Oncolysis by SARS-CoV-2: modeling and analysis

Abbreviation

$N$: nutrient, it is produced from a source at a fixed rate, cells grow as a result of consuming nutrient; $M$: epithelial cells, the type of cells in the lungs that are infected by SARS-CoV-2; $C$ cancer cells: the cells characterized by uncontrolled growth that become infected by SARS-CoV-2; $V$: the free SARS-CoV-2 particles, the virus responsible for COVID-19, these particles infect epithelial and cancer cells; $T$: anti-cancer CTLs, the immune cells that specifically target and eliminate cancer cells; $A$: antibodies, they are used by the immune system to eliminate virus particles from the body

1.   Introduction

COVID-19 is a disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). While there has been a decline in reported cases and deaths globally, SARS-CoV-2 is still spreading in many countries ^[1]. SARS-CoV-2 passes into host cells by way of a transmembrane protein known as the angiotensin-converting enzyme 2 receptor ^[2]. It principally causes infection in alveolar epithelial type-II cells of the lungs ^[3,4]. However, other organs can be infected by SARS-CoV-2. The impact of COVID-19 on cancer patients has opened up a wide field of research. This group of patients is vulnerable to COVID-19 due to weakened immune system or ongoing anti-cancer treatments ^[5]. The question of whether SARS-CoV-2 induces remission in cancer patients or exacerbates the severity of the disease remains controversial ^[5,6].

The interconnection between cancer and viruses has become one of the most important topics in oncology and virology ^[7]. Some viruses, called oncolytic viruses (OVs), have the ability to infect cancer cells. These viruses can be found in nature or genetically modified to replicate in cancer cells without infecting normal cells ^[5,8]. OVs kill cancer cells after massive replication inside them and by inducing a specific antitumor immune response ^[5,8]. Examples of OVs include adenovirus, vaccinia virus, Coxsackievirus, and herpes simplex virus ^[8,9]. Talimogene laherparepvec is the only approved oncolytic virotherapy ^[6,10]. Talimogene laherparepvec is an engineered herpes virus used to treat advanced melanoma through immediate injection into the tumor ^[10].

The impact of COVID-19 on cancer patients is bidirectional. It has been indicated that SARS-CoV-2 infection can enhance cancer progression ^[5,7,8,11]. On the other hand, cancer remission after SARS-CoV-2 infection has been reported in many patients ^[6,12,13,14]. For example, Pasin et al. ^[15] reported the case of a patient with refractory natural killer (NK)/T-cell lymphoma who experienced a transient remission during SARS-CoV-2 infection. As angiotensin-converting enzyme 2 is expressed in NK cells, the authors in ^[15] proposed that SARS-CoV-2 could own some oncolytic properties. Challenor and Tucker ^[16] presented the case of a remission in a patient with classical Hodgkin lymphoma after SARS-CoV-2 infection. Another case was reported by Sollini et al. ^[17] involving a patient with follicular lymphoma who achieved full remission after SARS-CoV-2 infection. The authors in ^[16,17] supposed that the infection stimulated an antitumor immune response. Kandeel et al. ^[18] reported the remission of two cases with acute leukemia. Antwi-Amoabeng et al. ^[19] presented a case in which a patient with multiple myeloma had remission following SARS-CoV-2 infection. The patient received a single dose of chemotherapy. However, the authors mentioned that the remission in this case was parallel to the remission in patients who got four doses of chemotherapy. Ohadi et al. ^[20] reported a case of mycosis fungoides that went into remission after the coronavirus infection. Other cases of remission were reported in ^[6].

The above remission cases suggest that SARS-CoV-2 could have an oncolytic role in many types of cancer. It may infect and destroy cancer cells to expose the tumor-associated antigens. These antigens stimulate an immune response against cancer cells, leading to cancer remission ^[18]. Hence, there is an urgent need to understand the relationship between SARS-CoV-2 infection and cancer. This understanding may enable the engineering of SARS-CoV-2 for use as an efficient therapy against certain types of cancer. Mathematical modeling is a strong tool that is often employed to assist experiments and medical research ^[21,22]. Mathematical models have been used to understand the dynamics of many infectious diseases and test hypotheses that may be challenging to assess experimentally. The analysis of these models can offer predictions of outcomes and aid in identifying optimal treatment strategies.

Many mathematical models about SARS-CoV-2 ^[23,24,25], cancer ^[26], SARS-CoV-2/cancer ^[27], and oncolytic virotherapy ^{[28,29,30,31,32]} have been constructed and studied. However and to the best of our knowledge, no oncolytic SARS-CoV-2 models have been established yet. Such models are important to understand the effect of the infection of cancer cells by SARS-CoV-2 and the role of different immune responses during this coinfection. In this work, we propose an oncolytic SARS-CoV-2 virotherpy model. The construction of this model follows similar principles of those used in ^[33]. We conduct a comprehensive mathematical analysis of this model including assessments of boundedness, nonnegativity, and global stability of equilibrium points. In addition, we implement some numerical simulations.

This paper is structured as follows. Section 2 introduces the model under consideration. Section 3 demonstrates that all solutions are bounded and have zero or positive values. Furthermore, it computes the equilibrium solutions of the proposed model. Section 4 verifies the global properties of these solutions. Section 5 is dedicated to numerical simulations. The last section discusses the results and provides a glimpse of the future vision.

2.   Oncolytic SARS-CoV-2 model with immune responses

In formulating the model, we consider the following assumptions:

(i) The nutrient is produced from a source at a fixed rate, while it is depleted due to its consumption by epithelial cells and cancer cells. Additionally, depletion occurs as a result of natural death;

(ii) Epithelial cells proliferate as a result of nutrient consumption, and their numbers decrease due to either viral infection or natural death;

(iii) Cancer cells replicate as a result of nutrient utilization, and their numbers decline due to viral infection, attacks by cytotoxic T lymphocytes (CTLs), or natural cell death;

(iv) Free virus particles increase as a consequence of infecting epithelial cells and cancer cells, but they diminish due to the removal by antibodies or natural death;

(v) CTLs are stimulated by infected cancer cells, while antibodies are stimulated by free virus particles;

(vi) CTLs and antibodies undergo decay through natural processes;

(vii) The induction of CTLs by SARS-CoV-2 infection is implied in the stimulation rate of CTLs;

(viii) The model does not contain infected components for epithelial cells and cancer cells.

The proposed ordinary differential equation (ODE) model consists of six nonlinear equations and takes the form

where $N(t)$, $M(t)$, $C(t)$, $V(t)$, $T(t)$, and $A(t)$ typify the concentrations of nutrient, epithelial cells, cancer cells, SARS-CoV-2 particles, anti-cancer CTLs, and antibodies. The nutrient is released from its source at rate $\alpha$ and declines at rate $\omega N$. Epithelial cells expend nutrient at rate $\lambda_{1}NM$, reproduce at rate $\theta \lambda_{1}NM$, and get infected by SARS-CoV-2 at rate $\epsilon_{1}MV$. Cancer cells expend nutrient at rate $\lambda_{2}NC$, grow at rate $\theta \lambda_{2}NC$, and become infected at rate $\epsilon_{2}CV$. SARS-CoV-2 replicates as a result of infecting epithelial cells and cancer cells at rates $p\epsilon_{1}MV$ and $p\epsilon_{2}CV$, respectively. CTLs kill cancer cells at rate $\xi_{1}CT$ and reproduce at rate $s_{1}\xi_{1}CT$. Antibodies eliminate SARS-CoV-2 at rate $\xi_{2}VA$ and get stimulated at rate $s_{2}\xi_{2}VA$. Epithelial cells, cancer cells, SARS-CoV-2 particles, CTLs, and antibodies die at natural rates $\omega_{1}M$, $\omega_{2}C$, $\omega_{3}V$, $\omega_{4}T$, and $\omega_{5}A$, respectively. The parameters in model (2.1) are postulated to take positive values. Figure 1 provides a schematic diagram of the model. For simplicity, we consider the following

3.   Basic properties

The following theorem demonstrates the existence and uniqueness of the solutions of model (2.1).

Theorem 1. Assume that the initial values $\left(N_{0}, M_{0}, C_{0}, V_{0}, T_{0}, A_{0}\right)\in \mathbb{R}^{6}$ are given. There exists $t_{0} > 0$ and continuously differentiable functions $N, M, C, V, T, A$: $[0, t_{0})\rightarrow \mathbb{R}$ such that $\left(N, M, C, V, T, A\right)$ satisfies model (2.1) and

Proof. As the system of ODEs given in (2.1) is autonomous, it is enough to prove that the function $f$: $\mathbb{R}^{6}\rightarrow \mathbb{R}^{6}$ defined by

is locally Lipschitz in its $z$ argument. We observe that the Jacobian matrix

is linear in $z$ and consequently locally bounded for all $z\in \mathbb{R}^{6}$. Therefore, $f$ has a continuous and bounded derivative on any compact subset of $\mathbb{R}^{6}$, and so $f$ is locally Lipschitz in $z$. According to the classical Picard-Lindelöf theorem ^[34], there exists a unique solution $z(t)$ to the ODE

on the time interval $[0, t_{0}]$ for some $t_{0} > 0$ ^[35]. □

Next, we prove the nonnegativity and boundedness of the solutions of model (2.1).

Theorem 2. Let $\tau_{i} > 0$ ($i = 1, 2, 3, 4, 5$), then the set

is positively invariant set for system (2.1).

Proof. For system (2.1), we obtain

This shows that

for $t\geq0$ whenever

To prove the boundedness, we consider the function

By computing $\dfrac{d\chi (t)}{dt}$, we get

where

This implicates that

where

Consequently, we have $N(t)\leq \tau_{1}$, $M(t)\leq \tau_{2}$, $C(t)\leq \tau_{2}$, $V(t)\leq \tau_{3}$, $T(t)\leq \tau_{4}$, and $A(t)\leq \tau_{5}$, where $\tau_{2} = \theta \tau_{1}$, $\tau_{3} = \theta p \tau_{1}$, $\tau_{4} = \theta s_{1}\tau_{1}$, and $\tau_{5} = \theta p s_{2}\tau_{1}$. Thus, the set $\Omega$ is positively invariant ^[36]. □

Theorem 3. Model (2.1) has thirteen equilibrium points as follows:

(1) The trivial equilibrium $E_{0}$ always exists;

(2) The uninfected-epithelial equilibrium $E_{1}$ exists if $R_{0} > 1$;

(3) The uninfected-cancer equilibrium $E_{2}$ exists if $R_{1} > 1$;

(4) The infected-epithelial equilibrium $E_{3}$ exists if

(5) The infected-cancer equilibrium $E_{4}$ exists if

(6) The uninfected-cancer equilibrium with CTLs $E_{5}$ exists if

(7) The infected epithelial-cancer equilibrium without immunity $E_{6}$ exists if

(8) The uninfected epithelial-cancer equilibrium with CTLs $E_{7}$ exists if

(9) The infected-epithelial equilibrium with antibodies $E_{8}$ exists if

(10) The infected-cancer equilibrium with antibodies $E_{9}$ exists if

(11) The infected-cancer equilibrium with CTLs and antibodies $E_{10}$ exists if

(12) The infected epithelial-cancer equilibrium with CTLs $E_{11}$ exists if

and

(13) The infected epithelial-cancer equilibrium with CTLs and antibodies $E_{12}$ exists if

and

Proof. To get the equilibria of system (2.1), we solve the following system:

Then, we obtain

(1) The trivial equilibrium

This point always exists. This point has no biological meaning as all components vanish except for the nutrient.

(2) The uninfected-epithelial equilibrium $E_{1} = (N_{1}, M_{1}, 0, 0, 0, 0)$. The components $N_{1}$ and $M_{1}$ are defined as:

where

As $N_{1} > 0$, the equilibrium $E_{1}$ exists if $R_{0} > 1$. This equilibrium represents a healthy individual without cancer or SARS-CoV-2 infection.

(3) The uninfected-cancer equilibrium $E_{2} = (N_{2}, 0, C_{2}, 0, 0, 0)$, where

where

As $N_{2} > 0$, the point $E_{2}$ exists if $R_{1} > 1$. This equilibrium represents the case of a person who has cancer, but without SARS-CoV-2 infection.

(4) The infected-epithelial equilibrium $E_{3} = (N_{3}, M_{3}, 0, V_{3}, 0, 0)$, where

and

Clearly, $N_{3} > 0$, $M_{3} > 0$, and $V_{3} > 0$ if

Hence, $E_{3}$ exists if

The person here suffers from SARS-CoV-2 infection, but he is cancer-free.

(5) The infected-cancer equilibrium $E_{4} = (N_{4}, 0, C_{4}, V_{4}, 0, 0)$, where

and

Notably, $N_{4} > 0$, $C_{4} > 0$, and $V_{4} > 0$ if

Thus, $E_{4}$ exists if

In this scenario, the cancer patient is experiencing a SARS-CoV-2 infection with the disappearance of healthy epithelial cells.

(6) The uninfected-cancer equilibrium with CTLs $E_{5} = (N_{5}, 0, C_{5}, 0, T_{5}, 0)$, where

and

Clearly, $N_{5} > 0$, $C_{5} > 0$, and $T_{5} > 0$ if

Hence, $E_{5}$ exists if

CTLs are activated to eliminate cancer cells in a patient who has experienced the disappearance of healthy epithelial cells.

(7) The infected epithelial-cancer equilibrium without immunity $E_{6} = (N_{6}, M_{6}, C_{6}, V_{6}, 0, 0)$, where

We note that the components are positive if

Thus, $E_{6}$ is defined when the above conditions are met. In this scenario, the cancer patient has SARS-CoV-2 infection with inactive immune responses.

(8) The uninfected epithelial-cancer equilibrium with CTLs $E_{7} = (N_{7}, M_{7}, C_{7}, 0, T_{7}, 0)$, where

Accordingly, $N_{7} > 0$, $C_{7} > 0$, $M_{7} > 0$ if

and $T_{7} > 0$ if

Therefore, $E_{7}$ exists when

Here, the cancer patient has active CTL immunity that works on killing cancer cells.

(9) The infected-epithelial equilibrium with antibodies $E_{8} = (N_{8}, M_{8}, 0, V_{8}, 0, A_{8})$, where

Thus, the components are positive and $E_{8}$ exists if

Notably, the first condition is naturally satisfied when the second condition is met.

The SARS-CoV-2 patient has active antibody immunity against the virus.

(10) The infected-cancer equilibrium with antibodies $E_{9} = (N_{9}, 0, C_{9}, V_{9}, 0, A_{9})$, where

We note that $E_{9}$ is defined when

where the other condition

is naturally met when the previous condition is satisfied.

The infected cancer patient who suffers from the disappearance of epithelial cells has active antibody immunity against the virus.

(11) The infected-cancer equilibrium with CTLs and antibodies $E_{10} = (N_{10}, 0, C_{10}, V_{10}, T_{10}, A_{10})$, where

Thus, $E_{10}$ is biologically accepted when

This point represents the case of a cancer patient who is infected, with active immune responses, but experiencing the disappearance of epithelial cells.

(12) The infected epithelial-cancer equilibrium with CTLs $E_{11} = (N_{11}, M_{11}, C_{11}, V_{11}, T_{11}, 0)$, where

It is easy to observe that $N_{11}$, $V_{11}$, and $T_{11}$ are positive and $E_{11}$ exists when

and

At this point, the infected cancer patient has active CTLs and inactive antibody immune response.

(13) The infected epithelial-cancer equilibrium with CTLs and antibodies $E_{12} = (N_{12}, M_{12}, C_{12}, V_{12}, T_{12}, A_{12})$, where

We observe that the components are positive and $E_{12}$ is defined if

and

This point imitates the case of cancer patient with SARS-CoV-2 infection and active immune responses. □

In the next sections, we will focus our analysis on the equilibria $E_{0}$, $E_{1}$, $E_{3}$, $E_{6}$, $E_{7}$, $E_{8}$, $E_{11}$, and $E_{12}$ as we are interested in the points where the epithelial cells component (M) does not vanish.

4.   Global properties

The following theorems are aimed to establish the global stability of equilibria through nominating Lyapunov functions. Let $Y_{i}^{^{\prime}}$ be the largest invariant subset of

where $i = 0, 1, 3, 6, 7, 8, 11, 12$.

Theorem 4. The equilibrium $E_{0}$ is globally asymptotically stable (GS) when $R_{0}\leq 1$ and $R_{1}\leq 1$.

Proof. We consider

Then, we get

We see that

if $R_{0}\leq 1$ and $R_{1}\leq 1$. Furthermore,

when $N = N_{0}$ and $M = C = V = A = 0.$ This gives

From the third equation of model (2.1), we obtain $T = 0$. Hence, $Y_{0}^{^{\prime} } = \{E_{0}\}$ and by LaSalle's invariance principle (LP) ^[37], $E_{0}$ is GS if $R_{0}\leq 1$ and $R_{1}\leq 1$. □

Theorem 5. Let $R_{0} > 1$. Then, the equilibrium $E_{1}$ is GS if

and

Proof. We opt

Then, we get

At equilibrium, $E_{1}$ solves the system

Applying the above equations to collect (4.1) gives

Hence,

if

In addition,

if $N = N_{1}$ and $C = V = T = A = 0$. Consequently,

and the first equation of (2.1) gives $M = M_{1}$. Therefore, $Y_{1}^{^{\prime} } = \{E_{1}\}$ and $E_{1}$ is GS when

according to LP ^[37]. □

Theorem 6. Let

Then, the equilibrium $E_{3}$ is GS if

and

Proof. We construct

Then, we get

The equilibrium conditions at $E_{3}$ are

By using (6), Eq (4.2) becomes

We observe that

if

and

In addition, $\dfrac{d\Sigma_{3}}{dt} = 0$ when $N = N_{3}$, $M = M_{3}$, $V = V_{3}$ and $C = T = A = 0$. Thus, $Y_{3}^{^{\prime}} = \{E_{3}\}$ and LP ^[37] implies that $E_{3}$ is GS when

with the above conditions. □

Theorem 7. Let the existence conditions of $E_{6}$ be satisfied. Then, the equilibrium $E_{6}$ is GS if

and

Proof. See Appendix A. □

Theorem 8. Let

Then, the equilibrium $E_{7}$ is GS if

Proof. See Appendix B. □

Theorem 9. Let

Then, $E_{8}$ is GS if

Proof. See Appendix C. □

Theorem 10. Let

and

Then, $E_{11}$ is GS if

Proof. See Appendix D. □

Theorem 11. Let

and

Then, the equilibrium $E_{12}$ is GS.

Proof. We nominate

Then, we get

By using the conditions of equilibrium state at $E_{12}$

Equation (4.4) is transformed into

We note that $\dfrac{d\Sigma_{12}}{dt}\leq 0$ and $\dfrac{d\Sigma_{12}}{dt} = 0$ at $E_{12}$. Based on LP ^[37], $E_{12}$ is GS when the existence conditions are met. □

5.   Numerical simulations

The ode45 solver of Matlab is utilized to effectuate the numerical simulations. ode45 is the default solver for ODEs in Matlab. It utilizes an explicit Runge-Kutta formula and generally performs well with a wide range of ODE problems. Nevertheless, when dealing with stiff problems or situations demanding high accuracy, alternative solvers like ode15s, ode23s, and ode23t may prove more efficient. We consider three different groups of initial conditions:

(1) $\left(N(0), M(0), C(0), V(0), T(0), A(0) \right) = \left(0.0001, 0.01, 0.03, 0.01, 0.001, 0.001 \right)$;

(2) $\left(N(0), M(0), C(0), V(0), T(0), A(0) \right) = \left(0.1, 0.03, 0.06, 0.05, 0.002, 0.003 \right)$;

(3) $\left(N(0), M(0), C(0), V(0), T(0), A(0) \right) = \left(0.6, 0.06, 0.1, 0.06, 0.03, 0.01 \right)$.

The chosen sets of initial conditions are arbitrary, as the global stability is guaranteed for any initial values. To affirm the global stability of $E_{0}$, $E_{1}$, $E_{3}$, $E_{6}$, $E_{7}$, $E_{8}$, $E_{11}$, and $E_{12}$, we partition the numerical simulations into eight classes. We change the values of $\lambda_{1}$, $\lambda_{2}$, $s_{1}$, $s_{2}$, $\epsilon_{1}$, $\epsilon_{2}$, $\xi_{2}$, $\omega_{2}$, $\omega_{3}$, and $\omega_{5}$ to obtain the global stability of the equilibrium in each case. The other values are fixed and given in Table 1.

Thus, we have

(i) We opt $\lambda_{1} = 0.03$, $\lambda_{2} = 0.03$, $s_{1} = 0.1$, $s_{2} = 0.2$, $\epsilon_{1} = 0.55$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 0.17$, $\omega_{3} = 0.6$, and $\omega_{5} = 0.05$. This yields $R_{0} = 0.8 < 1$ and $R_{1} = 0.1263 < 1$. Thus, $E_{0} = \left(1, 0, 0, 0, 0, 0\right)$ is GS as indicated in Theorem 4 (see Figure 2). This point has no significant biological interpretation as all populations, except the nutrient's component, tend to zero.

(ii) We choose $\lambda_{1} = 0.05$, $\lambda_{2} = 0.03$, $s_{1} = 0.1$, $s_{2} = 0.2$, $\epsilon_{1} = 1\times 10^{-5}$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 0.17$, $\omega_{3} = 0.9$, and $\omega_{5} = 0.05$. The corresponding thresholds are

Thus, $E_{1} = (0.75, 0.1333, 0, 0, 0, 0)$ is GS which matches with Theorem 5 (see Figure 3). This simulates the case of an individual who neither has SARS-CoV-2 infection nor cancer.

(iii) We nominate $\lambda_{1} = 0.07$, $\lambda_{2} = 0.03$, $s_{1} = 0.1$, $s_{2} = 0.2$, $\epsilon_{1} = 0.5$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 0.17$, $\omega_{3} = 1\times 10^{-4}$, and $\omega_{5} = 0.2$. This gives

and

This causes $E_{3} = (0.6304, 0.1675, 0, 0.01061, 0, 0)$ to be GS as verified in Theorem 6 (see Figure 4). This imitates the case of a patient with SARS-CoV-2 infection but without cancer.

(iv) We pick out $\lambda_{1} = 0.06$, $\lambda_{2} = 0.05$, $s_{1} = 0.1$, $s_{2} = 0.2$, $\epsilon_{1} = 1\times 10^{-5}$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 0.001$, $\omega_{3} = 0.001$, and $\omega_{5} = 0.05$. This corresponds to

and

In parallel with Theorem 7, $E_{6} = (0.625, 0.06742, 0.1591, 0.0073, 0, 0)$ is GS (see Figure 5). This simulates the case of a cancer patient who has SARS-CoV-2 infection with inactive immune responses.

(v) We opt $\lambda_{1} = 0.06$, $\lambda_{2} = 0.05$, $s_{1} = 1.2$, $s_{2} = 0.2$, $\epsilon_{1} = 1\times 10^{-5}$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 1\times 10^{-4}$, $\omega_{3} = 0.9$, and $\omega_{5} = 0.05$. The resultant thresholds are

and

In agreement with Theorem 8, $E_{7} = (0.625, 0.0333, 0.2, 0, 0.0098, 0)$ is GS (see Figure 6). In this scenario, the cancer patient does not have SARS-CoV-2 infection, and the CTL immunity against cancer cells is active.

(vi) We select $\lambda_{1} = 0.07$, $\lambda_{2} = 0.03$, $s_{1} = 0.4$, $s_{2} = 1.6$, $\epsilon_{1} = 0.5$, $\epsilon_{2} = 0.55$, $\xi_{2} = 1.6$, $\omega_{2} = 0.17$, $\omega_{3} = 1\times 10^{-6}$, and $\omega_{5} = 1\times 10^{-8}$. This gives

and

As indicated in Theorem 9, $E_{8} = (0.6055, 0.1862, 0, 0.0078, 0, 0.0015)$ is GS (see Figure 7). Here, the patient is solely affected by SARS-CoV-2 infection with an active immune response against the virus.

(vii) We consider $\lambda_{1} = 0.06$, $\lambda_{2} = 0.06$, $s_{1} = 1.6$, $s_{2} = 0.2$, $\epsilon_{1} = 1.4\times 10^{-1}$, $\epsilon_{2} = 0.55$, $\xi_{2} = 4.88\times 10^{-8}$, $\omega_{2} = 0.001$, $\omega_{3} = 0.001$, and $\omega_{5} = 0.05$. This gives

and

As proved in Theorem 10, $E_{11} = (0.6422, 0.0357, 0.15, 0.0059, 0.01316, 0)$ is GS (Figure 8). The CTL immunity against cancer cells is activated in the cancer patient infected with SARS-CoV-2.

(viii) We pick up $\lambda_{1} = 0.06$, $\lambda_{2} = 0.06$, $s_{1} = 1.6$, $s_{2} = 1.9$, $\epsilon_{1} = 1.4\times 10^{-1}$, $\epsilon_{2} = 1.2$, $\xi_{2} = 1.9$, $\omega_{2} = 0.001$, $\omega_{3} = 0.001$, and $\omega_{5} = 0.000001$. The corresponding thresholds are

and

As indicated in Theorem 11, $E_{12} = (0.641, 0.03657, 0.1499, 0.006, 0.006, 0.0123)$ is GS (see Figure 9). The CTL and antibody immune responses are activated in the cancer patient infected with SARS-CoV-2.

To observe the impact of $\epsilon_{2}$ (the infection rate of cancer cells by SARS-CoV-2) on the concentration of cancer cells before stimulating any immune responses, we increase the value of $\epsilon_{2}$ in case (iv). When we set $\epsilon_{2} = 0.7$, we get $C_{6} = 0.125$. Additionally, if we raise $\epsilon_{2}$ to 0.9, we find $C_{6} = 0.097$. Figure 10 shows the impact of increasing $\epsilon_{2}$ on the decrease in the concentration of cancer cells for other values of $\epsilon_{2}$. Thus, the infection of cancer cells by SARS-CoV-2 can lead to a reduction in cancer cells concentration, consequently resulting in a remission or an improvement in the patient's situation. Similarly, when we increase the value of $\xi_{1}$ (the killing rate of cancer cells by CTLs) in case (viii), the concentration of cancer cells decreases to lower values (See Figure 11). In fact, these results align with many studies that suggest the ability of SARS-CoV-2 to infect cancer cells and induce immune responses, leading to cancer remission ^{[15,16,17,18,19]}.

6.   Conclusions and discussion

Cancer remission after SARS-CoV-2 infection has been observed in many patients. This remission has been transient or complete, and it has been recorded with various types of cancer such as NK/T-cell lymphoma ^[15], Hodgkin lymphoma ^[16], follicular lymphoma ^[17], acute leukemia ^[18], and other types of cancer ^[6]. This has raised an urgent need to understand the relationship between cancer and SARS-CoV-2. This paper proposes and analyzes an oncolytic SARS-CoV-2 model. The model has 13 equilibrium points, and we focused our analysis on the points with the most important biological significance as follows:

(1) The trivial equilibrium $E_{0}$ which is GS if $R_{0}\leq 1$ and $R_{1}\leq 1$. At this point, all populations disappear except for the nutrient.

(2) The uninfected-epithelial equilibrium $E_{1}$ is GS if $R_{0} > 1$, $R_{0}\leq 1+\dfrac{\lambda_{1}a_{3}}{\omega p \epsilon_{1}}$, and $\dfrac{a_{1}\lambda_{2}}{a_{2}\lambda_{1}}\leq 1$. Here, the patient is free from both SARS-CoV-2 infection and cancer.

(3) The infected-epithelial equilibrium $E_{3}$ is GS if

and

The patient here has only SARS-CoV-2 infection.

(4) The infected epithelial-cancer equilibrium without immunity $E_{6}$ is GS when

and

Here, the cancer patient has SARS-CoV-2 infection with inactive immunity.

(5) The uninfected epithelial-cancer equilibrium with CTLs $E_{7}$ is GS when

and

Here, the cancer patient with active CTL immunity does not have SARS-CoV-2 infection.

(6) The infected-epithelial equilibrium with antibodies $E_{8}$ is GS if

and

The patient is cancer-free and suffers only from SARS-CoV-2 infection with active immunity against the virus.

(7) The infected epithelial-cancer equilibrium with CTLs $E_{11}$ is GS when

and

In this case, the cancer patient has SARS-CoV-2 with active CTLs against the cancer cells.

(8) The infected epithelial-cancer equilibrium with CTLs and antibodies $E_{12}$ is GS if

and

The cancer patient has SARS-CoV-2 infection with active immune responses against the cancer cells and virus particles.

We found complete agreement between the theoretical contributions and numerical simulations. The global stability conditions of equilibrium points determine various infection scenarios, such as patients having only SARS-CoV-2, cancer, both SARS-CoV-2 and cancer, or no infections. These conditions are dependent on the parameters of model (2.1), emphasizing the importance of carefully selecting their values. Furthermore, our findings indicate that the infection rate of cancer cells by SARS-CoV-2 ($\epsilon_{2}$) and the killing rate of these cells by CTLs ($\xi_{1}$) contribute to the reduction in the concentration of cancer cells. Based on these results, SARS-CoV-2 has the potential to lead to cancer remission or improve health conditions by either infecting cancer cells or inducing an anti-cancer immune response. This outcome aligns with recent studies suggesting an oncolytic role of SARS-CoV-2 ^{[15,16,17,18,19]}. In comparison to existing works, our model is the first to propose and analyze the oncolytic effect of SARS-CoV-2 in cancer patients. As such, these results warrant further investigation and comparison with the outcomes of experimental studies. Then, the model can be utilized in studies aiming to employ SARS-CoV-2 as oncolytic virotherapy to target cancer cells. However, a main limitation of this work is the absence of real data to estimate the values of the parameters in model (2.1), given the limited availability of such data in this direction. We utilized values from the literature and made assumptions for some parameters. Consequently, model (1) can be developed by:

(i) Estimating parameter values through fitting with real data once sufficient information becomes available;

(ii) Testing the model results against real data;

(iii) Studying the effect of immune responses on the oncolytic role of SARS-CoV-2 and when they can be supportive;

(iv) Including the direct induction of CTLs by SARS-CoV-2;

(v) Adding more components to the model, such as infected cancer cells and infected epithelial cells, for a deeper understanding of the model's dynamics;

(vi) Considering time delays that occur during different biological processes;

(vii) Accounting for parameters and model uncertainties by performing sensitivity analysis and other methods once experimental or real data becomes available.

These enhancements would contribute to a better understanding of the model and facilitate improved predictions.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors acknowledge that this project was funded by the Deanship of Scientific Research (DSR), University of Business and Technology, Jeddah 21361, Saudi Arabia. The authors, therefore, gratefully acknowledge the DSR technical and financial support.

Conflict of interest

The authors declare that there are no conflicts of interest.

Appendix

Appendix A

Proof of Theorem 7. We nominate

By evaluating $\dfrac{d\Sigma_{6}}{dt}$, we get

At equilibrium, $E_{6}$ fulfills the equations:

Thus, Eq (A.1) can be collected as

Thus,

if

and

Also, it is easy to observe that $\dfrac{d\Sigma_{6}}{dt} = 0$ when $\left(N, M, C, V, T, A \right) = \left(N_{6}, M_{6}, C_{6}, V_{6}, 0, 0 \right)$. Hence, $Y_{6}^{^{\prime}} = \{E_{6}\}$ and $E_{6}$ is GS when the existence conditions and the global stability conditions are met based on LP ^[37]. □

Appendix B

Proof of Theorem 8. We select

Then, we obtain

By applying the equilibrium conditions at $E_{7}$

Equation (B.1) can be collected as

Thus,

if

Also, $\dfrac{d\Sigma_{7}}{dt} = 0$ when $\left(N, M, C, V, T, A \right) = \left(N_{7}, M_{7}, C_{7}, 0, T_{7}, 0 \right)$. Hence, $Y_{7}^{^{\prime}} = \{E_{7}\}$ and by LP ^[37], $E_{7}$ is GS if

and

□

Appendix C

Proof of Theorem 9. We nominate

Then, we have

At equilibrium, $E_{8}$ fulfills the following:

Thus, Eq (C.1) can be collected as

Hence,

if

when $\left(N, M, C, V, T, A \right) = \left(N_{8}, M_{8}, 0, V_{8}, 0, A_{8} \right)$. Therefore, $Y_{8}^{^{\prime}} = \{E_{8}\}$ and based on LP ^[37], $E_{8}$ is GS when it exists and

Appendix D

Proof of Theorem 10. We select

Then, we have

At equilibrium, $E_{11}$ fulfills the system

Thus, Eq (D.1) can be collected as

Thus,

if

Also,

when $\left(N, M, C, V, T, A \right) = \left(N_{11}, M_{11}, C_{11}, V_{11}, T_{11}, 0 \right)$. Therefore, $Y_{11}^{^{\prime}} = \{E_{11}\}$ and by LP ^[37], $E_{11}$ is GS when the existence and stability conditions are met. □