Modeling and Stability Analysis of Within-Host IAV/SARS-CoV-2 Coinfection with Antibody Immunity

: Studies have reported several cases with respiratory viruses coinfection in hospitalized patients. Inﬂuenza A virus (IAV) mimics the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) with respect to seasonal occurrence, transmission routes, clinical manifestations and related immune responses. The present paper aimed to develop and investigate a mathematical model to study the dynamics of IAV/SARS-CoV-2 coinfection within the host. The inﬂuence of SARS-CoV-2-speciﬁc and IAV-speciﬁc antibody immunities is incorporated. The model simulates the interaction between seven compartments, uninfected epithelial cells, SARS-CoV-2-infected cells, IAV-infected cells, free SARS-CoV-2 particles, free IAV particles, SARS-CoV-2-speciﬁc antibodies and IAV-speciﬁc antibodies. The regrowth and death of the uninfected epithelial cells are considered. We study the basic qualitative properties of the model, calculate all equilibria and investigate the global stability of all equilibria. The global stability of equilibria is established using the Lyapunov method. We perform numerical simulations and demonstrate that they are in good agreement with the theoretical results. The importance of including the antibody immunity into the coinfection dynamics model is discussed. We have found that without modeling the antibody immunity, the case of IAV and SARS-CoV-2 coexistence is not observed. Finally, we discuss the inﬂuence of IAV infection on the dynamics of SARS-CoV-2 single-infection and vice versa.


Introduction
Coronavirus disease 2019  was detected in December 2019, in Wuhan, China during the season when influenza was still circulating [1]. COVID-19 is caused by a dangerous type of virus called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). According to the update provided by the World Health Organization (WHO) on 21 August 2022 [2], over 593 million confirmed cases and over 6.4 million deaths have been reported globally. SARS-CoV-2 is transmitted to people when they are exposed to respiratory fluids carrying infectious viral particles. The implementation of preventive measures such as physical and social distancing, using face masks, hand washing, disinfection of surfaces and getting vaccinated can reduce SARS-CoV-2 transmission. Eleven vaccines for COVID-19 have been approved by WHO for emergency use. These include Novavax, CanSino, Bharat Biotech, Pfizer/BioNTech, Moderna, Serum Institute of India (Novavax formulation), Janssen (Johnson & Johnson), Oxford/AstraZeneca, Serum Institute of India (Oxford/AstraZeneca formulation), Sinopharm and Sinovac [3]. SARS-CoV-2 is a singlestranded positive-sense RNA virus that infects the epithelial cells. SARS-CoV-2 can cause an acute respiratory distress syndrome (ARDS), which has high mortality rates, particularly in patients with immunosenescence [4]. Immunosenescence renders vaccination less effective and increases the susceptibility to viral infections [5].
Influenza viruses are members of the family of Orthomyxoviridae, which are negativesense RNA viruses. There are four distinct influenza viruses, A, B, C and D. Influenza A virus (IAV) can infect a wide range of species. IAV is a significant public health threat, resulting in 15-65 million infections and over 200,000 hospitalizations every year during seasonal epidemics in the United States [6]. IAV infects the uninfected epithelial cells of the host respiratory tract [7]. Both SARS-CoV-2 and IAV have analogous transmission ways, moreover, they have common clinical manifestations including dyspnea, cough, fever, headache, rhinitis, myalgia and sore throat [1]. Viral shedding usually takes place 5 to 10 days in influenza, whereas it does 2 to 5 weeks in COVID-19 [1]. Acute respiratory distress is less common in influenza than COVID-19 [1]. Deaths in influenza cases are less than 1%, while in cases of COVID-19 it ranges from 3% to 4% [1].
It was reported in [8] that 94.2% of individuals with COVID-19 were also coinfected with several other microorganisms, such as fungi, bacteria and viruses. Important viral copathogens include the respiratory syncytial virus (RSV), human enterovirus (HEV), human rhinovirus (HRV), influenza A virus (IAV), influenza B virus (IBV), human metapneumovirus (HMPV), parainfluenza virus (PIV), human immunodeficiency virus (HIV), cytomegalovirus (CMV), dengue virus (DENV), Epstein Barr virus (EBV), hepatitis B virus (HBV) and other coronaviruses (COVs), among which the HRV, HEV and IAV are the most common copathogens [9]. Several coinfection cases of COVID- 19 and influenza have been reported in [1,8,[10][11][12] (see also the review papers [13][14][15][16]). Based on two separate studies presented in [10,11], COVID-19-influenza coinfection did not result in worse clinical outcomes [10]. In addition, this condition reduced the mortality rate among COVID-19influenza coinfected patients. Coinfection with influenza virus in COVID-19 patients might render them less vulnerable to morbidities associated with COVID-19, and therefore, a better prognosis overall [11]. In [16], it was found that, although patients with IAV and SARS-CoV-2 coinfection did not experience longer hospital stays compared with those with a SARS COV-2 single-infection, they usually presented with more severe clinical conditions. Viral interference phenomenon can appear in case of infections with multiple competitive respiratory viruses. One virus may be able to suppress the growth of another virus [17][18][19]. Disease progression and outcome in SARS-CoV-2 infection are highly dependent on the host immune response, particularly in the elderly in whom immunosenescence may predispose to increased risk of coinfection [17].
Over the years, mathematical models have demonstrated their ability to provide useful insight to gain a further understanding of the dynamics and mechanisms of the viruses within a host level. These models may assist in the development of viral therapies and vaccines as well as the selection of appropriate therapeutic and vaccine strategies. Moreover, these models are helpful in determining the sufficient number of factors to analyze the experimental results and explain the biological phenomena [7]. Stability analysis of the model's equilibria can help researchers (i) to expect the qualitative features of the model for a given set of values of the model's parameters, (ii) to establish the conditions that ensure the persistence or deletion of this infection, and (iii) to determine under what conditions the immune system is stimulated against the infection. Mathematical models of within-host IAV single-infection have been developed in several works. Baccam et al. [20] presented the following IAV-single-infection with limited target cells: where X = X(t), I = I(t) and P = P(t) are the concentrations of uninfected epithelial cells, IAV-infected epithelial cells and free IAV particles, at time t, respectively. The model was fitted using real data from six patients infected with influenza [20]. Several works have been devoted to study IAV single-infection dynamics models (see the review papers [21][22][23][24]) by including the effect of innate immune response [20,25], adaptive immune response [26,27] and both innate and adaptive immune responses [5,7,[28][29][30]. Handel et al. [31] presented a mathematical model for within-host influenza infection under the effect of neuraminidase inhibitors drugs. The effect of a combination of neuraminidase inhibitors and anti-IAV therapies was addressed in [26]. In [26], the first equation of model (1) was modified by considering the target cell production and death as: where X(0) is the initial concentration of the uninfected epithelial cells. Model (1) was utilized to characterize the dynamics of SARS-CoV-2 within a host in [32]. Li et al. [33] used Equation (2) for the SARS-CoV-2 infection dynamics. A model with target-cell limited and a model with regrowth and death of the uninfected epithelial cells presented, respectively, in [32,33] were extended and modified by including (i) latently infected epithelial cells [32,[34][35][36], (ii) effect of immune response [37][38][39][40][41][42], (iii) effect of different drug therapies [35,43,44], and (iv) effect of time delay [45].
The model presented in [18,19] describes the competition between two respiratory viruses. However, the impact of the immune response against the two viruses was not modeled. Further, the regeneration and death of the uninfected epithelial cells were neglected. Furthermore, mathematical analysis of the model was not studied. Therefore, the aim of the present paper is to develop a within-host IAV/SARS-CoV-2 coinfection model with immune response. The model is a generalization of the model presented in [18,19] by incorporating (i) the regrowth and death of the uninfected epithelial cells, and (ii) the impact of SARS-CoV-2-specific antibody and IAV-specific antibody. We study the basic qualitative properties of the proposed model, calculate all equilibria and investigate the global stability of the equilibria. We support our theoretical results via numerical simulations. Finally, we discuss the obtained results.
Our proposed model can be useful to describe the within-host dynamics of coinfection with two or more viral strains, or coinfection of SARS-CoV-2 (or IAV) and other respiratory viruses. Moreover, the model may help to predict new treatment regimens for viral coinfections.

Model Formulation
In this section, we present an IAV/SARS-CoV-2 coinfection dynamics model. The dynamics of IAV/SARS-CoV-2 coinfection is presented in the diagram  The uninfected epithelial cells are the target for both SARS-CoV-2 and IAV [18,20,32]. A3 The uninfected epithelial cells are regenerated and die at rates λ and αX, respectively [33,40,42,56]. A4 The SARS-CoV-2-specific antibodies proliferate at rate σ Z VZ, decay at rate µ Z Z and neutralize the SARS-CoV-2 particles at rate κ V VZ [45,57]. A5 The IAV-specific antibodies proliferate at rate σ M PM, decay at rate µ M M and neutralize the IAV particles at rate κ P PM [56].

Basic Qualitative Properties
In this section, we study the basic qualitative properties of system (3). We establish the nonnegativity and boundedness of the system's solutions to ensure that our model is biologically acceptable. Particularly, the concentrations of the model's compartments should not become negative or unbounded. Proof. We have thaṫ This guarantees that (X(t), Y(t), I(t), V(t), P(t), Z(t), M(t)) ∈ R 7 ≥0 for all t ≥ 0 when (X(0), Y(0), I(0), V(0), P(0), Z(0), M(0)) ∈ R 7 ≥0 . Let us define Then, This proves the boundedness of the solutions.

Equilibria
In this section, we are interested in the conditions of existence of the system's equilibria. Moreover, we derive a set of threshold parameters which govern the existence of equilibria. At any equilibrium Ξ = (X, Y, I, V, P, Z, M), the following equations hold: Solving Equations (4)-(10), we obtain eight equilibria.
In terms of 2 , we can write Therefore, Ξ 2 exists if 2 > 1.
In summary, we have eight threshold parameters which determine the existence of the model's equilibria

Global Stability
Stability analysis is at the heart of dynamical analysis. Only stable solutions can be noticed experimentally. Therefore, in this section we examine the global asymptotic stability of all equilibria by establishing suitable Lyapunov functions [58] and applying the Lyapunov-LaSalle asymptotic stability theorem (L-LAST) [59][60][61]. The following arithmetic-mean-geometric-mean inequality will be utilized: Let a function Λ j (X, Y, I, V, P, Z, M) andΩ j be the largest invariant subset of The following result suggests that when 1 ≤ 1 and 2 ≤ 1, both IAV and SARS-CoV-2 infections are predicted to die out regardless of the initial conditions (any disease stages).
The following result suggests that, when 1 > 1, 2 / 1 ≤ 1 and 3 ≤ 1, the SARS-CoV-2 single-infection with inactive immune response is always established regardless of the initial conditions.
The result of the following theorem suggests that, when 2 > 1, 1 / 2 ≤ 1 and

Proof. Define
We calculate dΛ 3 dt as: Then, simplifying Equation (16), we obtain: Using the equilibrium conditions for Ξ 3 : we obtain, Using inequality (12) and 5 ≤ 1, we obtain dΛ 3 dt ≤ 0 for all X, Y, V, P, M > 0. Further, Further, the trajectories of system (3) tend toΩ 3 which has elements with V = V 3 and P = 0. Then,V = 0 andṖ = 0. The fourth and fifth equations of system (3) provide In the following theorem, we show that when 4 > 1 and 6 ≤ 1, the IAV singleinfection with active immune response is always established regardless of the initial conditions.

Proof.
Define a function Λ 7 as: Calculating dΛ 7 dt as: We collect the terms of Equation (22) as: Using the equilibrium conditions for Ξ 7 : we obtain, Using inequality (12), we obtain dΛ 7 dt ≤ 0 for all X, Y, I, V, P > 0, where dΛ 7 dt = 0 when X = X 7 , Y = Y 7 , I = I 7 , V = V 7 and P = P 7 . The solutions of system (3) tend toΩ 7 which includes element with V = V 7 and P = P 7 which givesV =Ṗ = 0, and from the fourth and fifth equations of system (3), we obtain: Therefore,Ω 7 = {Ξ 7 } and by employing L-LAST, we obtain Ξ 7 is G.A.S. Based on the above findings, we summarize the existence and global stability conditions for all equilibrium points in Table 1. Table 1. Conditions of existence and global stability of the system's equilibria.

Numerical Simulations
The global stability of the system's equilibria will be illustrated numerically. In addition, we make a comparison between single-infection and coinfection. We use the values of the parameters presented in Table 2. Some values of parameters are taken from studies for SARS-CoV-2 single-infection and IAV single-infection, while other values are assumed just to perform the numerical simulations. To the best of our knowledge, until now there is no available data (e.g., the concentrations of SARS-CoV-2, IAV, antibodies, etc.) from SARS-CoV-2 and IAV coinfection patients. Therefore, estimating the parameters of the coinfection model is still open for future work.

Stability of the Equilibria
In this subsection, we support our global stability results provided in Theorems 1-8 by showing that the solutions of system (3) with any chosen initial conditions (any IAV/SARS-CoV-2 coinfection stage) will tend to one of the eight equilibria. Let us solve system (3) with three different initial conditions (states) as: Selecting the values of β V , β P , σ Z and σ M leads to the following situations: Situation 1 (Stability of Ξ 0 ): β V = 0.001, β P = 0.001, σ Z = 0.01 and σ M = 0.02. For these values of parameters, we have 1 = 0.0909 < 1 and 2 = 0.2 < 1. Figure 2 shows that the trajectories tend to the equilibrium Ξ 0 = (10, 0, 0, 0, 0, 0, 0) for all initials C1-C3. This demonstrates that Ξ 0 is G.A.S. based on Theorem 1. In this situation, both SARS-CoV-2 and IAV will be removed.
Situation 7 (Stability of Ξ 6 ): β V = 0.04, β P = 0.05, σ Z = 0.01 and σ M = 0.05. We compute 6 = 2.0202 > 1, 8 = 0.627 < 1 and 2 / 1 = 2.75 > 1. We find that the equilibrium Ξ 6 = (2.75, 2.3, 0.55, 2.3, 0.8, 0, 4.38) exists. Further, the numerical solutions outlined in Figure 8 show that Ξ 6 is G.A.S., and this boosts the result of Theorem 7. In this situation, a coinfection with SARS-CoV-2 and IAV is attained where only the IAV-specific antibody is activated. In this case, the concentration of the SARS-CoV-2 particles tends to a value less than or equal to µ Z σ Z = 5, and then the SARS-CoV-2-specific antibody will be deactivated. On the other hand, the activity of IAV-specific antibodies reduces the growth of IAV, and this leads to the coexistence of the two viruses.
For more confirmation, we investigate the local stability of the system's equilibria. Calculating the Jacobian matrix J = J(X, Y, I, V, P, Z, M) of system (3) as: At each equilibrium, we compute the eigenvalues λ j , j = 1, 2, . . . , 7 of J. If Re(λ j ) < 0, j = 1, 2, . . . , 7, then the equilibrium point is locally stable. We select the parameters β V , β P , σ Z and σ M as given in situations 1-8; then, we compute all nonnegative equilibria and the accompanying eigenvalues. Table 3 outlined the nonnegative equilibria, the real parts of the eigenvalues and whether or not the equilibrium point is stable. We found that the local stability agrees with the global one. Table 3. Local stability of nonnegative equilibria Ξ i , i = 0, 1, . . . , 7.

Comparison Results
In this subsection, we present a comparison between the single-infection and coinfection.

Influence of IAV infection on the dynamics of SARS-CoV-2 single-infection
Here, we compare the solutions of model (3) and the following SARS-CoV-2 singleinfection model: We fix parameters β V = 0.09, β P = 0.05, σ Z = 0.5 and σ M = 0.9 and select the initial state as:

Influence of SARS-CoV-2 infection on the dynamics of IAV single-infection
To examine the impact of SARS-CoV-2 infection on IAV single-infection, we compare the solutions of model (3) and the following IAV single-infection model: We fix parameters β V = 0.095, β P = 0.08, σ Z = 0.9 and σ M = 0.95 and consider the following initial condition:

Discussion
IAV and SARS-CoV-2 coinfection cases were reported in some works (see [1,8,10,11]). Therefore, it is important to understand the within-host dynamics of this coinfection. In this paper, we develop and examine a within-host IAV/SARS-CoV-2 coinfection model. We studied the basic and global properties of the model. We find that the system has eight equilibria, and their existence and global stability are governed by eight threshold parameters ( i , i = 1, . . . , 8). We proved the following: (I) The infection-free equilibrium Ξ 0 always exists. It is G.A.S. when 1 ≤ 1 and 2 ≤ 1. In this case, the patient is recovered from both IAV and SARS-CoV-2 infections. From a control viewpoint, making 1 ≤ 1 and 2 ≤ 1 will be a good strategy. This can be achieved by reducing the parameters β V and β P (or κ V and κ P ). Let V ∈ [0, 1] and P ∈ [0, 1] be the effectiveness of the antiviral drugs for SARS-CoV-2 and IAV, respectively. Then, the parameters β V and β P will be changed to (1 − V )β V and (1 − P )β P . Moreover, 1 and 2 become To make 1 ≤ 1 and 2 ≤ 1, the effectiveness V and P have to satisfy (II) The SARS-CoV-2 single-infection equilibrium without antibody immunity Ξ 1 exists if 1 > 1. It is G.A.S. when 1 > 1, 2 / 1 ≤ 1 and 3 ≤ 1. This case leads to the situation of a patient who is only infected by SARS-CoV-2 with inactive immune response. As we will see below, if both SARS-CoV-2-specific antibody and IAV-specific antibody immunities are not activated against the two viruses, then according to the competition between the two viruses, SARS-CoV-2 may be able to block the IAV infection.
(III) The IAV single-infection equilibrium without antibody immunity Ξ 2 exists if 2 > 1. It is G.A.S. when 2 > 1, 1 / 2 ≤ 1 and 4 ≤ 1. This case leads to the situation of a patient who is only infected by IAV with unstimulated immune response. Then, IAV may be able to block the SARS-CoV-2 infection.
(IV) The SARS-CoV-2 single-infection equilibrium with stimulated SARS-CoV-2-specific antibody immunity Ξ 3 exists if 3 > 1. It is G.A.S. when 3 > 1 and 5 ≤ 1. This point represents the situation of a SARS-CoV-2 single-infection patient with active SARS-CoV-2-specific antibody immunity. Despite the activity of antibodies against the SARS-CoV-2 particles, the SARS-CoV-2 may be able to block the IAV.
(V) The IAV single-infection equilibrium with stimulated IAV-specific antibody immunity Ξ 4 exists if 4 > 1. It is G.A.S. when 4 > 1 and 6 ≤ 1. This point represents the case of an IAV single-infection patient with active IAV-specific antibody immunity. Despite the activity of antibodies against the IAV particles, the IAV may be able to block the SARS-CoV-2.
We discussed the influence of IAV infection on SARS-CoV-2 single-infection dynamics and vice versa. We found that the concentration of free IAV or SARS-CoV-2 particles cells tend to be the same value in both single-infection and coinfection. This agrees with the work Ding et al. [10] which reported that IAV/SARS-CoV-2 coinfection did not result in worse clinical outcomes [10]. In addition, the spread of seasonal influenza can increase the likelihood of coinfection in patients with COVID-19 [8].
We note that the case of IAV and SARS-CoV-2 coexistence does not appear. In the recent studies presented in [1,8,10,11], it was recorded that some COVID-19 patients were coinfected with IAV. Therefore, neglecting the immune response may not describe the coinfection dynamics accurately. This supports the idea of including the immune response into the IAV/SARS-CoV-2 coinfection model, where the case of IAV and SARS-CoV-2 coexistence is observed.

Conclusions
Mathematical models are frequently used to understand the complex behavior of biological systems. In this paper, we formulated an IAV and SARS-CoV-2 coinfection model within a host. The model is a seven-dimensional nonlinear ODEs which describes the interaction between uninfected epithelial cells, SARS-CoV-2-infected cells, IAV-infected cells, free SARS-CoV-2 particles, free IAV particles, SARS-CoV-2-specific antibodies and IAVspecific antibodies. The regrowth and death of the uninfected epithelial cells are considered. We first examined the nonnegativity and boundedness of the solutions; then we calculated the model's equilibria and established their existence in terms of eight threshold parameters. We proved the global stability of all equilibria by constructing Lyapunov functions and applying the Lyapunov-LaSalle asymptotic stability theorem. We performed numerical simulations and demonstrated that they are in good agreement with the theoretical results. We discussed the effect of including the antibody immunity into the coinfection dynamics model. We found that including the antibody immunity in the coinfection model plays an important role in establishing the case of IAV and SARS-CoV-2 coexistence which is practically detected in many patients. Finally, we discussed the influence of IAV infection on the dynamics of SARS-CoV-2 single-infection and vice versa.
The model proposed in this research and its analysis shows three main biological states, (i) clearance of both IAV and SARS-CoV-2 particles, (ii) appearance of interference phenomenon, where one virus may be able to suppress the growth of another virus, and (iii) coexistence of the two viruses.
The model developed in this work can be improved by (i) utilizing real data to find a good estimation of the parameters' values, (ii) studying the effect of time delays that occur during infection or production of IAV and SARS-CoV-2 particles [45], (iii) considering viral mutations [65,66], (iv) considering the effect of treatments on the progression of both viruses, and (v) including the influence of Cytotoxic T-Lymphocytes (CTLs) in killing SARS-CoV-2infected and IAV-infected cells [40]. These research points need further investigations so we leave them to future works.