key: cord-0790269-dkt4xnjb
authors: Elaiw, A.M.; Al Agha, A.D.
title: Global dynamics of SARS-CoV-2/cancer model with immune responses
date: 2021-05-12
journal: Appl Math Comput
DOI: 10.1016/j.amc.2021.126364
sha: e15b70823e448ef54b160e05323ec1c36a7da046
doc_id: 790269
cord_uid: dkt4xnjb

The world is going through a critical period due to a new respiratory disease called coronavirus disease 2019 (COVID-19). This disease is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Mathematical modeling is one of the most important tools that can speed up finding a drug or vaccine for COVID-19. COVID-19 can lead to death especially for patients having chronic diseases such as cancer, AIDS, etc. We construct a new within-host SARS-CoV-2/cancer model. The model describes the interactions between six compartments: nutrient, healthy epithelial cells, cancer cells, SARS-CoV-2 virus particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies. We verify the nonnegativity and boundedness of its solutions. We outline all possible equilibrium points of the proposed model. We prove the global stability of equilibria by constructing proper Lyapunov functions. We do some numerical simulations to visualize the obtained results. According to our model, lymphopenia in COVID-19 cancer patients may worsen the outcomes of the infection and lead to death. Understanding dysfunctions in immune responses during COVID-19 infection in cancer patients could have implications for the development of treatments for this high-risk group.

Coronavirus disease 2019 (COVID-19) is a new respiratory and highly infectious disease. It appeared in Wuhan, China in late 2019 and spread rapidly to a large number of countries around the world. On March 11, 2020, the World Health Organization (WHO) upgraded COVID-19 to a pandemic [1] . According to the WHO report of August 2020, over 1.7 million new COVID-19 cases and 39,000 new deaths were recorded in late August [2] . The total number of cases and deaths from the start of the pandemic to the date of this report reached over 23 million cases and 800,000 deaths [2] . Despite the tremendous effort and great competition between countries, no effective cure or vaccine has been proven yet. The WHO posted many guidelines to reduce the spread of COVID-19 such as wearing masks, washing hands, and monitoring a safe distance between you and others [3] . In response to these recommendations and to avoid the collapse of healthcare systems, countries imposed rules to follow in public places and workplaces.

COVID-19 is caused by a single-stranded RNA virus called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [4, 5] . It belongs to the family Coronaviridae, which also involves SARS-CoV that emerged in 2002 and the Middle East respiratory syndrome coronavirus (MERS-CoV) that emerged in 2012 [6, 7] . SARS-CoV-2 virus targets cells with angiotensin-converting enzyme 2 (ACE2) receptor, which makes these cells more susceptible to virus entry [8] . ACE2 is expressed in many cells such as myocardial epithelial cells, kidney tubular epithelial cells, and gastrointestinal epithelial cells [7] . However, type II alveolar epithelial cells have a higher expression rate of ACE2 and therefore are the main target for SARS-CoV-2 virus [7, 9] . The most common symptoms of COVID-19 include fatigue, cough, fever, shortness of breath, sore throat, headache, and diarrhea [7, 10] . Although a large percentage of patients show mild symptoms and do not need to stay in hospitals, about 20% of them develop pneumonia, multiple organ failure, and ultimately death [8, 10] . Lymphopenia, which is defined as a lymphocyte count of less than 1.5 × 10 3 /mm 3 , was also reported in severe COVID-19 patients [4, 11, 12] .

There are many risk factors that cause disease progression in COVID-19 patients and rise the need for intensive care unit (ICU) admission or using mechanical ventilation [4, 13] . These factors include hypertension, older age, diabetes, chronic kidney disease, obesity, cardiovascular disease, and cancer [5, 13, 14] . Cancer patients are at higher risk of serious complications and death due to COVID-19 infection [15] [16] [17] . This risk increases further in patients with lymphopenia [1, 10] . COVID-19/cancer patients with lymphopenia have a 10 times higher risk of death than the patients infected only by COVID-19 [7] . Therefore, several measures have been proposed to protect cancer patients during the pandemic [1] . In general, the role of immune dysregulation in COVID-19 cancer patients is being investigated [9, 18] .

Mathematical models have been used to understand the transmission of COVID-19 between individuals and provide useful insights into the development of control strategies [19] . Crucial and important decisions can be taken based on these models. Most of the recent COVID-19 models are epidemiological models based on SEIR models (see for example, [19] [20] [21] [22] [23] [24] ). These models depict the transmission dynamics of the virus between susceptible, exposed, infectious, and recovered individuals. Different computations were performed to estimate the basic reproductive number and to predict the impact of control measures on reducing the transmission rates of the disease [19] [20] [21] [22] [23] [24] .

Within-host models, which study the interactions between SARS-CoV-2 and human cells, have received less attention than epidemiological models. Most of within-host SARS-CoV-2 models are based on Nowak and Bangham's model [25] that was extended to study the dynamics of several viruses such as human immunodeficiency virus (HIV) [26] [27] [28] , hepatitis B virus (HBV) [29] [30] [31] , chikungunya virus (CHIKV) [32] , and other viruses. For example, Du and Yuan [8] studied a within-host model of COVID-19 infection. They explored the effect of the interaction between innate and adaptive immunity on the peak of viral load in COVID-19 patients. Li et al. [33] used a within-host viral model to study the dynamics of SARS-CoV-2 virus in host, and they estimated the values of model's parameters. Ghosh [34] proposed a mathematical model to study the interaction between healthy cells, virus particles, and immune response in SARS-CoV-2 infection. He tested the efficacy of different antiviral drugs and estimated the parameters by fitting with real data. Hattaf and Yousfi [35] proposed a within-host model to study the interactions between host epithelial cells, SARS-CoV-2 virus, and cytotoxic T lymphocytes (CTLs) with virus-to-cell and cell-to-cell transmission modes. Pinky and Dobrovolny [36] used a mathematical model to investigate SARS-CoV-2 coinfections with many other respiratory viruses. They argued that SARS-CoV-2 virus replication is suppressed by other viruses when the infections occur simultaneously.

As mentioned above, cancer patients who get infected by SARS-CoV-2 are at increased risk of severe infection and death. To design effective treatments that can target both SARS-CoV-2 and cancer, we need to understand the interactions between healthy cells, cancer cells, virus particles, and immune responses. Mathematical modeling is a powerful tool that can help us understand the within-host interactions. To the best of our knowledge, no mathematical models have been developed so far to study the effect of COVID-19 on cancer patients. In this paper, we construct a new within-host SARS-CoV-2/cancer model. This model is different from other within-host models investigated before because it is an adjustment of the oncolytic virotherapy models studied in [37] and [38] . For our proposed model, we (i) show the nonnegativity and boundedness of model's solutions; (ii) discuss all biologically acceptable equilibrium points of the suggested model; (iii) prove the global stability of equilibria; (iv) support the theoretical results by performing some numerical simulations; (v) present the effect of lymphopenia on the growth of cancer cells and SARS-CoV-2 particles.

This paper is organized as follows. Section 2 provides a full description of the new model under study. Section 3 shows that the solutions of the developed model are nonnegative and bounded. In addition, it lists all possible equilibrium points. Section 4 proves the global asymptotic stability of equilibria considered in Section 3. Section 5 presents some numerical simulations to support the theoretical results of the previous section. Finally, Section 6 discusses the results and some possible future works.

In this section, we investigate a SARS-CoV-2/cancer model with two types of immune responses: SARS-CoV-2specific antibody immune response and cancer-specific CTL immune response. Thus, we propose the following ordinary differential equation model

where A(t), N (t), C(t), V (t), W (t), and Z(t) denote the concentrations of nutrient, healthy epithelial cells, cancer cells, SARS-CoV-2 virus particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies at time t, respectively. The model is considered in the chemostat in which there is a competition between healthy epithelial cells and cancer cells on a restricted nutrient source [37] . The nutrient is recruited from a source at rate µ and decays at rate θA. It is consumed by healthy epithelial cells at rate η 1 AN , while it is consumed by cancer cells at rate η 2 AC. The healthy epithelial cells and cancer cells grow after consuming the nutrient at rates σ 1 η 1 AN and σ 2 η 2 AC, respectively. SARS-CoV-2 virus infects epithelial cells at rate η 3 N V , while it replicates at rate σ 3 η 3 N V . Cancer-specific CTLs kill cancer cells at rate η 4 CW , and they are stimulated at rate σ 4 η 4 (1 − ρ 1 )CW . SARS-CoV-2-specific antibodies neutralize virus particles at rate η 5 V Z, while they are produced at rate σ 5 η 5 (1 − ρ 2 )V Z. The natural death rates of epithelial cells, cancer cells, virus particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies are given by θ 1 , θ 2 , θ 3 , θ 4 , and θ 5 , respectively. The parameter ρ 1 measures the effect of lymphopenia on reducing the stimulation rate of cancer-specific CTL immune response, where 0 ≤ ρ 1 < 1. On the other hand, the parameter ρ 2 measures the effect of lymphopenia on the production rate of SARS-CoV-2-specific antibodies, where 0 ≤ ρ 2 < 1. All parameters of model (1) are assumed to be positive. The description of all parameters is provided in Table 2 . For simplicity, we will use the following notations in the next sections:

In this section, we establish the existence, nonnegativity, and boundedness of the solutions of model (1) . In addition, we determine all biologically acceptable equilibrium points of system (1) and identify the conditions needed for their existence.

Theorem 1. Let π i > 0, i = 1, 2, ..., 6, and define the compact set Π = (A, N, C, V, W, Z) ∈ R 6

Then, the set Π is positively invariant for model (1) .

This guarantees that A(t), N (t), C(t), V (t), W (t), Z(t) ∈ R 6 + for all t ≥ 0 when the initial conditions

To show the boundedness, we define

Then, we get

where π 1 = µ θ . As A(t), N (t), C(t), V (t), W (t), and Z(t) are nonnegative, we have

where π 2 = σ 1 π 1 , π 3 = σ 2 π 1 , π 4 = σ 1 σ 3 π 1 , π 5 = σ 2 σ 4 (1 − ρ 1 )π 1 , and π 6 = σ 1 σ 3 σ 5 (1 − ρ 2 )π 1 . This proves that the set Π is positively invariant.

and ψ 3 > 0 such that model (1) has ten equilibrium points under the following conditions:

(1) The trivial equilibrium E 0 = (A 0 , 0, 0, 0, 0, 0) always exists;

(2) The healthy-cell equilibrium E 1 = (A 1 , N 1 , 0, 0, 0, 0) exists if R N > 1;

(3) The cancer-cell equilibrium E 2 = (A 2 , 0, C 2 , 0, 0, 0) exists if R C > 1;

(6) The virus-free equilibrium E 5 = (A 5 , N 5 , C 5 , 0, W 5 , 0) exists if R C R N > 1 and R N > R CW ;

(7) The immune-free equilibrium E 6 = (A 6 , N 6 , C 6 , V 6 , 0, 0) exists if R N R C > 1 and R C > R N V ;

(10) The coexistence equilibrium E 9 = (A 9 , N 9 , C 9 , V 9 , W 9 ,

Proof. Any equilibrium point E = (A, N, C, V, W, Z) of system (1) satisfies the following system of equations:

By finding the solutions of algebraic system (2), we get the following equilibrium points:

(1) The trivial equilibrium is given by E 0 = (A 0 , 0, 0, 0, 0, 0), where A 0 = µ θ > 0. Thus, E 0 always exists.

(2) The healthy-cell equilibrium takes the form E 1 = (A 1 , N 1 , 0, 0, 0, 0), where

where R N = µσ1η1 θΘ1 . As A 1 > 0, the equilibrium E 1 exists when N 1 > 0 and this holds if R N > 1. Hence, R N is a threshold number needed for the persistence of only healthy epithelial cells in the presence of nutrient.

(3) The cancer-cell equilibrium takes the form E 2 = (A 2 , 0, C 2 , 0, 0, 0), where

where R C = µσ2η2 θΘ2 . As A 2 > 0, the equilibrium point E 2 is defined when C 2 > 0 which corresponds to the condition R C > 1. Hence, R C is a threshold number needed for the persistence of only cancer cells in the presence of nutrient.

(4) The infection equilibrium is given by 

Here, R C R CW is a threshold number which determines the activation of CTL immune response against cancer cells when the epithelial cells do not exist.

(6) The virus-free equilibrium is given by

It is clear that

On the other hand, we have

The immune-free equilibrium has the form E 6 = (A 6 , N 6 , C 6 , V 6 , 0, 0), where

Thus, the equilibrium point

The cancer-free equilibrium is given by

R is a threshold number needed for the activation of antibody immune response against SARS-CoV-2 in cancer-free patients. We note that

(9) The antibodies-free equilibrium is given by

On the other hand, we have

and

.

The coexistence equilibrium 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,

.

In this section, we prove the global stability of the equilibrium points of model (1) by constructing Lyapunov functions following the method presented in [39] [40] [41] . From now on, the following simplifications will be considered:

Theorem 3. The equilibrium E 0 is globally asymptotically stable when R N ≤ 1 and R C ≤ 1. It becomes unstable when R N > 1 or R C > 1.

Proof. Define a Lyapunov function P 0 (A, N, C, V, W, Z) as

Then, we obtain

We note that dP0

Then, we discuss four cases:

We need to show that N = C = 0. As A = A 0 , we get from the first equation of system (1) that

We obtain from Eq.

For these four cases the singleton {E 0 } is the largest invariant subset of (A, N, C, V, W, Z) | dP0 dt = 0 . According to LaSalle's invariance principle [42] , E 0 is globally asymptotically stable if R N ≤ 1 and R C ≤ 1.

To check the local instability of E 0 when R N > 1 or R C > 1, we compute the characteristic equation. The Jacobian matrix evaluated at E 0 is given by

The associated characteristic equation is

We note that two of the eigenvalues of Eq. (4) are given by

This implies that E 0 is unstable when R N > 1 or R C > 1.

Theorem 4. Suppose that R N > 1 and R C ≤ R N ≤ R N V . Then, the healthy-cell equilibrium E 1 is globally asymptotically stable.

Proof. Define a Lyapunov function P 1 (A, N, C, V, W, Z) as

Then, we get

By using the equilibrium conditions at

the time derivative of P 1 in (5) is reduced to

Hence, the solutions of system (1) converge to Γ 1 which includes elements with A = A 1 and C = V = W = Z = 0. From system (1) we have

From the proof of Theorem 3, we see that E 0 is the only stable equilibrium when R N ≤ 1 and R C ≤ 1. When R N > 1 and R C ≤ R N ≤ R N V , E 0 loses its stability and E 1 becomes globally asymptotically stable according to Theorems 3 and 4. This means that a transcritical bifurcation occurs at R N = 1.

Theorem 5. Suppose that R C > 1 and R N ≤ R C ≤ R CW . Then, the cancer-cell equilibrium E 2 is globally asymptotically stable.

By using the equilibrium conditions at

we obtain

Thus, the singleton {E 2 } is the largest invariant subset of (A, N, C, V, W, Z) | dP2 dt = 0 . Accordingly, the global asymptotic stability of E 2 is guaranteed by LaSalle's invariance principle [42] when

According to Theorem 3, E 0 is the only stable equilibrium when R N ≤ 1 and R C ≤ 1. When R C > 1 and R N ≤ R C ≤ R CW , E 0 becomes unstable and E 2 appears and it is globally asymptotically stable as stated in Theorems 3 and 5. Hence, we have a transcritical bifurcation at R C = 1.

Then, the infection equilibrium E 3 is globally asymptotically stable.

Proof. Define a Lyapunov function P 3 (A, N, C, V, W, Z) as

From Eq. (2), E 3 at the equilibrium state satisfies the conditions

By using the above conditions, the time derivative of P 3 can be written as:

By using the value of A 3 of the equilibrium E 3 and the value of A 6 of the equilibrium E 6 computed in the proof of Theorem 2, we get

Also, from the equilibrium points E 3 and E 7 computed in the proof of Theorem 2 we obtain

Hence, Eq. (6) can be rewritten as

We see that dP3 Theorem 7. Suppose that R N ≤ R CW < R C . Then, the cancer-CTL equilibrium E 4 is globally asymptotically stable.

Proof. Consider the following Lyapunov function

By using the equilibrium conditions at

we get

Z.

Using the same arguments as in Theorems 3-6, we conclude that E 4 is globally asymptotically stable if R N ≤ R CW .

. Then, the virus-free equilibrium E 5 is globally asymptotically stable.

Proof. Take a Lyapunov function

At the equilibrium state, E 5 satisfies the following system of equations

By using the above conditions, we have

Thus, dP5 dt ≤ 0 if R N ≤ R CW + η1Θ3 θσ3η3 . Also, dP5 dt = 0 when A = A 5 , V = 0, and Z = 0. Let Γ 5 be the largest invariant subset of Γ 5 = (A, N, C, V, W, Z) | dP5 dt = 0 . Hence, the solutions of system (1) converge to Γ 5 which includes elements with A = A 5 , V = 0, and Z = 0. From the first equation of system (1), we find

From the second equation of system (1), we have

Hence, N = constant = N . Substituting this value of N in Eq. (7), we get

By solving Eq. (8) for C, we find that C = constant = C. Consequently, we have dC dt = 0. From the third equation of system (1), we obtain

By solving Eq. (9) for W, we find that W = W 5 and so dW dt = 0. Then, from the fifth equation of system (1) we get C = C 5 . Accordingly, from Eq. (7) we get N = N 5 . It follows that Γ 5 = {E 5 }. Based on LaSalle's invariance principle [42] , the equilibrium E 5 is globally asymptotically stable if R N ≤ R CW + η1Θ3 θσ3η3 . ρ1) . Then, the immune-free equilibrium E 6 is globally asymptotically stable.

The equilibrium conditions at E 6 are given by

After using the equilibrium conditions, we obtain 

From the third equation of system (1), we get

This implies that C = constant = C. Substituting this value of C in Eq. (10) gives

By solving Eq. (11) for N , we get N = constant = N . This gives dN dt = 0, and from the second equation of system (1), we have

Solving Eq. (12) for V gives V = V 6 and as a result we have dV dt = 0. Then, from the fourth equation of (1) we obtain N = N 6 . By substituting N 6 into Eq. (11), we get C = C 6 . It follows that Γ 6 = {E 6 }. Based on LaSalle's invariance principle [42] , the equilibrium E 6 is globally asymptotically stable if R C ≤ R N V + η2Θ4 θσ4η4(1−ρ1) and ρ2) . Then, the cancer-free equilibrium E 7 is globally asymptotically stable ifR ≤ R N R C .

By using the following equilibrium conditions at

we get 

Then, the fourth equation of system (1) gives

Based on LaSalle's invariance principle [42] , the equilibrium E 7 is globally asymptotically stable ifR ≤ R N R C .

Theorem 11. Suppose that R N > R CW + η1Θ3 θσ3η3 and R C > R N V + η2Θ4 θσ4η4(1−ρ1) . Then, the antibodies-free

Proof. Define a Lyapunov function P 8 (A, N, C, V, W, Z) as follows

The equilibrium conditions at E 8 are provided by the following equations

After rearranging and using the above conditions, we obtain ρ2) . Then, the coexistence equilibrium E 9 is stable.

Proof. Define a Lyapunov function P 9 (A, N, C, V, W, Z) as follows

After rearranging and using equilibrium conditions, the time derivative of P 9 is given by

We see that dP9 dt ≤ 0 which implies the stability of E 9 . As it is not easy to show that dP9 dt = 0 at E 9 , we will show the local asymptotic stability of E 9 numerically in the next section.

In this section, we carry out some numerical simulations to advocate the results obtained in the previous section. Also, we discuss the effect of lymphopenia on the growth of tumor and SARS-CoV-2 virus. For this purpose, we choose the following initial conditions of system (1) According to Theorems 3-12, the stability is ensured for any other initial conditions. We divide the numerical simulations into ten cases corresponding to the stability of each equilibrium point computed in Theorem 2. We get these cases by varying the values of η 1 , η 2 , η 4 , η 5 , θ 1 , θ 2 , θ 3 , θ 4 , and θ 5 . The values of all other parameters are fixed and given in Table 2 . The values of ρ 1 and ρ 2 are fixed to ρ 1 = ρ 2 = 0 in all cases. Accordingly, we get the following cases:

Case 1: We choose the values η 1 = 0.03, η 2 = 0.03, η 4 = 0.03, η 5 = 0.3, θ 1 = 0.1, θ 2 = 0.08, θ 3 = 0.5, θ 4 = 0.9, and θ 5 = 0.07. This gives R N = 0.2 < 1 and R C = 0.24 < 1. According to Theorem 3, the equilibrium E 0 = (1, 0, 0, 0, 0, 0) is globally asymptotically stable as shown in Figure 1a . At this point, there is no competition between healthy epithelial cells and cancer cells. As a result, the concentrations of SARS-CoV-2 particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies vanish.

Case 2: We select the values of parameters as η 1 = 0.1, η 2 = 0.03, η 4 = 0.03, η 5 = 0.3, θ 1 = 0.02, θ 2 = 0.08, θ 3 = 0.5, θ 4 = 0.9, and θ 5 = 0.07. These values give R N = 2 > 1, R C R N = 0.12 < 1, and R N < 20.697 = R N V . These thresholds lead to the global asymptomatic stability of E 1 = (0.5, 0.2, 0, 0, 0, 0), which agrees with Theorem 4 (see Figure 1b) . This point represents the ideal situation in which the patient is cancer-free and coronavirus-free at the same time. This situation can be reached under functional treatments or effective immune responses. Of note, treating cancer and COVID-19 simultaneously is one of the most active research areas [18] .

Case 3: We take the values η 1 = 0.03, η 2 = 0.1, η 4 = 0.03, η 5 = 0.3, θ 1 = 0.1, θ 2 = 0.01, θ 3 = 0.5, θ 4 = 0.9, and θ 5 = 0.07. Then, we have R C = 2.6667 > 1, R N R C = 0.075 < 1, and R C < 1.5343 × 10 3 = R CW . Under these conditions and in agreement with Theorem 5, the equilibrium E 2 = (0.3750, 0, 0.3333, 0, 0, 0) is globally asymptotically stable as shown in Figure 1c . At this point, the epithelial cells become extinct under the competition with cancer cells and so there is no infection with COVID-19 in this situation. ρ2) . This causes the solutions of system (1) to asymptotically converge the equilibrium E 3 = (0.2268, 0.2273, 0, 0.0262, 0, 0) as exhibited in Figure 1d and supported by Theorem 6. At this point, the number of healthy epithelial cells decreases due to SARS-CoV-2 infection with no immune response. Also, the cancer cells are eliminated, which is not likely to happen in COVID-19 patients with progressive cancer.

Case 5: We select the values η 1 = 0.03, η 2 = 0.1, η 4 = 0.9, η 5 = 0.3, θ 1 = 0.1, θ 2 = 0.0005, θ 3 = 0.5, θ 4 = 0.0005, and θ 5 = 0.07. For this set of parameters, we get R C = 3.9024 > 2.1389 = R CW and R N = 0.2 < R CW . This causes the equilibrium E 4 = (0.4675, 0, 0.2278, 0, 0.0188, 0) to be globally asymptotically stable as shown in Figure 1e , which supports Theorem 7. In this case, the cancer-specific CTL immune response is activated to eliminate cancer cells, while the healthy epithelial cells are not present. Figure 2c ). In this case, the patient has only SARS-CoV-2 mono-infection with active SARS-CoV-2-specific antibody immune response.

Case 9: We consider the values η 1 = 0.2, η 2 = 0.3, η 4 = 1.2, η 5 = 0.3, θ 1 = 0.01, θ 2 = 0.008, θ 3 = 0.0005, θ 4 = 0.0001, and θ 5 = 0.07.

The corresponding thresholds are R N = 5.3333 > 5.0655 = R CW + η1Θ3 θσ3η3 , R C = 8.5714 > 5.0655 = R N V + η2Θ4 θσ4η4(1−ρ1) , and (1−ρ2) . In agreement with Theorem 11, the equilibrium E 8 = (0.1974, 0.1553, 0.1675, 0.0029, 0.0161, 0) is stable as shown in Figure 2d . In this situation, the SARS-CoV-2-specific antibody immune response in a COVID-19 cancer patient has not been activated yet. This may allow a rapid replication of SARS-CoV-2 particles and cause disease progression.

Case 10: We select the values η 1 = 0.9, η 2 = 0.5, η 4 = 1.7, η 5 = 1.7, θ 1 = 0.0001, θ 2 = 0.0003, θ 3 = 0.0003, θ 4 = 0.0001, and θ 5 = 0.0001. This givesR = 2.6176 > 1.8179 ρ2) . Accordingly, the equilibrium E 9 = (0.0731, 0.2162, 0.1182, 0.0591, 0.0053, 0.0048) is stable as shown in Figure  2e , which agrees with Theorem 12. Here, both immune responses are active. The cancer-specific CTL immune response works on killing cancer cells, while the SARS-CoV-2-specific antibody immune response works on clearing the virus. However, the effectiveness of these roles depends on the functionality of immune responses.

To further confirm the asymptotic stability of E 8 and E 9 in Cases 9 and 10, we compute the Jacobian matrix of system (1) as follows

After that, we compute the eigenvalues λ i (i = 1, 2, ..., 6) of the Jacobian matrix J at all possible equilibrium points. For the stability of E 8 and E 9 , we need to show that

while all other equilibria have eigenvalues with positive real parts. The results are given in Table ( 1)

The effect of lymphopenia on SARS-CoV-2/cancer patients A functional exhaustion of immune responses due to lymphopenia can be seen by increasing the values of ρ 1 and ρ 2 in Case 10 while keeping all other parameters fixed. Increasing the values of ρ 1 and ρ 2 means decreasing the efficacies of cancer-specific CTL immune response and SARS-CoV-2-specific antibody immune response, respectively. Figure 3 shows the effect of increasing values from ρ 1 = ρ 2 = 0 to ρ 1 = ρ 2 = 0.1 and ρ 1 = ρ 2 = 0.2. As we can see, the concentration of cancer cells increases with the increase in ρ 1 . Similarly, the concentration of SARS-CoV-2 particles increases with the increase in ρ 2 . This can worsen the state of tumor and cause severe COVID-19 outcomes. The patients with this condition might be at a high risk of death. 

COVID-19 is a new respiratory disease caused by SARS-CoV-2 virus. The virus has reached most countries of the world, and the total number of deaths is increasing everyday [2] . Mathematical modeling is a promising tool that can support laboratory experiments and clinical trials. We studied a within-host model of six ordinary differential equations. The model studies the interactions between nutrient, healthy epithelial cells, cancer cells, SARS-CoV-2 virus particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies. It has ten possible equilibrium points corresponding to the following ten cases:

(a) The trivial equilibrium E 0 always exists, and it is globally asymptotically stable if R N ≤ 1 and R C ≤ 1. Here, the SARS-CoV-2 particles and epithelial target cells are eliminated. The complete elimination of target cells and its validity were discussed in [37] .

(f) The virus-free equilibrium E 5 exists if R C R N > 1 and R N > R CW , while it is globally asymptotically stable if R N ≤ R CW + η1Θ3 θσ3η3 . In this situation, the cancer patient becomes COVID-19 free. Hence, the parameters used in this case could help in drug discovery experiments.

(g) The immune-free equilibrium E 6 is defined if R N R C > 1 and R C > R N V , while it is globally asymptotically stable if R C ≤ R N V + η2Θ4 θσ4η4(1−ρ1) and R N R C ≤R. At this point, the immune responses in COVID-19 cancer patient are not active. This may lead to the growth of tumor and cause severe COVID-19 infection.

(h) The cancer-free equilibrium E 7 is defined if R N >R and R N > R N V + ψ1Θ5 θΘ1σ3σ5η5(1−ρ2) , and it is globally asymptotically stable ifR ≤ R N R C . Here, the cancer cells are eliminated, and the SARS-CoV-2-specific antibody immune response is activated to fight the viral infection.

(i) The antibodies-free equilibrium E 8 exists if R N > R CW + η1Θ3 θσ3η3 and R C > R N V + η2Θ4 θσ4η4 (1−ρ1) . It is stable if R N ≤ R N V + ψ1Θ5 θΘ1σ3σ5η5(1−ρ2) + η2Θ4 θσ4η4(1−ρ1) + η2η3Θ4Θ5 θΘ1σ4η4σ5η5(1−ρ1)(1−ρ2) . The SARS-CoV-2-specific antibody immune response is not active at this point.

(j) The coexistence equilibrium E 9 is defined and stable ifR > R N R C , R N >R + ρ2) . The cancer-specific CTL immunity is activated to eradicate cancer cells, while the SARS-CoV-2-specific antibody immunity is activated to clear the virus. Notably, the immune responses are not able to completely eliminate cancer cells or virus particles.

We found that the numerical results are fully aligned with the theoretical results. Lymphopenia in COVID-19 cancer patients increases the concentrations of cancer cells and SARS-CoV-2 particles, which may cause severe complications and lead to death. Moreover, we noted that the immune responses in Case 10 (see numerical simulation section) are not able to remove cancer cells or virus particles even if there is no lymphopenia (ρ 1 = ρ 2 = 0). Thus, the values of these two parameters should be carefully controlled. Recent studies has shown that cancer and COVID-19 may be concomitantly aggravated by lymphopenia [1, 7, 10, 43] . It has been found that SARS-CoV-2/cancer patients with lymphopenia are at four times higher risk for hospitalization and ten times higher risk for death compared with COVID-19 patients without cancer [7] . In fact, the possibility of developing effective immune responses during SARS-CoV-2 infection in cancer patients is an active area of research. Comparing with the existing mathematical models of COVID-19, the model developed in this paper is the first within-host model that studies COVID-19 infection in cancer patients with a full analysis. Treating COVID-19 cancer patients is one of the most difficult challenges for the health sector [5, 18] . Therefore, our results can be tested clinically to measure its accuracy. The results can be used to understand the interactions between healthy cells and cancer cells in COVID-19 cancer patients. Also, the results can be used to explore the importance of effective immune responses in this group of patients. Understanding complicated issues that may occur during COVID-19 infection in cancer patients can help in (i) developing more effective ways to deal with this group; (ii) finding effective treatments that may target both cancer and COVID-19 infection [43] . The model studied in this paper can be developed in many ways. First, by adding a coinfection term to model (1) as the following                    

where SARS-CoV-2 particles infect cancer cells at rateη 3 CV and are produced by cancer cells at ratẽ σ 3η3 C(t)V (t). However, we were cautious about adding these terms for two reasons: (i) We mentioned earlier that SARS-CoV-2 infects certain types of cells in some organs [7, 43] and thus the location of viral infection might be different from the site of tumor; (ii) A limited amount of research has been done to clarify whether and how SARS-CoV-2 infection impacts cancer cells [7] . In fact, if SARS-CoV-2 is able to target cancer cells, it can be used to treat cancer. Nevertheless, there is no available data about whether SARS-CoV-2 can be used as oncolytic virotherapy like many other viruses [7, 38] . Hence, adding a coinfection term can lead to very interesting results, but we need to be careful as the data on patients with cancer who develop COVID-19 is still very limited [1, 7, 44] . Second, by fitting the model with real data and using more realistic values of the parameters. Third, by considering time delays that will convert model (1) into delay differential equations model. Finally, by considering space variations and consequently studying partial differential equations model.

Not applicable

Conflict of interest: The authors declare that they have no conflict of interest.

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