Free Virus Research Articles

HIV-AN INFECTIOUS DISEASE THAT IMPOSES THREAT TO LIFE

HIV/AIDS is one of the fatal diseases within the world and is caused by human immunodeficiency virus. Aids initially cause immune system failure resulting in infection and risk of cancers. HIV infection caused by blood transfusion and by using contaminated syringes, HIV is present in body in 2 forms – Free virus & within infected immune cells. CD4 T- cells are infected by HIV and reduce its numbers by breaking down T-cells. Most of the symptoms of HIV are opportunistically caused by bacteria, virus, fungi etc. The infection is smaller than 1% per annum in couples who used condoms. Female condoms may provide equal level of protection. The treatment of antiretroviral drugs has changed AIDS from a life-threatening disease to a feasible disease. Research has known how the virus replicates, manipulates, and hides in infected persons. Although our understanding is increased about the virus yet, a cure and vaccine remain tough.

Optimal control of an HIV model with a trilinear antibody growth function

<p style='text-indent:20px;'>We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.</p>

Mechanism of Antiviral Activities of Nanoviricide's Platform Technology based Biopolymer (NV-CoV-2).

NV-CoV-2 is a nanoviricide that is covalently bonded with polyethylene glycol (PEG) and alkyl pendants. This molecular design is used to attack many strains of coronaviruses in a broad-spectrum manner. The ligand works by competitive inhibition and binds to the same site on the S-protein of SARS-CoV that attaches to the cognate cellular receptor, ACE2. This prevents SARS-CoV from binding and infecting the cell. NV-CoV-2 is designed to bind to the free virion particles at multiple points encapsulate the virus and disable its ability to infect the cells. The multi-point binding interaction, like a nano-velcro-tape, may lead to lipid-lipid fusion of the alkyl chains in the nanoviricide micelle with the lipid envelope of the virus. The virus becomes dismantled to a capsid form before the host immune system becomes involved. This putative mechanism is orthogonal to many other anti-coronavirus agents in development. Thus, it maybe possible to produce a stronger antiviral effect when combining NV-CoV-2 therapy with other anti-coronavirus therapies such as Remdesivir (RDV). NV-CoV-2 can encapsulate other antiviral compounds as well. In this study, RDV was encapsulated and protected from serum-mediated degradation in vivo. As a result, RDV was available for a longer period of time to interact with RNA polymerase and inhibit.

Dynamics of a stochastic HBV infection model with drug therapy and immune response.

Hepatitis B is a disease that damages the liver, and its control has become a public health problem that needs to be solved urgently. In this paper, we investigate analytically and numerically the dynamics of a new stochastic HBV infection model with antiviral therapies and immune response represented by CTL cells. Through using the theory of stochastic differential equations, constructing appropriate Lyapunov functions and applying Itô's formula, we prove that the disease-free equilibrium of the stochastic HBV model is stochastically asymptotically stable in the large, which reveals that the HBV infection will be eradicated with probability one. Moreover, the asymptotic behavior of globally positive solution of the stochastic model near the endemic equilibrium of the corresponding deterministic HBV model is studied. By using the Milstein's method, we provide the numerical simulations to support the analysis results, which shows that sufficiently small noise will not change the dynamic behavior, while large noise can induce the disappearance of the infection. In addition, the effect of inhibiting virus production is more significant than that of blocking new infection to some extent, and the combination of two treatment methods may be the better way to reduce HBV infection and the concentration of free virus.

Primary Peripheral Epstein-Barr Virus Infection Can Lead to CNS Infection and Neuroinflammation in a Rabbit Model: Implications for Multiple Sclerosis Pathogenesis.

Epstein-Barr virus (EBV) is a common herpesvirus associated with malignant and non-malignant conditions. An accumulating body of evidence supports a role for EBV in the pathogenesis of multiple sclerosis (MS), a demyelinating disease of the CNS. However, little is known about the details of the link between EBV and MS. One obstacle which has hindered research in this area has been the lack of a suitable animal model recapitulating natural infection in humans. We have recently shown that healthy rabbits are susceptible to EBV infection, and viral persistence in these animals mimics latent infection in humans. We used the rabbit model to investigate if peripheral EBV infection can lead to infection of the CNS and its potential consequences. We injected EBV intravenously in one group of animals, and phosphate-buffered saline (PBS) in another, with and without immunosuppression. Histopathological changes and viral dynamics were examined in peripheral blood, spleen, brain, and spinal cord, using a range of molecular and histopathology techniques. Our investigations uncovered important findings that could not be previously addressed. We showed that primary peripheral EBV infection can lead to the virus traversing the CNS. Cell associated, but not free virus in the plasma, correlated with CNS infection. The infected cells within the brain were found to be B-lymphocytes. Most notably, animals injected with EBV, but not PBS, developed inflammatory cellular aggregates in the CNS. The incidence of these aggregates increased in the immunosuppressed animals. The cellular aggregates contained compact clusters of macrophages surrounded by reactive astrocytes and dispersed B and T lymphocytes, but not myelinated nerve fibers. Moreover, studying EBV infection over a span of 28 days, revealed that the peak point for viral load in the periphery and CNS coincides with increased occurrence of cellular aggregates in the brain. Finally, peripheral EBV infection triggered temporal changes in the expression of latent viral transcripts and cytokines in the brain. The present study provides the first direct in vivo evidence for the role of peripheral EBV infection in CNS pathology, and highlights a unique model to dissect viral mechanisms contributing to the development of MS.

Antiviral Activities of Eucalyptus Essential Oils: Their Effectiveness as Therapeutic Targets against Human Viruses.

Given the limited therapeutic management of infectious diseases caused by viruses, such as influenza and SARS-CoV-2, the medicinal use of essential oils obtained from Eucalyptus trees has emerged as an antiviral alternative, either as a complement to the treatment of symptoms caused by infection or to exert effects on possible pharmacological targets of viruses. This review gathers and discusses the main findings on the emerging role and effectiveness of Eucalyptus essential oil as an antiviral agent. Studies have shown that Eucalyptus essential oil and its major monoterpenes have enormous potential for preventing and treating infectious diseases caused by viruses. The main molecular mechanisms involved in the antiviral activity are direct inactivation, that is, by the direct binding of monoterpenes with free viruses, particularly with viral proteins involved in the entry and penetration of the host cell, thus avoiding viral infection. Furthermore, this review addresses the coadministration of essential oil and available vaccines to increase protection against different viruses, in addition to the use of essential oil as a complementary treatment of symptoms caused by viruses, where Eucalyptus essential oil exerts anti-inflammatory, mucolytic, and spasmolytic effects in the attenuation of inflammatory responses caused by viruses, in particular respiratory diseases.

A Comprehensive Analysis of Human Endogenous Retroviruses HERV-K (HML.2) from Teratocarcinoma Cell Lines and Detection of Viral Cargo in Microvesicles.

About 8% of our genome is composed of sequences from Human Endogenous Retroviruses (HERVs). The HERV-K (HML.2) family, here abbreviated HML.2, is able to produce virus particles that were detected in cell lines, malignant tumors and in autoimmune diseases. Parameters and properties of HML.2 released from teratocarcinoma cell lines GH and Tera-1 were investigated in detail. In most experiments, analyzed viruses were purified by density gradient centrifugation. HML.2 structural proteins, reverse transcriptase (RT) activity, viral RNA (vRNA) and particle morphology were analyzed. The HML.2 markers were predominantly detected in fractions with a buoyant density of 1.16 g/cm3. Deglycosylation of TM revealed truncated forms of transmembrane (TM) protein. Free virions and extracellular vesicles (presumably microvesicles—MVs) with HML.2 elements, including budding intermediates, were detected by electron microscopy. Viral elements and assembled virions captured and exported by MVs can boost specific immune responses and trigger immunomodulation in recipient cells. Sequencing of cDNA clones demonstrated exclusive presence of HERV-K108 env in HML.2 from Tera-1 cells. Not counting two recombinant variants, four known env sequences were found in HML.2 from GH cells. Obtained results shed light on parameters and morphology of HML.2. A possible mechanism of HML.2-induced diseases is discussed.

Stability study of virus replication model

An HIV infection model with time delay in which uninfected cells become infected cells is analysed. We studied conditions under which steady states will be asymptotically stable. We also examined that for endemically infected equilibrium a critical value of time delay may occur. The steady state will be asymptotically stable when delay is less than a critical value. Else the uninfected cells, infected cells, free virus, and CTLs may undergo cyclic oscillations. We estimate the delay length to maintain stability. Numerical simulations are done to aid mathematical findings.

Nanobiotechnological Approaches to Enhance Potato Resistance against Potato Leafroll Virus (PLRV) Using Glycyrrhizic Acid Ammonium Salt and Salicylic Acid Nanoparticles

The present research was aiming to study In-Silico the effect of Glycyrrhizic Acid ammonium salt (GAS) and Salicylic acid (SA) on the coat protein of potato leafroll virus (PLRV). In addition, in-vitro studying the effect of (GAS NPs) and Salicylic acid (SA NPs) nanoparticles at concentrations 0.15, 0.30, 1.25 and 2.5 mM, respectively, to control, decline or reduce the presence of PLRV in potato plants Solanum tuberosum L. selena. (GAS NPs) and (SA NPs) were applied in the MS medium at concentrations 0.15, 0.30, 1.25 and 2.5 mM, respectively. Results revealed that, enhancement or decline the PLRV according to the initiation of specific pathways. The expression level of Kinase 3 gene increased significantly due to the two used concentrations of GAS NPs. While the expression of callose gene was upregulated significantly in response to treatment of PLRV infected plant with (GAS NPs) with concentration (0.30 mM). Treatment with (SA NPs) caused upregulation significance only of callose gene at (2.5 mM) concentration. The molecular modeling results of used compounds (glycyrrhizic acid ammonium salt and salicylic acid) showed highest score of binding and the best rms define value with a very good binding mode and perfect interactions with amino acids of the three subunits (A, B and C) forming the protein coat of leaf roll virus. Glycyrrhizic acid ammonium salt and salicylic acid nanoparticles could be perfect solution to produce potato plant free virus in-vitro. Further larger studies are needed to investigate the role of the studied compounds in vivo.

Semen-Derived Exosomes Mediate Immune Escape and Transmission of Reticuloendotheliosis Virus.

Reticuloendotheliosis virus (REV) causes immune-suppression disease in poultry, leading to a significant economic burden worldwide. Recent evidence demonstrated that the REV can enter the semen and then induce artificial insemination, but how the virus gets into semen was little known. Accumulating studies indicated that exosomes serve as vehicles for virus transmission, but the role of exosomes in viral shedding through the semen remains unclear. In this study, exosomes purified from the REV-positive semen were shown with reverse transcription-PCR and mass spectrometry to contain viral genomic RNA and viral proteins, which could also establish productive infections both in vivo and in vitro and escape from the REV-specific neutralizing antibodies. More importantly, compared with the infection caused by free virions, the exosome is more efficient for the virus to ensure effective infection and replication, which can also help the REV compromise the efficacy of the host immune response. In summary, this study demonstrated that semen-derived exosomes can medicate the transmission and immune escape of REV, implicating a novel mechanism for REV entering the semen and leading to vertical transmission.

Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD4+ T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD4+ T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD4+ T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2×2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Lectin from Triticum vulgaris (WGA) Inhibits Infection with SARS-CoV-2 and Its Variants of Concern Alpha and Beta.

Even in the face of global vaccination campaigns, there is still an urgent need for effective antivirals against SARS-CoV-2 and its rapidly spreading variants. Several natural compounds show potential as antiviral substances and have the advantages of broad availabilities and large therapeutic windows. Here, we report that lectin from Triticum vulgaris (Wheat Germ Agglutinin) displays antiviral activity against SARS-CoV-2 and its major Variants of Concern (VoC), Alpha and Beta. In Vero B4 cells, WGA potently inhibits SARS-CoV-2 infection with an IC50 of <10 ng/mL. WGA is effective upon preincubation with the virus or when added during infection. Pull-down assays demonstrate direct binding of WGA to SARS-CoV-2, further strengthening the hypothesis that inhibition of viral entry by neutralizing free virions might be the mode of action behind its antiviral effect. Furthermore, WGA exhibits antiviral activity against human coronavirus OC43, but not against other non-coronaviruses causing respiratory tract infections. Finally, WGA inhibits infection of the lung cell line Calu-3 with wild type and VoC viruses with comparable IC50 values. Altogether, our data indicate that topical administration of WGA might be effective for prophylaxis or treatment of SARS-CoV-2 infections.

Variation in human herpesvirus 6B telomeric integration, excision, and transmission between tissues and individuals.

Human herpesviruses 6A and 6B (HHV-6A/6B) are ubiquitous pathogens that persist lifelong in latent form and can cause severe conditions upon reactivation. They are spread by community-acquired infection of free virus (acqHHV6A/6B) and by germline transmission of inherited chromosomally integrated HHV-6A/6B (iciHHV-6A/6B) in telomeres. We exploited a hypervariable region of the HHV-6B genome to investigate the relationship between acquired and inherited virus and revealed predominantly maternal transmission of acqHHV-6B in families. Remarkably, we demonstrate that some copies of acqHHV-6B in saliva from healthy adults gained a telomere, indicative of integration and latency, and that the frequency of viral genome excision from telomeres in iciHHV-6B carriers is surprisingly high and varies between tissues. In addition, newly formed short telomeres generated by partial viral genome release are frequently lengthened, particularly in telomerase-expressing pluripotent cells. Consequently, iciHHV-6B carriers are mosaic for different iciHHV-6B structures, including circular extra-chromosomal forms that have the potential to reactivate. Finally, we show transmission of an HHV-6B strain from an iciHHV-6B mother to her non-iciHHV-6B son. Altogether, we demonstrate that iciHHV-6B can readily transition between telomere-integrated and free virus forms.

Modeling Dengue Immune Responses Mediated by Antibodies: A Qualitative Study.

Simple SummaryWith more than one-third of the world population at risk of acquiring the disease, dengue fever is a major public health problem. Caused by four antigenically distinct but related serotypes, disease severity is associated with the immunological status of the individual, seronegative or seropositive, prior to a natural dengue infection. While a primary natural dengue infection is often asymptomatic or mild, individuals experiencing a secondary dengue infection with a heterologous serotype have higher risk of developing the severe form of the disease, linked to the antibody-dependent enhancement (ADE) process. We develop a modeling framework to describe the dengue immune responses mediated by antibodies. Our model framework can describe qualitatively the dynamic of the viral load and antibodies production for scenarios of primary and secondary infections, as found in the empirical immunology literature. Studies such as the one described here serve as a baseline to further model extensions. Future refinements of our framework will be of use to evaluate the impact of imperfect dengue vaccines.Dengue fever is a viral mosquito-borne infection and a major international public health concern. With 2.5 billion people at risk of acquiring the infection around the world, disease severity is influenced by the immunological status of the individual, seronegative or seropositive, prior to natural infection. Caused by four antigenically related but distinct serotypes, DENV-1 to DENV-4, infection by one serotype confers life-long immunity to that serotype and a period of temporary cross-immunity (TCI) to other serotypes. The clinical response on exposure to a second serotype is complex with the so-called antibody-dependent enhancement (ADE) process, a disease augmentation phenomenon when pre-existing antibodies to previous dengue infection do not neutralize but rather enhance the new infection, used to explain the etiology of severe disease. In this paper, we present a minimalistic mathematical model framework developed to describe qualitatively the dengue immunological response mediated by antibodies. Three models are analyzed and compared: (i) primary dengue infection, (ii) secondary dengue infection with the same (homologous) dengue virus and (iii) secondary dengue infection with a different (heterologous) dengue virus. We explore the features of viral replication, antibody production and infection clearance over time. The model is developed based on body cells and free virus interactions resulting in infected cells activating antibody production. Our mathematical results are qualitatively similar to the ones described in the empiric immunology literature, providing insights into the immunopathogenesis of severe disease. Results presented here are of use for future research directions to evaluate the impact of dengue vaccines.

Effects of toxicity and zooplankton selectivity on plankton dynamics under seasonal patterns of viruses with time delay

Intricate functional diversities in marine ecosystems often lead to exhibit a variety of dynamical behaviors, including chaos. To bring order into such chaotic situations, the system develops different mechanisms of self‐adaptability. Our present investigation is based on the study of an eco‐epidemiological model for virally infected toxic phytoplankton and zooplankton system. Both healthy and infected phytoplankton are assumed to be exposed to grazer zooplankton with a varying degree of exposure. An extension is made by introducing the fact that viral replication process in a lysis event is mediated by some time lag. To achieve more realistic and explicit outcomes of the existing phenomena correlated with our model, we consider the force of infection and replication of free viruses as seasonally forced. We find that time delay accounts for recurrent stability switching event in the system. The chaotic behavior is observed in the seasonally forced delayed system, which indicates the emergence of harmful blooms. The main motive is to gain a clear insight into the field of synchronization and stabilization of the irregular behaviors of the system. Our investigations suggest that increased strength of toxic compounds exuded by phytoplankton species may suppress the prevalence of chaotic disorder and drives the system into a zooplankton‐free zone. The intensity of selective grazing of zooplankton changes the state of chaos to order and promotes the disease‐free system.

The "oral" history of COVID-19: Primary infection, salivary transmission, and post-acute implications.

Severe acute respiratorysyndrome coronavirus 2 (SARS‐CoV‐2), the causative agent of COVID‐19, has led to more than 3.25 million recorded deaths worldwide as of May 2021. COVID‐19 is known to be clinically heterogeneous, and whether the reported oral signs and symptoms in COVID‐19 are related to the direct infection of oral tissues has remained unknown. Here, we review and summarize the evidence for the primary infection of the glands, oral mucosae, and saliva by SARS‐CoV‐2. Not only were the entry factors for SARS‐CoV‐2 found in all oral tissues, but these were also sites of SARS‐CoV‐2 infection and replication. Furthermore, saliva from asymptomatic individuals contained free virus and SARS‐CoV‐2‐infected oral epithelial cells, both of which were found to transmit the virus. Collectively, these studies support an active role of the oral cavity in the spread and transmission of SARS‐CoV‐2 infection. In addition to maintaining the appropriate use of personal protective equipment and regimens to limit microbial spread via aerosol or droplet generation, the dental community will also be involved in co‐managing COVID‐19 “long haulers”—now termed Post‐Acute COVID‐19 Syndrome. Consequently, we propose that, as SARS‐CoV‐2 continues to spread and as new clinical challenges related to COVID‐19 are documented, oral symptoms should be included in diagnostic and prognostic classifications as well as plans for multidisciplinary care.

Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell

Replication of Human Immunodeficiency Virus type 1 (HIV) in infected CD4+ T cells represents a key driver of HIV infection. The HIV life cycle is characterised by the heterogeneity of infected cells with respect to multiplicity of infection and the variability in viral progeny. This heterogeneity can result from the phenotypic diversity of infected cells as well as from random effects and fluctuations in the kinetics of biochemical reactions underlying the virus replication cycle. To quantify the contribution of stochastic effects to the variability of HIV life cycle kinetics, we propose a high-resolution mathematical model formulated as a Markov chain jump process. The model is applied to generate the statistical characteristics of the (i) cell infection multiplicity, (ii) cooperative nature of viral replication, and (iii) variability in virus secretion by phenotypically identical cells. We show that the infection with a fixed number of viruses per CD4+ T cell leads to some heterogeneity of infected cells with respect to the number of integrated proviral genomes. The bottleneck factors in the virus production are identified, including the Gag-Pol proteins. Sensitivity analysis enables ranking of the model parameters with respect to the strength of their impact on the size of viral progeny. The first three globally influential parameters are the transport of genomic mRNA to membrane, the tolerance of transcription activation to Tat-mediated regulation, and the degradation of free and mature virions. These can be considered as potential therapeutical targets.

A New Infectious Unit: Extracellular Vesicles Carrying Virus Populations.

Viral egress and transmission have long been described to take place through single free virus particles. However, viruses can also shed into the environment and transmit as populations clustered inside extracellular vesicles (EVs), a process we had first called vesicle-mediated en bloc transmission. These membrane-cloaked virus clusters can originate from a variety of cellular organelles including autophagosomes, plasma membrane, and multivesicular bodies. Their viral cargo can be multiples of nonenveloped or enveloped virus particles or even naked infectious genomes, but egress is always nonlytic, with the cell remaining intact. Here we put forth the thesis that EV-cloaked viral clusters are a distinct form of infectious unit as compared to free single viruses (nonenveloped or enveloped) or even free virus aggregates. We discuss how efficient and prevalent these infectious EVs are in the context of virus-associated diseases and highlight the importance of their proper detection and disinfection for public health.

Diffusion effects in nonlinear dynamics of hepatitis B virus

A dynamic system of viral hepatitis B with healthy, infected cells and free virus is investigated in this work taking into account diffusion effects. The model is shown to be governed by a biological system of equations of prey-predation, competition and commensalism species. The existence of exact traveling waves solutions is established through the (G’/G) expansion method. To further understand the dynamic mechanism of hepatitis B virus infection, we get the Kink, Bright and Dark type profiles and rise out the influence of diffusion parameter on the localization and the amplitude of the solutions which permits us to improve and give more information on the transmission and control of the disease. Numerical simulations are done to give evidence that hepatitis B virus model can be studied by considering the mobility of three species scenarios and also, indicated which dynamic specie cell is more stable than the other.

Modeling the dynamics of viral–host interaction during treatment of productively infected cells and free virus involving total immune response

Virus dynamics models are useful in interpreting and predicting the change in viral load over the time and the effect of treatment in emerging viral infections like HIV/AIDS, hepatitis B virus (HBV).We propose a mathematical model involving the role of total immune response (innate, CTL, and humoral) and treatment for productively infected cells and free virus to understand the dynamics of virus–host interactions. A threshold condition for the extinction or persistence of infection, i.e. basic reproductive number, in the presence of immune response (RI ) is established. We study the global stability of virus-free equilibrium and interior equilibrium using LaSalle’s principle and Lyapunov’s direct method. The global stability of virus-free equilibrium ensures the clearance of virus from the body, which is independent of initial status of subpopulations. Central manifold theory is used to study the behavior of equilibrium points at RI = 1, i.e. when the basic reproductive number in the presence of immune response is one. A special case, when the immune response (IR) is not present, has also been discussed. Analysis of special case suggests that the basic reproductive number in the absence of immune response R0 is greater than that of in the presence of immune response RI , i.e. R0&gt; RI . It indicates that infection may be eradicated if RI &lt; 1. Numerical simulations are performed to illustrate the analytical results using MatLab and Mathematica.