A Global Analysis of Delayed SARS-CoV-2/Cancer Model with Immune Response

: Coronavirus disease 2019 (COVID-19) is a respiratory disease caused by SARS-CoV-2. It appeared in China in late 2019 and rapidly spread to most countries of the world. Cancer patients infected with SARS-CoV-2 are at higher risk of developing severe infection and death. This risk increases further in the presence of lymphopenia affecting the lymphocytes count. Here, we develop a delayed within-host SARS-CoV-2/cancer model. The model describes the occurrence of SARS-CoV-2 infection in cancer patients and its effect on the functionality of immune responses. The model considers the time delays that affect the growth rates of healthy epithelial cells and cancer cells. We provide a detailed analysis of the model by proving the nonnegativity and boundedness of the solutions, ﬁnding steady states, and showing the global stability of the different steady states. We perform numerical simulations to highlight some important observations. The results indicate that increasing the time delay in the growth rate of cancer cells reduced the size of tumors and decreased the likelihood of deterioration in the condition of SARS-CoV-2/cancer patients. On the other hand, lymphopenia increased the concentrations of SARS-CoV-2 particles and cancer cells, which worsened the condition of the patient.


Introduction
Coronavirus disease 2019 (COVID-19) is a respiratory disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It is one of the worst epidemics that we have witnessed in the modern era. It has changed our social, economic, and health lives since its first appearance in China in late 2019. It is highly contagious and has affected millions of people around the world [1,2]. According to the World Health Organization (WHO) report of 7 February 2021 [3], the total number of confirmed cases reached over 105 million, and the total number of deaths reached over 2 million. The highest numbers of cases were reported in the United States of America, Brazil, France, the United Kingdom of Great Britain and Northern Ireland, and the Russian Federation [3].
There are more than 60 COVID-19 vaccine candidates in clinical development and over 70 in pre-clinical development [4]. Two COVID-19 vaccines were authorized by the U.S. Food and Drug Administration (FDA) for emergency use: the Pfizer-BioNTech COVID-19 vaccine and Moderna COVID-19 vaccine [5]. Pfizer was authorized on 11 December 2020 for use in individuals 16 years of age and older [5]. Moderna was authorized on 18 December 2020 for use in individuals 18 years of age and older [5]. In addition to vaccination, it is essential to understand the biology of COVID-19 to develop effective treatments for patients who have already become infected [6]. virus-specific antibodies; (ii) investigating the effect of time delays on the growth rates of healthy cells and cancer cells; and (iii) examining the effect of lymphopenia on the activation rates of CTLs and antibodies.
The paper is organized as follows. Section 2 presents the model under consideration and defines the meanings of the different parameters and rates. Section 3 establishes the well-posedness of the model. In addition, it lists all possible steady states and the associated existence conditions. Section 4 proves the global stability of the steady states computed in Section 3. Section 5 displays some numerical simulations and outlines some important observations. The results and future works are presented in Section 6.

The Proposed Model
The model developed in this work was inspired by the oncolytic virotherapy models investigated in [23,40]. These models depict the cancer cell killing effect of the oncolytic M1 virus when the virus infects cancer cells. Here, we reformulate the models to measure the effect of SARS-CoV-2 infection in cancer patients, where SARS-CoV-2 infects healthy epithelial cells. Hence, our model takes the form: where A(t), N(t), C(t), V(t), W(t), and Z(t) represent the concentrations of nutrient, healthy epithelial cells, cancer cells, SARS-CoV-2 particles, cancer-specific CTLs, and virusspecific antibodies at time t, respectively. The nutrient is produced at a constant rate ϑ and decays at rate κA. The healthy cells consume nutrients at rate η 1 AN, grow at rate σ 1 η 1 e −b 1 τ 1 A(t − τ 1 )N(t − τ 1 ), and die at a natural death rate κ 1 N. The cancer cells consume nutrients at rate η 2 AC, grow at rate σ 2 η 2 e −b 2 τ 2 A(t − τ 2 )C(t − τ 2 ), and die at rate κ 2 C. The delay τ 1 > 0 is the time needed for nutrients to contribute to the biomass of healthy cells, while τ 2 > 0 is the time needed for nutrients to contribute to the biomass of cancer cells [40]. e −b 1 τ 1 and e −b 2 τ 2 are the survival probabilities of healthy and cancer cells during the delay period, respectively. SARS-CoV-2 particles infect healthy cells at rate η 3 NV, proliferate at rate σ 3 η 3 NV, and die at rate κ 3 V. CTLs kill cancer cells at rate η 4 CW, decay at rate κ 4 W, and are stimulated by cancer cells at rate σ 4 η 4 (1 − ρ 1 )CW. Antibodies neutralize virus particles at rate η 5 VZ, die at rate κ 5 Z, and are produced by B cells at rate σ 5 η 5 (1 − ρ 2 )VZ. The parameter ρ 1 measures the impact of lymphopenia on the activation rate of the CTL immune response, where 0 ≤ ρ 1 < 1. Furthermore, the parameter ρ 2 measures the impact of lymphopenia on the production rate of antibodies, where 0 ≤ ρ 2 < 1. For simplicity, we will utilize the following shortcuts in the next parts of the paper:

Nonnegativity and Boundedness of the Solution
In this subsection, we establish the nonnegativity and boundedness of the solutions of model (1). Theorem 1. All solutions of model (1) subject to condition (2) remain nonnegative and ultimately bounded.

Steady States
In this subsection, we compute all the biologically acceptable steady states of model (1) and determine their existence conditions.
We define the following parameters: The meaning of the threshold numbers and the usage of the above parameters are given in the next theorem. Theorem 2. Model (1) has ten steady states as follows: 1.
The healthy-cell steady state The cancer-cell steady state The infection cancer-immune-free steady state The cancer-CTL steady state E 4 = (A 4 , 0, C 4 , 0, W 4 , 0) exists if R C > R CW ; 6.
The cancer-free steady state E 7 = (A 7 , N 7 , 0, V 7 , 0, Z 7 ) exists if R N >R and The antibody-free steady state 10. The coexistence steady state E 9 = (A 9 , N 9 , C 9 , V 9 , W 9 , Proof. Any steady state E = (A, N, C, V, W, Z) of system (1) fulfills the following algebraic system: By solving system (3), we obtain the following steady states: 1.

2.
The healthy-cell steady state comes in the form E 1 = (A 1 , N 1 , 0, 0, 0, 0), where As A 1 > 0, the steady state E 1 exists when R N > 1. Here, R N is a threshold number required for the persistence of healthy epithelial cells with nutrient. 3.
The cancer-cell steady state is given by E 2 = (A 2 , 0, C 2 , 0, 0, 0), where As A 2 > 0, the steady state E 2 is defined when C 2 > 0, which corresponds to the condition R C > 1. Therefore, R C is a threshold number required for the persistence of cancer cells with nutrient.

4.
The infection steady state is given by is a threshold number, which determines the establishment of SARS-CoV-2 infection in cancer-free patient.

5.
The cancer-CTL steady state has the form E 4 = (A 4 , 0, C 4 , 0, W 4 , 0), where We see that A 4 > 0, C 4 > 0, and Here, R C R CW is a threshold number that determines the activation of cancer-specific CTL immune response when the healthy cells are extinct. 6.
The virus-free steady state is given by E 5 = (A 5 , N 5 , C 5 , 0, W 5 , 0), where It is clear that On the other hand, we have Accordingly, The immune-free steady state has the form E 6 = (A 6 , N 6 , C 6 , V 6 , 0, 0), where Thus, the steady state E 6 exists if R N > R C > R NV . 8.

9.
The antibody-free steady state is given by It is clear that A 8 > 0, N 8 > 0, and C 8 > 0. On the other hand, we have 10. The coexistence steady state has the form E 9 = (A 9 , N 9 , C 9 , V 9 , W 9 , Z 9 ), where It is easy to note that A 9 > 0, C 9 > 0, V 9 > 0, and Similarly, Hence,

Global Stability
This section is devoted to show the global stability of the steady states of model (1) by choosing appropriate Lyapunov functions. Hereafter, the following shortcuts will be applied: Theorem 3. The steady state E 0 is globally asymptotically stable when R N ≤ 1 and R C ≤ 1.
Proof. Choose a Lyapunov function P 0 as Then, we obtain We note that Depending on LaSalle's invariance principle [42], E 0 is globally asymptotically stable (GAS) if R N ≤ 1 and R C ≤ 1.

Theorem 4.
Suppose that R N > 1 and R C ≤ R N ≤ R NV . Then, the healthy-cell steady state E 1 is GAS.
Proof. Choose a Lyapunov function P 1 as Then, we obtain By using the steady state conditions at E 1 and using the following relation: the derivative of P 1 in (4) is converted to Theorem 5. Suppose that R C > 1 and R N ≤ R C ≤ R CW . Then, the cancer-cell steady state E 2 is GAS.
Proof. See Appendix A.
and R C ≤ R NV . Then, the infection steady state E 3 is GAS.
Proof. See Appendix B.
Theorem 7. Suppose that R N ≤ R CW < R C . Then, the cancer-CTL steady state E 4 is GAS.
Proof. See Appendix C.
. Then, the virus-free steady state E 5 is GAS.
Proof. See Appendix D.
. Then, the immune-free steady state E 6 is GAS.
Proof. See Appendix E.
. Then, the cancer- Proof. See Appendix F.
. Then, the Proof. See Appendix G.
Then, the coexistence steady state E 9 is GAS.
Proof. See Appendix H.

Numerical Simulations
In this section, we present some numerical simulations to visualize the theoretical results obtained in Theorems 1-12. We use the MATLAB solver dde23 to solve system (1). We show the effect of time delays and lymphopenia on the dynamics of model (1). To achieve this goal, we choose the initial conditions as follows: where τ = max{τ 1 , τ 2 }. According to the results of Theorems 3-12, the stability of steady states is guaranteed for any other choice of initial conditions. We vary the values of η 1 , η 2 , η 4 , η 5 , κ 1 , κ 2 , κ 3 , κ 4 , and κ 5 while fixing the values of all other parameters. The values of the fixed parameters are given in Table 1. As a result, we obtain ten cases corresponding to the global stability of the ten steady states as follows: 1.

2.
We consider , and κ 5 = 0.07. These selected values give R N = 1.8097 > 1 and R C = 0.2172 < R N < R NV = 20.697. In agreement with Theorem 4, the steady state E 1 = (0.5526, 0.1619, 0, 0, 0, 0) is GAS (Figure 1b). In this case, the cancer cells and SARS-CoV-2 particles are eliminated from the body. This situation would be reached with effective treatments that can target both cancer cells and virus particles. Finding effective ways to target cancer and COVID-19 is still under investigation [12]. 3.
Here, the concentration of healthy epithelial cells tends to zero. This situation might be reached after a strong competition with cancer cells. Thus, there are no healthy cells exposed to SARS-CoV-2 infection.
We take η 1 = 0.03,  Figure 2b and supported by Theorem 8. This represents an ideal situation where the virus is completely eliminated from the body of the cancer patient. Thus, the parameters used to reach this situation can be of special benefit. 7.

The Impact of Time Delays on Healthy and Cancer Cells
Increasing or decreasing time delays can have a strong impact on the concentrations of healthy and cancer cells and on the dynamics of model (1). For example, if we take the same values of the parameters considered in case (6) and increase only the value of τ 1 from 0.1 to 0.3, we find that E 4 = (0.4675, 0, 0.2278, 0, 0.0148, 0) is GAS. This means that increasing τ 1 causes a bifurcation in the system, where E 5 loses its stability and E 4 becomes stable. Figure 4a shows how increasing τ 1 causes a reduction in the concentration of healthy epithelial cells.
Alternatively, if we consider the same parameter values considered in case (7) and take τ 2 = 0.3 instead of τ 2 = 0.1, we find that the steady state E 3 = (0.3003, 0.1553, 0, 0.0459, 0, 0) is GAS. Hence, increasing τ 2 changes the stability of the steady states E 6 and E 3 . Figure 4b shows the impact of increasing the value of τ 2 on decreasing the concentration of cancer cells.
Similarly, decreasing the values of τ 1 and τ 2 will increase the concentrations of healthy cells and cancer cells, respectively. Increasing or decreasing the concentrations of these cells could be related to the severity of SARS-CoV-2 infection.

The Impact of Lymphopenia on Cancer Cells and SARS-CoV-2 Particles
To see the impact of dysfunction in the CTL immune response during SARS-CoV-2 infection, we take the same values of parameters considered in case (10) and increase only the value of ρ 1 . The result is shown in Figure 5a. We see that increasing the value of ρ 1 causes a rise in the concentration of cancer cells. Similarly, decreasing the functionality of the antibody immune response (by increasing the value of ρ 2 ) causes an increase in the concentration of SARS-CoV-2 particles (Figure 5b). Therefore, lymphopenia can worsen the state of cancer and allow the virus to replicate faster.

Discussion
Although many vaccines have been authorized or are under development [4], COVID-19 is still spreading and causing daily deaths. Mathematical modelling is an efficient tool that can contribute to both understanding the disease and finding better ways to defeat it [27]. Cancer patients are at greater risk for hospitalization and death due to SARS-CoV-2 infection compared to other patients who do not have cancer [2].
In this paper, we developed a within-host SARS-CoV-2 cancer model. This model consists of a system of delay differential equations and depicts the interactions between nutrients, healthy epithelial cells, cancer cells, SARS-CoV-2 virus particles, cancer-specific CTLs, and virus-specific antibodies. The model has ten steady states that have only positive components and are stable under the following conditions:

1.
The trivial steady state E 0 is GAS if R N ≤ 1 and R C ≤ 1.

2.
The healthy-cell steady state E 1 is defined and GAS if R N > 1 and R C ≤ R N ≤ R NV . This point represents the case when both cancer cells and viral particles are eliminated from the body.

3.
The cancer-cell steady state E 2 is GAS if R C > 1 and R N ≤ R C ≤ R CW . Here, all compartments tend to zero except for cancer cells and nutrients.

4.
The infection cancer-immune-free steady state and R C ≤ R NV . SARS-CoV-2 infection is established while cancer cells and immune responses are not present.

5.
The cancer-CTL steady state E 4 is GAS if R N ≤ R CW < R C . Here, the CTL immune response is activated to kill cancer cells in the absence of healthy cells.

The virus-free steady state
This point corresponds to the case when SARS-CoV-2 is eliminated from SARS-CoV-2/cancer patient.

7.
The immune-free steady state E 6 is GAS if 1 < R N R C ≤R and R NV < R C ≤ . The cancer patient here fights SARS-CoV-2 infection with inactive immune responses.

The cancer-free steady state
At this point, antibody immunity is activated to eliminate the virus, while cancer cells are removed.

9.
The antibody-free steady state . Here, the concentration of antibodies tends to zero, while all other compartments have positive values.
10. The coexistence steady state E 9 is GAS ifR > We found that the time delays affected the growth rates of healthy epithelial cells and cancer cells and could cause a bifurcation in the system. Increasing the time delay τ 2 , which represents the delay in the utilization of nutrients by cancer cells, decreases the concentration of cancer cells. This can have a positive effect by preventing the situation of SARS-CoV-2 cancer patient from getting worse.
We observed that lymphopenia, which affects the functionality of immune responses, increased the concentrations of both cancer cells and SARS-CoV-2 particles. This leads to the presence of high viral loads and the progression of cancer. As a result, the condition of SARS-CoV-2/cancer patient will worsen and may lead to death. This result agrees with many studies that correlated lymphopenia with severe infection in cancer patients [2,13,20,44].
The model studied in this paper can be used (i) to estimate the parameters needed to clear the virus from the body of cancer patient (see case (6) in Section 5); (ii) to test the effect of time delays on the concentrations of healthy cells and cancer cells during SARS-CoV-2 infection; (iii) to observe the effect of lymphopenia on the activation rates of immune responses; accordingly, on the severity of SARS-CoV-2 infection in cancer patients; and (iv) to check the possibility of eliminating cancer cells and virus particles at the same time (see case (2) in Section 5)).
The main limitation of this work is that we did not use real data to estimate the values of the parameters in model (1). We assumed that SARS-CoV-2 does not infect cancer cells. This is because the data on SARS-CoV-2 infection in cancer patients are very limited, and it is not yet clear whether and how SARS-CoV-2 affects cancer cells at the cellular level [10]. The model and the theoretical results of this paper can be tested and developed depending on the availability of real data.
We believe that this work can help to provide a better understanding of SARS-CoV-2 infection in one of the groups that is the most vulnerable to severe infection and death. A better understanding will facilitate finding more effective ways to treat SARS-CoV-2/cancer patients. Model (1) can be developed by (i) fitting the model with real data, (ii) performing a detailed bifurcation analysis, and (iii) applying a multiscale approach where the within-host dynamics are connected with the between-hosts dynamics. Proof. Choose a Lyapunov function By using the steady state conditions at E 2 we obtain It is clear that dP 2 dt ≤ 0 if R N ≤ R C and R C ≤ R CW . One can show that dP 2 dt = 0 if A = A 2 , C = C 2 , and N = W = V = Z = 0. Thus, the singleton {E 2 } is the largest invariant subset of (A, N, C, V, W, Z) | dP 2 dt = 0 . Consequently, the global stability of E 2 is confirmed by LaSalle's invariance principle [42] when R C > 1 and R N ≤ R C ≤ R CW .

Appendix B. Proof of Theorem 6
Proof. Select a Lyapunov function P 3 as From Equation (3), E 3 at the equilibrium state fulfills the conditions By utilizing the above conditions, the time derivative of P 3 can be provided as: By using the value of A 3 and A 6 computed in the proof of Theorem 2, we obtain Additionally, from the steady states E 3 and E 7 computed in the proof of Theorem 2, we obtain Hence, Equation (A1) can be rewritten as: We see that . One can show that and R C ≤ R NV .

Appendix C. Proof of Theorem 7
Proof. Take the Lyapunov function By using the steady state conditions at E 4 we find By following the same reasoning given in Theorems 3-6, we find that E 4 is GAS if R N ≤ R CW < R C .

Appendix D. Proof of Theorem 8
Proof. Choose a Lyapunov function At the equilibrium state, E 5 fulfils the following system of equations By utilizing the above conditions, we find Thus, Therefore, the solutions of system (1) converge to Γ 5 , which comprises elements with A = A 5 , N = N 5 , C = C 5 , and V = Z = 0. From the third equation of system (1), we conclude that W = W 5 . It follows that Γ 5 = {E 5 }. Depending on LaSalle's invariance principle [42], the steady state

Appendix E. Proof of Theorem 9
Proof. Take a Lyapunov function The steady state conditions at E 6 are given by After using the steady state conditions, we obtain Thus, This implies that Γ 6 = {E 6 }. In accordance with LaSalle's invariance principle [42], the

Appendix F. Proof of Theorem 10
Proof. Consider a Lyapunov function By using the following steady state conditions at E 7 We obtain Thus, dP 7 dt ≤ 0 ifR ≤ R N R C . dP 7 dt = 0 when A = A 7 , N = N 7 , and C = W = 0. Then, from system (1) we find that V = V 7 and Z = Z 7 . Assume that Γ 7 is the largest invariant subset of Γ 7 = (A, N, C, V, W, Z) | dP 7 dt = 0 . As a consequence, we obtain Γ 7 = {E 7 }. Based on LaSalle's invariance principle [42], the steady state E 7 is GAS if

Appendix G. Proof of Theorem 11
Proof. Choose a Lyapunov function P 8 as follows The steady state conditions at E 8 are given by the following equations After rearranging and using the above conditions, we obtain Thus, . It is easy to show that dP 8 dt = 0 when A = A 8 , N = LaSalle's invariance principle [42], the steady state E 8 is GAS if R N > R CW + η 1 Θ 3 κσ 3 η 3 , , and R N ≤ R NV + ψ 1 Θ 5 κΘ 1 σ 3 σ 5 η 5 (1 − ρ 2 ) .
After rearranging and utilizing the steady state conditions, the time derivative of P 9 is given by We see that dP 9 dt ≤ 0. It is easy to show that dP 9 dt = 0 at E 9 . Let Γ 9 be the largest invariant subset of Γ 9 = (A, N, C, V, W, Z) | dP 9 dt = 0 . It follows that Γ 9 = {E 9 }.
Depending on LaSalle's invariance principle [42], the steady state E 9 is GAS.