In this paper, we prove some random fixed point theorems for generalized random $\alpha-\psi-$contractive mappings in a Polish space and, as some applications, we show the existence of random solutions of second order random differential equation.

Introduction: Adverse events related to vaccines are reactions that can lead to serious complications, in the present that we live at the publication of this article with the need to generate high rates of vaccination have been presented in the case of vaccine agents against COVID-19 adverse events, with thrombotic events being the most important. Case Presentation: This is a 37-year-old female patient who presents 11 days after ChAdOx1-S vaccination against COVID-19 presents with intense headache associated with symptoms of intracranial hypertension, she was initially taken to simple cranial tomography showing presence of cerebral edema and indirect signs of cerebral venous thrombosis, it is proposed to carry out cerebral angiography which showed the presence of thrombosis of both transverse venous sinuses, no hypercoagulability states were found in the laboratories, the immunological tests carried out for antiphospholipid syndrome were negative, the thrombotic event was associated with the ChAdOx1-S vaccine, later the patient was taken to thrombectomy by Neuroradiology, achieving complete channeling of both transverse venous sinuses. Conclusions: Adverse events associated with vaccination have been described with multiple vaccine agents, this component against COVID-19 is also an event to consider, knowing the pathophysiology of these events is essential to be able to reduce these risks and select the least risky agent. Despite this risk, vaccination remains the only cost-effective measure to reduce the morbidity and mortality associated with COVID-19.

Background: Objectives were to determine the real-life effectiveness and safety of DT with dolutegravir (50 mg/QD) plus lamivudine (300 mg/QD) in multiple-tablet regimen (MTR) in naïve PLHIV followed up for 48 weeks and to evaluate the compliance and satisfaction of patients. Material and methods: Open, single-arm, multicenter, non-randomized clinical trial from May 2019 through September 2020 with 48-week follow-up. Results: The study included 88 PLHIV (91% male) with mean age of 35.9 years; 76.1% were MSM. Mean baseline CD4 was 516.4 cells/uL, with viral load (VL) of 104,828 cop/mL, and 11.4% were in AIDS stage. DT started within 7 days of first specialist consultation in all patients and the same day in 84.1%; 3.4% had baseline resistance mutations (K103N, V106I+E138A, and V108I); 12.5% were lost to follow-up. At week 48, 86.3% had VL< 50 cop/uL by intention-to-treat analysis and 98.7% by per-protocol (PP) analysis. Virological failure (VF) was recorded in 1.1%, with no resistance mutation. One blip was detected in 5.2%, without VF. Three reported anxiety, dizziness, and cephalgia, respectively, at week 4 and one insomnia at week 24; none reported adverse events at week 48. Mean weight was 4 kg higher at 48 weeks (p=0.0001) and abdominal circumference 3 cm larger at 24 weeks (p=0.022). No forgetfulness occurred in 98.7% of patients. Patient satisfaction was 90/100 at 4, 24, and 48 weeks. Conclusion: Real-world data demonstrate that dolutegravir plus lamivudine in MTR is effective, safe, and satisfactory, moderately increasing weight and abdominal circumference and administrable on a test-and-treat strategy.