As a consequence of the COVID-19 pandemic, the early diagnosis and constant monitoring of respiratory issues have emerged as crucial public health goals. Although the respiratory system is the primary target of the disease's acute phase, subsequent complications of SARS-CoV-2 infection might trigger enduring respiratory problems and symptoms, according to new research. These signs and symptoms, which collectively inflict considerable strain on healthcare systems and people's quality of life, include but are not limited to congestion, shortness of breath, tightness in the chest, and a decrease in lung function. Wearable technology offers a promising remedy to this persistent issue by offering continuous respiratory parameter monitoring, facilitating early control of and intervention in post-COVID-19 respiratory issues. In an effort to enhance patient outcomes and reduce expenses related to healthcare, this paper examines the possibility of using wearable technology to provide remote surveillance and early diagnosis of respiratory problems in individuals suffering from COVID-19. In this work, a fog computing-based and cost-effective smart health monitoring device for infectious disease applications is proposed. The proposed device consists of three different biosensor modules, namely a MAX90614 infrared temperature sensor, a MAX30100 pulse oximeter, and a microphone sensor. All these sensor modules are connected to a fog computing device, i.e., a Raspberry PI microcontroller. Also, the three different sensor modules are integrated with the Raspberry PI microcontroller. The wearer's physiological parameters, such as oxygen saturation (SPO2), heart rate, and cough sounds, are recorded by the computing device. Additionally, a Convolutional Neural Network (CNN)-based deep learning algorithm, trained with normal and COVID-19 cough sounds from the KAGGLE database, is coded inside the Raspberry PI microcontroller. This work appears to be of high clinical significance since the fog computing-based smart heath monitoring device developed herein is capable of identifying the presence of infectious disease with individual physiological parameters.