goal can be reached.By 2030 not only that at least 90％ of PLHIV are expected to be under ART but also both new HIV infections and AIDS related deaths are expected to decrease above 90％ since their highest in 2010.These expectations are based on the simulation of the records of PLHIV for the past four years.And therefore, we are assuming that the Botswana government keeps up or makes it better on HIV control.　　Since February 2016 ZIKV spread and its associated disease became one of a major world concern kind of epidemic.Since then, researches are continuously being conducted.In this thesis, we have formulated basically two mathematical models of ZIKV spread.One model is just the refinement of the other.ZIKV records have been scarce, at least some records were obtained in Brazil by the time so the saved for some simulation.That have helped at least to get a grasp of the study.The ZIKV spread model basically has a disease free equilibrium with no backward bifurcation. This has been found to be a strong point to control the epidemic, that is when R0＜1 disease dies out from the community.Otherwise when R0＞1 the epidemic grows.In addition, unlike AIDS, individual infected with ZIKV may recover.Basing on some researches, recovered individual are resistant to reinfection.By ZIKV spread model we have simulated how this help to protect the spread of the epidemic and also explains why countries ever infected by ZIKV are not prone to zika virus disease.In chapter 4, ZIKV spread model has been refined in the sense of having a more reliable model to explain and control the epidemic.The epidemic is still controllable in this case, though no specific country has been simulated.Some random records were selected in order to simulate some of the phenomenon in the model.Yet, when R0＜1 the disease free equilibrium is stable and no backward bifurcation analysed.The epidemic is catastrophic when R0＞1 and the endemic equilibrium is simulated to be stable.