Global stability of secondary DENV infection models with non-specific and strain-specific CTLs

Dengue virus (DENV) is a highly perilous virus that is transmitted to humans through mosquito bites and causes dengue fever. Consequently, extensive efforts are being made to develop effective treatments and vaccines. Mathematical modeling plays a significant role in comprehending the dynamics of DENV within a host in the presence of cytotoxic T lymphocytes (CTL) immune response. This study examines two models for secondary DENV infections that elucidate the dynamics of DENV under the influence of two types of CTL responses, namely non-specific and strain-specific responses. The first model encompasses five compartments, which consist of uninfected monocytes, infected monocytes, free DENV particles, non-specific CTLs, and strain-specific CTLs. In the second model, latently infected cells are introduced into the model. We posit that the CTL responsiveness is determined by a combination of self-regulating CTL response and a predator-prey-like CTL response. The model's solutions are verified to be nonnegativity and bounded and the model possesses two equilibrium states: the uninfected equilibrium EQ0 and the infected equilibrium EQ⁎. Furthermore, we calculate the basic reproduction number R0, which determines the existence and stability of the model's equilibria. We examine the global stability by constructing suitable Lyapunov functions. Our analysis reveals that if R0≤1, then EQ0 is globally asymptotically stable (G.A.S), and if R0>1, then EQ0 is unstable while EQ⁎ is G.A.S. To illustrate our findings analytically, we conduct numerical simulations for each model. Additionally, we perform sensitivity analysis to demonstrate how the parameter values of the proposed model impact R0 given a set of data. Finally, we discuss the implications of including the CTL immune response and latently infected cells in the secondary DENV infection model. Our study demonstrates that incorporating the CTL immune response and latently infected cells diminishes R0 and enhances the system's stability around EQ0.


Introduction
The Dengue virus (DENV) is an arbovirus transmitted by mosquitoes, particularly Aedes albopictus and Aedes aegypti, leading to the onset of dengue fever.Symptoms encompass severe headache, high fever (40 • C), rash, muscle and joint pains, nausea, pain behind the eyes, swollen glands, and vomiting [1].With almost half of the global population at risk, an estimated 100 to 400 million infections occur annually [1], predominantly in tropical and subtropical regions.
DENV targets various cells, including monocytes, macrophages, dendritic cells, endothelial, and epithelial cells [2].The interplay of innate and adaptive immune responses is crucial in combating DENV [3].The innate immune response, acting as the initial defense line, involves early activation of Interferon (IFN) and natural killer (NK) cells, contributing to the containment of DENV spread and clearance of infected cells.
The adaptive immune response, albeit slower to activate, ultimately eliminates DENV from the body [4].B cells and cytotoxic T lymphocytes (CTLs) are integral components of this response.B cells produce antibodies to neutralize DENV particles, while CTLs target and eliminate DENV-infected cells.DENV comprises four distinct serotypes (DENV1-4), each with different genotypes [5].
Generally, infection with one DENV serotype confers protective immunity against that specific serotype but not against others [6].Primary infections result in lifelong immunity to the original DENV strain [7].In secondary infections with a different DENV serotype, two CTL immunities are triggered: non-specific CTLs from the primary infection and strain-specific CTLs against the new DENV serotype [7].

Mathematical models for DENV infection
Mathematical modeling serves as a valuable approach in comprehending DENV infection dynamics, offering a cost-effective alternative to experimental assessments.This method enables the assessment of interactions between viruses, host cells, and immune cells, providing insights into the intricate dynamics of viral infections.
Nikin-Beers and Ciupe [7] introduced a target cell-limited model for primary DENV infection under the effect of CTL immune response as: , where  = () is the concentration of CTLs.Parameters , , , , , , ,  and  are positive.
In [7], a target cell-limited model for the secondary DENV infection with non-specific and strain-specific CTLs was formulated as: Free DENV particles: Non-specific CTLs: Strain-specific CTLs: Here, it was assumed that CTLs specific to primary DENV infection are being produced by immunological memory.In model ( 1)-( 5) we noted the following: (i) The regeneration and death of the uninfected monocytes are not included.However, in the literature, several DENV infection models considered the regeneration and death of the uninfected monocytes (see e.g., [17], [19], [20], [21]).The population dynamics equation for the uninfected monocytes was given as: where  and  represent the regeneration and death rates of the uninfected monocytes, respectively.
(ii) Both non-specific CTLs and strain-specific CTLs have the same regeneration rate .However, these rates may be differ.
(iii) The death rates of the non-specific CTLs and strain-specific CTLs are equal.However, these rates may be not equal.
The paper aims to formulate two models for secondary DENV infection, incorporating non-specific and strain-specific CTLs.The second model extends the first one by introducing two classes of DENV-infected cells: latently infected cells (containing DENV but not actively producing them) and actively infected cells (producing DENV).

Model with non-specific and strain-specific CTLs
We formulate virus dynamics model with non-specific CTLs ( 1 ) and strain-specific CTLs ( 2 ): We note that in the absence of DENV infection, there are  1 ∕ 1 non-specific CTLs and  2 ∕ 2 strain-specific CTLs available with  1 and  2 being the sources [7].The initial conditions of system ( 6)- (10) are We can see that, the right-hand side functions of ( 6)-( 10) satisfy Lipschitz condition, then system ( 6)- (10) with initial (11) has a unique solution for  ≥ 0.

Preliminary results
First, we determine a compact region for the concentrations of the model's compartments to guarantee that our suggested model is biologically realistic.It is especially important to avoid negative or unbounded concentrations.
Then, the infected equilibrium  * ( * ,  * ,  * ,  * 1 ,  * 2 ) is present when  0 > 1.It is worth noting that the next-generation matrix approach [23] or the local stability of the uninfected equilibrium  0 can also be employed to determine  0 .This basic reproduction number is defined as the quantity of infected cells produced by one infected cell in its lifespan in a fully susceptible environment.□

Global stability
The global stability analysis is fundamental for comprehending and predicting the behavior of systems described by mathematical models.In this section we investigate the global stability of the two equilibria of system ( 6)- (10).The proof is derived from the method of constructing Lyapunov functions for viral infection models (see e.g., [24][25][26]).Define a function Θ  (, , ,  1 ,  2 ) and let Γ be the largest invariant subset of During this section, we utilize the arithmeticmean-geometric-mean inequality The following finding indicates that, independent of the beginning circumstances (any illness phases), the DENV infection is anticipated to die out when  0 ≤ 1.
(ii) The Jacobian matrix  1 =  1 (, , ,  1 ,  2 ) of system ( 6)-( 10) is calculated as: , and the characteristic equation at the equilibrium  0 is expressed as where  is the eigenvalue and Clearly if  0 > 1, then  0 < 0 and Eq. ( 21) has a positive root and hence  0 is unstable.□ According to the results shown below, regardless of the starting circumstances, the DENV infection always occurs when  0 > 1. )) )) .
along the trajectories of ( 6)- (10): Applying the equilibrium conditions we get and Then From the equilibrium conditions we have  * − ) .

Model with latency
Throughout this section, we consider that cells infected with the DENV virus can exist in two states, namely latent or active.We posit that a proportion  ∈ (0, 1) of DENV-infected cells transition to an active state, while the remaining fraction (1 − ) becomes latent.Let  = () represent the concentration of latently DENV-infected cells at time .The model incorporating latency can be expressed as: Latent DENV-infected cells activate at a rate of  and experience mortality at a rate of .

Sensitivity analysis
Sensitivity analysis holds a significant role in the study of dynamic systems, specifically in ecological and epidemiological studies [31].Among the important facets of this study is the sensitivity analysis of model parameters.Specific sensitivity indices for each parameter are provided, illustrating their roles in disease dynamics.A sensitivity analysis of various parameters is presented in this part versus   0 .
To conduct sensitivity analysis, we determine the normalized forward sensitivity index of a variable utilizing the formula which gives the sensitivity index of   0 with respect to the parameter , and we get sensitivity with respect to  ∶   = and It is apparent that parameters , , , , ,  1 and  2 have positive indices, indicating that   0 will increase as these parameters increase.In addition, the increase or decrease of the two parameters  1 and  2 has no effect on the value of   0 .Other parameters have negative sensitivity indices, indicating that as they increase,   0 decreases.
along the trajectories of ( 22)-( 27): The equilibrium conditions of  * imply Then, we get It follows that

Numerical simulations
In this section, we illustrate our theoretical findings for models (6)-( 10) and ( 22)-( 27) by conducting some numerical computations.To get the numerical solutions of the models, we utilize MATLAB's ODE45 solver.6)- (10) Within this subsection, we conduct numerical simulations to validate the global stability of equilibria for system (6)- (10), employing the values specified in Table 1.

Stability of equilibria
To demonstrate that the trajectories of the system with any starting point converge to an equilibrium point, the following various initial conditions are employed: is G.A.S.In this case, DENV will perish, and the number of uninfected monocytes will return to its normal level. .As per Lemma 2 and Theorem 2,  * exists and is G.A.S.
Let us compare the basic reproduction numbers for system (6)- (10) and system (40)-(42) as: Therefore, the presence of a CTL response diminishes the basic reproduction number,  0 , thereby enhancing the system's stabilizability around the uninfected equilibrium,  0 .To assess the impact of integrating latently infected cells into the model, we compare the basic reproduction numbers for systems ( 6)-( 10) and ( 22)-( 27  This implies that incorporating latently infected cells results in a reduction of the basic reproduction number, leading to increased system stabilizability around the uninfected equilibrium  0 .

Discussions and perspectives
In this paper, two secondary DENV infection models with non-specific and strain-specific CTLs were considered.The first model explains the interaction of five populations: uninfected monocytes, infected monocytes, free DENV particles, non-specific CTLs and strain-specific CTLs.Latently infected cells were included in the second model.We established the non-negativity and boundedness of solutions for the proposed model.Two equilibrium points emerged: the uninfected equilibrium,  0 , and the infected equilibrium,  * , which depend on the basic reproduction number,  0 , determining the model's dynamic behavior.For  0 ≤ 1,  0 is globally asymptotically stable (G.A.S), while for  0 > 1,  * exhibits G.A.S.Our theoretical findings were corroborated through numerical computations.Additionally, we conducted sensitivity analysis, examining how parameter values influence the basic reproduction number given a dataset.Furthermore, we examined the impact of incorporating CTL immune response and latently infected cells in the secondary DENV infection model, revealing a reduction in  0 and an increased stabilizability around  0 .
This paper has proposed a model where the in-host dynamics modeled at the small scale of particles is linked to the large scale of individuals.This multiscale modeling trend has been developed with a focus on COVID-19 epidemics, where the in-host dynamics

Fig. 3 .
Fig. 3.The sensitivities of the model parameters that influence the basic reproduction number   0 of the system (22)-(27).The parameters in Figure (a) exhibit sensitivity indexes higher than those in Figure (b).
The key contributions of our study encompass: (C1) Considering the regeneration rate and death rate of uninfected monocytes; (C2) Assuming distinct regeneration rates for non-specific CTLs and strain-specific CTLs; (C3) Considering different death rates for non-specific CTLs and strain-specific CTLs; (C4) Investigating the fundamental and global properties of the models; (C5) Conducting sensitivity analysis; (C6) Validating theoretical findings through numerical simulations.

Table 2
Sensitivity index of