  Facing a global challenge with over 6.9 million fatalities, severe acute
respiratory syndrome coronavirus 2 (SARS-CoV-2), the causative agent of
CoViD-19, demands novel and comprehensive approaches to understand its complex
dynamics. This paper introduces a non-integer order model, capturing the
intricate interplay between SARS-CoV-2 and the host's cytotoxic T lymphocytes
(CTLs) response. Our work reveals a unique parameter space, in which an endemic
state of SARS-CoV-2 and a CTL response-free equilibrium can coexist -- a
crucial finding in our quest to decipher this pervasive virus. We further
explore the basic reproduction number, assessing how different model parameters
can potentially inhibit or fuel the infection's progression. Through extensive
numerical simulations, we scrutinize the impact of varying the order of the
fractional derivative and employing diverse CTL proliferation functions. This
study significantly enriches our understanding of CoViD-19 immunopathology,
offering invaluable insights that could guide future research and therapeutic
strategies.
