We use a new mathematical model to describe the dynamics of RNA viruses such as SARS-CoV-2 that incorporates two modes of transmission (virus-to-cell and cell-to-cell), two classes of infected cells, humoral immunity and antiviral treatment. The model is formulated by the following nonlinear system of ordinary differential equations (ODEs):
where the uninfected cells
are generated at rate
, die at rate
and become infected either by free virus particles at rate
or by direct contact with infected cells at rate
. The two modes of transmission are inhibited by non-lytic humoral immune response at rate
and
, respectively. The latently infected cells
die at rate
and become productively infected cells rate
. Also, the latently infected cells are assumed to be cured at rate
, resulting from the clearance of virus through the non-cytolytic process as for HCV infection in [20 (
link)] and HIV in [21 (
link),22 (
link)]. The cure of infected epithelial cells was also considered in a recent work of SARS-CoV-2 [23 (
link)]. The productively infected cells
die at rate
. Free viruses
are produced by infected cells at rate
, cleared at rate
and neutralized by antibodies at rate
. Antibodies develop in response to free virus at rate
and decay at rate
. Here, the parameter
represents the effectiveness of the antiviral treatment which blocks the production of viral particles. The flow diagram of the model is shown in
Figure 1.
Most viruses can spread via two modes: by virus-to-cell infection and through direct cell-cell contact [24 (
link),25 (
link),26 (
link)]. A recent study provided evidence that SARS-CoV-2 spreads through cell-cell contact in cultures, mediated by the spike glycoprotein [27 (
link)]. Furthermore, it is known that antibodies neutralize free virus particles and inhibit the infection of susceptible cells [28 (
link)]. They also contribute significantly to non-lytic antiviral activity [29 ]. For this reason, both modes of transmission with the lytic and non-lytic immune response are considered into the model.
On the other hand, it is very important to note that the SARS-CoV-2 model presented by system (
1) includes many mathematical models for viral infection existing in the literature. For instance, we get the model of Rong et al. [21 (
link)] when
,
and both treatment and humoral immunity are ignored. The global stability of the model [21 (
link)] was investigated in [30 (
link)]. In addition, the model of Baccam et al. [31 (
link)] is a special case of system (
1), it suffices to neglect immunity and take
,
,
and
. The last model presented in [31 (
link)] was recently used by Rodriguez and Dobrovolny [32 (
link)] to determine viral kinetics parameters for young and aged SARS-CoV-2 infected macaques. In the case where latently infected cells not revert back to susceptible and when antibodies do not reduce cell-to-cell transmission, we have
,
and system (
1) reduces to the following model:
Hattaf K., El Karimi M.I., Mohsen A.A., Hajhouji Z., El Younoussi M, & Yousfi N. (2023). Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2. Vaccines, 11(2), 201.