Modeling and analysis of a periodic delays spatialdiffusion HIV model with three-stage infection process

Considering the antiviral drugs can act on the fusion, reverse transcription, and budding stages of HIV infected cells, in this paper, we formulate a two periodic delays heterogeneous space diﬀusion HIV model with three-stage infection process to study the eﬀects of periodic antiviral treat-ment and spatial heterogeneity on HIV infection process. We ﬁrst study the well-posedness of the full system incorporates the ultimate boundedness, the global existence of the solution and the existence of strong global attractor for ω -periodic semiﬂow for the model. We derive the basic reproduction number R 0 , which is deﬁned as the spectral radius of the next generation operator. We further prove that R 0 is a threshold for the elimination and persistence of HIV infection by comparison principle and persistence theory for nonautonomous system, i.e., the HIV infection will be eliminated when R 0 < 1 and will persist when R 0 > 1. In the spatial homogeneous case, we obtain the explicit expression of R 0 and show the global attractivity of the positive steady state by using the ﬂuctuation method. Some numerical simulations are conducted to illustrate the theoretical results and our works suggest that both spatial heterogeneity and periodic delays caused by periodic antiviral therapy have a remarkable impact on the progression of HIV infection and should not be overlooked in clinical treatment process.


Introduction
HIV (Human Immunodeficiency Virus) is a viral infection that affects the immune system, gradually leading to a weakened defense mechanism against various infections and diseases.It is primarily transmitted through sexual contact, blood transfusion, needle sharing, and from an infected mother to her child during pregnancy, childbirth, or breastfeeding [1].Upon entering the body, HIV targets and destroys a specific type of white blood cells called CD4 T lymphocytes, which play a crucial role in coordinating the immune response.As the virus replicates, the CD4 cell count declines, resulting in a weakened immune system.This progressive immune system deterioration leads to a condition known as AIDS (Acquired Immunodeficiency Syndrome), characterized by severe vulnerability to infections and certain types of cancers.One of the reasons HIV is a challenging infection to combat is its ability to evade the immune system and integrate its genetic material into the DNA of the infected cells [2].

P. Wu
This integration makes it difficult for the body's immune system to recognize and eliminate the virus effectively.Additionally, HIV has a high mutation rate, resulting in different strains of the virus with varying characteristics and response to treatments.Although there is currently no cure for HIV, significant advancements have been made in the development of antiretroviral therapy (ART) [3].ART involves a combination of medications that can suppress viral replication, allowing the immune system to recover and function more effectively.Early diagnosis and initiation of ART have been instrumental in improving the quality of life for people living with HIV and reducing the risk of transmitting the virus to others.Prevention remains a critical aspect of combating HIV infection.Strategies such as practicing safe sex, using sterile needles for injections, ensuring blood and organ donation screening, and providing antiretroviral prophylaxis to pregnant women can significantly reduce the transmission of HIV [4].Public awareness campaigns, education, and access to testing and treatment services are essential in controlling the spread of the virus and supporting individuals affected by HIV.
HIV undergoes several critical processes within the host cells to replicate and spread throughout the body.These processes include cell fusion, reverse transcription, and budding (see Figure1).Cell Fusion: HIV primarily infects CD4 T lymphocytes and macrophages, which play essential roles in the immune response.The first step in the HIV replication cycle is the process of cell fusion [5].The virus binds to specific receptors on the surface of the target cell, primarily the CD4 receptor and a co-receptor called CCR5 or CXCR4.This binding triggers a conformational change in the viral envelope glycoprotein (gp120), allowing it to interact with the host cell membrane [6].Upon binding, the viral envelope fuses with the host cell membrane, resulting in the release of the viral core into the cytoplasm of the host cell.This fusion process allows the viral genetic material to enter the host cell, initiating the subsequent steps of the replication cycle.Reverse Transcription: Once inside the host cell, the viral core contains two identical copies of the single-stranded RNA genome, along with several enzymes, including reverse transcriptase.Reverse transcription is the process by which the viral RNA genome is converted into double-stranded DNA [7].Reverse transcriptase synthesizes a complementary DNA strand using the viral RNA as a template.This process involves the synthesis of a complementary DNA strand (cDNA) from the viral RNA genome, followed by degradation of the RNA template and synthesis of the second DNA strand to generate a double-stranded DNA molecule.The newly synthesized viral DNA, known as the proviral DNA, is then transported into the nucleus of the host cell, where it integrates into the host cell's chromosomal DNA with the help of another viral enzyme called integrase [8].Integration allows the viral genetic material to become a permanent part of the host cell's genome.During budding, the viral structural proteins, such as Gag and Gag-Pol, are synthesized and transported to the host cell's plasma membrane.These structural proteins assemble at the inner surface of the plasma membrane, encapsulating the viral RNA and enzymes, forming a new viral particle called a virion.As the assembly process progresses, the host cell's plasma membrane wraps around the viral particle, acquiring its envelope from the cell membrane.This budding process eventually leads to the release of the mature virion from the host cell, allowing it to infect new target cells and continue the cycle of replication [9].A number of researchers have studied the HIV infection process through modeling and analysis.We mainly introduce some classic research work here.Wang and Li [10] formulated a mathematical model that describes HIV infection of CD4+ T cells and studied the global dynamics of the system.Perelson and Ribeiro [11] reviewed the developments in HIV modeling, emphasizing quantitative findings about HIV biology uncovered by studying acute infection.Considering the impact of latent infected cell on the process of HIV infection, Rong and Perelson [12] established a compartment model to study HIV persistence, the latent reservoir and viral blips.To investigate the influence of time delay on the HIV infection, Culshaw and Ruan [13] formulated a time-delayed HIV infection model to study the threshold dynamical behaviors of the system.Wu and Zhao [14] investigated the dynamics of a two infection routes and two viral strains HIV infection model.Guo et al [15] developed and analyzed a mathematical HIV infection model that includes sequential cell-free virus infection and cell-to-cell transmission.
Antiviral therapy is also an important research topic in the field of HIV dynamics modeling.Hosseoni and Ganhann formulated a multi-scale HIV infection model to study APOBEC3G-Based antiretroviral therapy [16].Suryawanshi and Hoffmann [17] presented a multi-scale mathematical modeling framework to study antiviral therapeutic opportunities in targeting HIV-accessory proteins.Rong and Perelson [18] developed a mathematical model that considers latently infected cell activation in response to stochastic antigenic stimulation to study viral blips and latent reservoir persistence in HIV-infected patients on potent therapy.Considering the age since infection and spatial factors, Wu and Zhao [19] formulated an age-structured HIV/AIDS epidemic model with HAART(highly active antiretroviral therapy) and spatial diffusion and studied the threshold dynamics of HIV transmission.It is worth mentioning that Wang et al [20] formulated and investigated the global dynamics of a spatial diffusion HIV model with 2-LTR and periodic therapy.In the paper, they assumed the drug efficacy function is periodic, which is characterized by a quick rise to a maximum soon after drug intake.Afterwards, Wu et al [21] investigated the evolution dynamics of a time delays heterogeneity spatial diffusion HIV latent infection model with periodic therapies and two strains.However, the above research works just consider the constant time delay, few time-dependent periodic delays spatial heterogeneity diffusion models are formulated to investigate the infection process of HIV within the host.In fact, on the one hand, antiviral drugs can act on the three stages of HIV infected cells: fusion, reverse transcription, and budding.On the other hand, HIV infected individuals periodically consume antiviral drugs to prevent the fusion of the HIV virus with cells in the body, reverse transcription of the virus, and virus budding.Due to the periodicity of antiviral drug efficacy, it is inevitable that the time delay in these three processes is time dependent, that is, the periodic function of time (more details are discussed in Sect.2).To the best of our knowledge, thus far, the global dynamics of a periodic delay reaction-diffusion disease model have only been explored by Zhao et al. [22,23].Building upon the insights from their work, the objective of this paper is to investigate the threshold dynamics of an HIV latent infection model in a heterogeneous environment, specifically focusing on the challenges associated with periodic nonlocal infection and periodic delays, which are relevant to HIV periodic antiviral therapy.
We organize the paper as follows.In Sect.2 and Sect.3, we formulate the model and study its wellposedness, respectively.In Sect.4,we derive the basic reproduction number R by the definition of the next generation operator.We first prove the threshold dynamics for the full system in terms of R 0 , and then show the corresponding spatial homogeneity diffusion system with constant time delays admits a globally attractive positive equilibrium in Sect.5.In Sect.6, we conduct some numerical simulations to verify the theoretical results and discuss the impact of some key factors on HIV infection process.The paper concludes with a brief summary in Sect.7.

Model simulation
According to the literature [24], the process of HIV infection can be mainly divided into three stages: the first stage is that the adsorption has been reverse transcribed; The second stage is integration and transcription; In the third stage, core particle assembly and budding (see Figure 1).In clinical treatment, antiviral therapy can directly address every step of virus replication and infection, thereby interrupting the replication and spread of the virus.Due to the periodicity of antiviral treatment, we can assume that all parameters of the model are continuous function with a time t period of 12h [25].
In order to finely characterize the infection process of HIV within-host, we first characterize the reverse transcription process.Let p be the completion of reverse transcription, and the time-dependent growth rate of reverse transcription process is c 1 (t).We assume that the reverse transcription completion degree p = p I 1 = 0 when susceptible cells T (x, t) turns into infected cells at the beginning, parameter P I represents the reverse transcription completion degree of the infected cells that have no ability to proliferate I 1 (x, t) into latent infected cells L(x, t) and infected cells that have the ability to proliferate I(x, t).Let ω 1 (p, x, t) be the density distribution of the infected cells with the reverse transcription completion degree p at location x and time t.Record Q 1 (x, t) = c 1 (t)ω 1 (p, x, t) as the density distribution of newly infected cells that have completed the reverse transcription process and successfully integrated viral RNA into cell DNA.Furthermore, consider J 1 (ω 1 , x, t) as the flux, indicating the movement of hosts with infection development level p, in the direction of increasing p, at a specific location x and time t.Similar to [26], we have the following where D I 1 and µ I 1 represent the diffusion coefficient and natural death rate of I 1 (x, t).Since J 1 (p, x, t) = c 1 (t)ω 1 (p, x, t), we have From the standpoint of biological views, we assume that the boundary condition of system (2.1) is ω 1 (0, x, t) = β(x, t)S(x, t)V 1 (x, t)/c 1 (t), where β(x, t) is the infection rate of infectious free virus V 1 (x, t).Now, we set η I := h I (t) := p I + t 0 c 1 (s)ds, and define ω1 (p, x, η , where h −1 (η I ) is the inverse function of h I (t).Then, we have Set w(x, s) = ω1 (x, s + p − η I , s), then we obtain where G I 1 (x, y, t, t 0 ) is the Green function with respect to ∂u ∂t = ∆u − µ I 1 (•, t)u subjects to no-flux boundary condition in the bounded domain Ω.Furthermore, we have ω1 (x, p, η I ) = Ω G I 1 (h −1 (η I ), h −1 (x, y, η I − p + p I )ω 1 (y, p I , η I − p + p I )dy.
We define τ 1 (p, t) as the time it takes for an infected cell to progress from reverse transcription completion level p I to reverse transcription completion level p at time t.It follows from dp/dt = c 1 (t) that Set s = h I (r), then we can obtain that µ I 1 (s)ds.

P. Wu
Similarly, we denote q as the budding level of free virus produced by the infected cells with proliferate ability, and the time-dependent growth rate of budding progress is denoted as c 2 (t).Assuming that the budding completion rate q = q I 2 = 0 when transitioning from I(x, t) to I 2 (x, t) at the beginning, and q = q v when budding infectious free virus V 1 (x, t) and no-infectious virus V 2 (x, t) from I 2 (x, t).Set ω 2 (q, x, t) be the density distribution of infected cells whose budding completion is q at time t.Record Q 2 (x, t) = c 2 (t)ω 2 (q v , x, t) as the distribution of infected cells that have completed the budding process and successfully released free viruses.Hence, we have where G I 2 (x, y, t, t 0 ) represents the Green function with respect to ∂u ∂t = D I 2 ∆u−µ I 2 (•, t)u with respect to no-flux boundary condition.Parameter K represents the burst size of I(x, t) and ϵ(t) represents the apoptosis rate of I(x, t), time-delayed τ 2 (t) is the time it takes the infected cells I(x, t) to complete the budding process, it satisfies with q 3), we have the following periodic delays reaction-diffusion HIV infection model with space heterogeneity where (2.4) are decoupled from the other equations.In addition, we denote (w 1 (x, t), w 2 (x, t), w 3 (x, t), w 4 (x, t)) = (S(x, t), L(x, t), I(x, t), V 1 (x, t)) for the sake of simplicity. it suffices to investigate the following system: 3 The well-posedness of system (2.5) In this section, we are devoted to studying the well-posedness of system (2.5).For this purpose, we give the following assumption Assumption 3.1.For system (2.5), we assume (1) The diffusion coefficients D S , D L , D I , D V 1 are all positive constants; (2) Function Λ(x, t), µ I (x, t), µ S (x, t), µ L (x, t), c(x, t), ϵ(x, t), β(x, t), γ(x, t) and η(x, t) are all Hölder continuous and positive on Ω × R, and ω−periodic in t, where ω = 24 h, and hence τ j (t)(j = 1, 2) are both ω−periodic functions with respect to time t.

Now, we define
Clearly, M is a Banach space.We define the positive cones of the spaces M and M as M+ := (R 4 + , Ω) and M+ := C([−τ , 0], M+), respectively.Thus, both (M, M+) and (M, M+) are ordered spaces.For a function w : [−τ , ξ) → M with ξ > 0, we set . Given that µ(x, t) is ω-periodic in time t, according to Lemma 6.1 [27], we know that T (s + ω, t + ω) = T (s, t) for (s, t) ∈ R 2 with t > s.Furthermore, we establish that the operator T (s, t) is strongly positive and compact.Define operator Then we can rewrite system (2.5) as the following abstract Cauchy problem Based on Theorem 1 [28], we can show that system (3.6)always admits a unique solution, w( Based on the definition of τ j (t)(j = 1, 2), we have the compatibility conditions of I 1 and I 2 as follow: Define the following space It is obvious that system (2.4) admits a unique solution Based on Corollary 4 [28], we can verified that U (•, t, ϕ) ≥ 0 on the maximal existence interval.According to (3.7), we have that and hence we know that I j (•, t) ≥ 0(j = 1, 2) for t ∈ [0, t f ].To prove the ultimate boundedness of the solution of system (2.5), we set N (t) = Ω (K(S(x, t) )dx, then it follows from system (2.5) that We have that N (t) ≤ ΛK|Ω|/µ := N for t > t 1 > 0 by using the comparison principle, where 4 The basic reproduction number of system (2.5) We are mainly arm to deriving the functional expression of the basic reproduction number of system (2.5) by applying the method mentioned in [22].
always admits a globally attractive positive ω−periodic solution S * (x, t).
Hence, the cumulative number of infectious cells at time t resulting from all the previously active viruses up to time t can be described as follows: In the next, we define Let E, F be two bounded linear operators on C ω (R, X) as follows Then we have S = E • F and S = F • E, it leads to S and S have the same spectral radius.Based on the definition of the next generation operator, we have the basic reproduction number of system (2.5) as follows R 0 := r(S ) = r(S), where r(•) is the spectral radius.
Define phase space and its positive cone as follow For ordered Banach space (H, H + ) and function v : By the method of steps as shown in Lemma 3.2 [26], we have the following result Lemma 4.4.If φ ∈ H + , then system (4.9) with initial values v 0 = φ has a unique non-negative solution v(•, t, φ) on [0, +∞) .
In the next, we show that lim t→∞ w 1 (x, t, φ) = S * (x, t) by applying the definition of internally chain transitive sets [34].Since w 1 (x, t, φ) is the solution of a nonautonomous system which is asymp-totic to system (4.8),we can verify that system (4.8)generates a solution semiflow P(t), t ≥ 0 on C([−τ (0), 0], Q + ).Then P is Poincáre map of system (4.8) and its has a global attractor in C([−τ (0), 0], Q + ).We rerrange ω-periodic semiflow L (t) as Based on the property of operator L (t), we know that 0 / ∈ J, 0 / ∈ J. Hence, we can obtain J is an internally chain transitive set for L(ω) from lemma 1.2.1 [34], this leads to that J is an internally transitive chain set for P(ω).Define is the stable set of w 0 1 .Hence, by Theorem 1.2.1 [34], we have This proof of statement ( 1) is finidhed.Now, we prove that statement (2).In this case, we know R > 1, r(P(ω)) > 1 and µ = ln r(P(ω) ω > 0. To complete the proof, we first introduce some denotations which will be used in later.We set ω(φ) := the omega limit set of the positive orbit γ + = {L n (ω)φ : ∀n ∈ N}.
We divide the proof into three steps.

The dynamical behaviors of system (2.5)
For the above set of model parameters, we can calculate the basic reproduction number numerically and have R 0 = 2.533 > 1 by using the method mentioned in [33], which implies that HIV infections will be persist within the host.In fact, in Fig. 2, we can see that the solution surface of system (2.5) moves away the zero plane as time approaches infinity.This is in lines with Theorem 5.2 (2).When we choose β(x) = 4.2 × 10 −6 (1 + 0.35 sin 2x) and the other parameters values are the same as in Table 1, we can have R 0 = 0.843 < 1.In Fig. 3, it is showed that the solution surface of system (2.5) approaches zero plane as time goes infinity.This indicates that the HIV infections will be eliminated within-host eventually, which is in lines with Theorem 5.2 (1).  1.  1.

The impact of time delays and spatial heterogeneity on HIV infection process
In this subsection, we are devoted to investigate the impact of time delays τ j (t) and spatial heterogeneity on HIV infection process.To this end, we first study the influence of time delays τ j (t) and spatial heterogeneity on R 0 .We set the space-averaged infection rate β 0 := 1 |Ω| Ω β(x)dx ≈ 0.0017 and the time-averaged time-delayed τ 0 j := 1 ω ω 0 τ j (t)dt, j = 1, 2. Then we have τ 0 1 ≈ 1.2632, τ 0 2 ≈ 1.3421.To investigate the influence of spatial heterogeneity on the progression of HIV infection, we depict in Figure 4 the impact of model parameters on the basic reproduction number R 0 under two scenarios: β(x) and β 0 .Specifically, as shown in Figure 4 (a), we can observe that the functional form of β(x) has a significant effect on the value of R 0 , indicating that the spatial factor cannot be ignored in influencing R 0 .Furthermore, from Figures 4 (b), (d), (e) and (f), it can be seen that R 0 is a decreasing function of c(x), η(x) and γ(x), while from Figure 4 (c), it can be seen that R 0 is an increasing function of ϵ(x).It is worth mentioning that a common phenomenon in Figures 4 (a)-(f) is that the value of R 0 under the β(x) scenario is significantly larger than that under the β 0 scenario.This implies that ignoring the impact of spatial heterogeneity on R 0 would seriously underestimate the risk of HIV infection outbreaks within the host, which is clearly detrimental to the treatment of HIV-infected individuals and the control of HIV transmission among the population.Similarly, in Figure 5, we demonstrate the influence of model parameters on the basic reproductive number R 0 under two scenarios: τ j (t) and τ 0 j , j = 1, 2. Likewise, from Figures 5 (a)-(f), we can observe that the value of R 0 under the τ j (t)(j = 1, 2) scenarios is significantly larger than that under the τ 0 j scenario.This indicates that compared to the constant time delay model, the model considering timedependent delays τ j (t) caused by periodic antiviral therapy can more accurately reflect the impact of periodic antiviral treatment on the progression of HIV infection.Moreover, compared to the timedependent delay τ j (t) scenario, the constant time delay scenario significantly underestimates the value of R 0 , which can lead to an underestimation of the possibility of HIV infection outbreaks within the host.This is highly unfavorable for HIV clinical treatment.
To visually demonstrate the impact of the two contrasting scenarios, β(x) vs. β 0 and τ j (t) vs. τ 0 j , on the progression of HIV infection within the host, we present the three-dimensional and twodimensional profile plots of the solutions to system (2.5) in Figures 6 and 7, respectively.In Figure 6, the red and green surfaces (curves) correspond to the solution dynamical behaviors under the β(x) and β 0 scenarios, respectively.Particularly, in the two-dimensional plot, we can observe that the ω 1 region is significantly larger than the Ω 2 region.Therefore, compared to the β(x) scenario, considering the infection rate as β 0 would lead to a substantial underestimation of the viral and cellular loads of HIV within the host.Similarly, in Figure 7, the pink and blue surfaces (curves) represent the dynamical evolution of the solutions to system (2.5) under the τ j (t) and τ 0 j scenarios, respectively.It can be observed that under the time-dependent delay scenario, the viral and cellular loads of HIV are significantly higher compared to the constant times delay scenario.This is mainly because the constant time delay scenario overlooks the periodic rebound of the HIV-infected cells and viruses within the host caused by periodic antiviral therapy.In conclusion, both spatial heterogeneity represented by β(x) and time delays induced by periodic antiviral therapy represented by τ j (t) have a notable impact on the progression of HIV infection.Therefore, these two factors should not be overlooked in the modeling and clinical treatment process.(f ) µI (x)

P. Wu
Figure 5.The impact of parameters on R 0 for cases τ j (t) (the pink color line) and τ 0 j (the green color line).(a) β = 1.5 × 10 −5 (1 + k sin(mx)), the other parameters values are the same as in Table 1; (b) c(x) = 20 + k, the other parameters values are the same as in Table 1; (c)-(f): The impact of ϵ(x), η(x), γ(x) and µ I (x) on R 0 , respectively.The other parameters values are the same as in Table 1.

P. Wu 7 Conclusion
As mentioned in Wang et al [20] and Wu [21], joint effects of incubation periods, periodic antiviral therapy and heterogeneity spatial diffusion are worth studying in the investigate of HIV infection within the host.Considering the antiviral drugs can act on the fusion, reverse transcription and budding stages of HIV infected cells, in this paper, we present and analyze a periodic delays spatial diffusion HIV infection model with three-stage infection process.
In theoretical part, we first study the well-posedness of the full system incorporates the global existence of the solution for system (2.5).Moreover, the existence of strong global attractor for ω-periodic semiflow is proved.Based on the theory mentioned in Liang et al [32], as a key threshold parameter that measure the risk of HIV infection within the host, we derive the basic reproduction number R 0 which is defined as the spectral radius of the next generation operator and can be calculated numerically.It is worth mentioning that two periodic delays are introduced in our model, which reflects the influences of periodic antiviral therapy on the prevention of HIV infection process.Inspired by Zhao [34], we ensure the linearized infectious compartments system generates an eventually strongly monotone periodic semiflow by defining a phase space.We further prove that R 0 is a threshold for the elimination and persistence of HIV infection by comparison principle and persistence theory for nonautonomous system, i.e., the HIV infection will be eliminated when R 0 < 1 and will persist when R 0 > 1.In the spatial homogeneous case, we obtain the explicit expression of R 0 and show the global attractivity of the positive equilibrium by using the fluctuation method.
In numerical simulation part, we display the dynamical behaviors of the solutions for system (2.5) associated with R 0 > 1 (and R 0 < 1) (see Figures 2-3).To discuss the impacts of time delays τ j (t)(j = 1, 2) and spatial heterogeneity on the HIV infection, we set the space-averaged infection rate β 0 and time-averaged delays τ 0 j .In Figures 4-5, we display the imapct of some model parameters on R 0 .In Figures 6-7, we discuss the capacity of infected cells and productive virus for β(x)(β 0 ) and τ j (t)(τ 0 j ) cases.Our simulation results suggest that both spatial heterogeneity and periodic delays caused by periodic antiviral therapy have a remarkable impact on the progression of HIV infection.However, there exists different antiviral therapy drug combination and HIV strains and their effects are worth investigating.Hence, it motivates us to introduce multi-strain into the model and discuss the effects of different antiviral drug treatment strategies.We leave it for further work.

Figure 1 .
Figure 1.The flowchart of the three stages (fusion, reverse transcription, budding )of HIV infection withinhost.

Figure 4 .
Figure 4.The impact of parameters on R 0 for cases β(x) (the red color line) and β 0 (the blue color line).(a) β = 1.5 × 10 −5 (1 + k sin(mx)), the other parameters values are the same as in Table 1; (b) c(x) = 20 + k, the other parameters values are the same as in Table1; (c)-(f): The impact of ϵ(x), η(x), γ(x) and µ I (x) on R 0 , respectively.The other parameters values are the same as in Table1.

Table 1 .
The values of some parameters in system(2.5)

Table 1 ;
(b) c(x) = 20 + k, the other parameters values are the same as in Table1; (c)-(f): The impact of ϵ(x), η(x), γ(x) and µ I (x) on R 0 , respectively.The other parameters values are the same as in Table1.