Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics

Abstract

The immune system is a vital defense network within the body, where lymph and lymph nodes play pivotal roles. Lymph is a transparent fluid containing white blood cells, specifically lymphocytes, which circulate through the lymphatic system, rapidly multiplying to fight viruses and bacteria. Lymph nodes function as filters, capturing and eliminating infections and abnormal cells found in the lymphatic fluid prior to its reentry into the circulatory system. A thorough meta-analysis of research publications in the topic is conducted utilizing Bibliometrix to comprehensively assess the current literature. The paper is centered on understanding the intricate relationship between interferons, dendritic cells, and macrophages in the lymphatic system during hepatitis virus infection. A nonlinear model for the development of the virus is used, together with the initial conditions, for a much better understanding of a hepatitis-C infection. The associated Cauchy problem is numerically solved and graphs are depicted. The interpretation of the figures explains the dynamics of interferons, dendritic cells, and macrophages as well as their interaction with other factors. By adopting an interdisciplinary approach, this study offers fresh perspectives and uncovers new research areas to better comprehend and battle hepatitis virus infections.

1. Introduction

The human body consists of two primary circulatory systems: the blood and lymphatic systems. Traditionally, the blood vascular system has been the focus of significant research, whereas the lymphatic system has been overlooked and regarded as subordinate. Advancements in molecular, cellular, and genetic research, together with imaging technologies, have led to tremendous progress in the understanding of the lymphatic system in recent decades [1]. Lymphatic fluid, also known as lymph, is a thin, clear, and yellowish-hued substance that leaks from blood capillary walls due to heart or cellular pressure. It is collected by lymphatic capillaries and carries through lymphatic arteries and lymph nodes before entering the venous circulation. Lymph consists of interstitial fluid with lymphocytes, bacteria, and other cells [2].

Bibliometrix is a software tool specifically developed for bibliometric analysis in R version 4.3.2 [3]. It allows the quantitative assessment of academic publications. This implementation’s purpose is to analyze the development of citations, authorship, and collaborative networks in scholarly literature. A Biblioshiny analysis in the Bibliometrix library shows that the keywords (i.e., immunology, mathematical modeling, lymph flow) have been used together in the scientific world by 289 authors since 1991.

Word clouds are an intriguing sort of diagram that illustrate the frequency of selected terms in a corpus of text data. These are a particularly interesting type of diagram. A Word cloud displays terms with a higher frequency with a larger font size than those with a lower frequency. One example of this can be found in Figure 1, where it is possible to see that dendritic cells, immune response, lymphagionesis, lymph nodes, and T-cells are utilized in a correlational manner and on a highly regular basis.

Another type of diagram is the three-field plot in Figure 2, which is generally utilized as a visual assessment tool to analyze the interconnection of several aspects, including sources, countries, affiliations, keywords, prominent authors, cited sources, and author-keywords. The one below includes the correlation between titles and different types of keywords [4]. With the diagram’s support, it is possible to notice that the relationships are numerous and intricate. This suggests that numerous scientists have made efforts to investigate and comprehend the immune system’s response, its functioning, and the precise mechanisms by which lymph transports and eliminates viruses and bacteria.

According to global data [5,6], the diagnosis and treatment coverage remains alarmingly low. For hepatits B, from the people who have been diagnosed with the disease, only 2.6% have received antiviral therapy. In the African region, only 0.2% of the hepatitis-infected people were treated. Similarly, in the Southeast Asia region, the treatment coverage of the diagnosed people is just 0.1%. The low diagnosis and therapeutic percentages exist due to many reasons, the most important one being the biological complexity of the hepatitis infection. Hepatitis B is marked by a long asymptomatic period and a fluctuating disease course, making eradication very difficult [7]. These numbers highlight a need for improvement in early detection and research in this area.

This article intends to provide an interdisciplinary assessment of the immune system’s response to hepatitis virus infections, with a focus on the interactions of interferons, dendritic cells, and macrophages within the lymphatic system. The study uses a nonlinear mathematical model to evaluate the dynamics of lymph flow and immune cell behavior in order to improve knowledge of the immune response mechanisms at work. As obtained in [8], lymphatic systems are closely connected with tumor progression. An easier and more effective detection approach is needed. Although the referenced model focuses on the evolution of cells in the lymph node for hepatitis, it can also be used for HIV or cancer. Better isolation of tumor cells and their evolution is necessary, and this can be achieved by also taking into account the connection with macrophage cells and with the immune system, respectively. Furthermore, the study provides new pathways for future research, contributing to the larger field of immunology and suggesting potential strategies for treating hepatitis infections. Lymph nodes are complex organs that intervene in the pathological and physiological processes of various diseases, one of the experimental methods of analysis being organ chips to better simulate the complex microstructure of lymph and the interactions between different immune cells [9].

The remainder of this paper is organized as follows: Section 2 provides the biological background necessary for understanding the lymphatic system and its role in immune defense, through materials and methods. A detailed analysis of the mathematical models and the nonlinear model is used to describe the interactions between immune system cells before and after infection, along with the coding process involved in developing the simulations. Section 3 delves into the theoretical study of initial conditions and their role in lymph flow analysis, as well as the analysis of parameter variation. At the end of the section, a graph analysis is performed and interpretations are made. Conclusions are drawn in Section 4.

2. Materials and Models

2.1. The Lymphatic System

The lymphatic system, an essential element of the circulatory system, has a pivotal function in the immune response and the removal of extracellular fluid. The lymphatic system consists of lymph tubes, lymph nodes, lymphatic cells, and other lymphoid organs [10,11]. The lymphatic system transports lymphatic fluid, purifies it via lymph nodes, and restores it to the circulatory system. The liver and intestine lymphatics account for 80% of the total volume of lymph. The lymphatic system plays a vital role in clinical settings, as it serves as a primary pathway for cancer spread and the development of inflammation in lymphatic vessels and lymph nodes [12]. This system consists of a linear arrangement of lymphatic veins and secondary lymphoid organs, illustrated in Figure 3, which is distinct from the circular layout of the blood circulatory system. Lymphatic capillaries are composed of a monolayer of lymphatic endothelial cells that partially overlap with one other. These capillaries lack basement membranes or pericytes. They are present in all organs and tissues that have blood vessels, except for the retina, bone, and brain. Lymphatic veins are connected to extracellular matrix through filament bundles, serving as direct anchors that regulate the drainage of lymph fluid [13]. Additionally, they function as a channel for transporting lymphocytes and antigen-presenting cells to nearby lymph nodes, where the immune system encounters infections, microorganisms, and other immunological triggers. Lymphatic vessels in lymph nodes absorb different antigens from tissues outside the lymph nodes and are stimulated by chemokines/cytokines released by B cells, macrophages, and dendritic cells during inflammation. Lymphatic arteries can also regulate inflammatory responses by inhibiting the development and function of dendritic cells through a mechanism that depends on the cell surface glycoproteins such as Mac-1/ICAM-1 interaction [14]. Thus, lymphatic-specific vascular endothelial growth factor receptor (VEGFR)-3 can suppress inflammation by reducing inflammatory edema development and the concentration of inflammatory cells. This indicates that inducing functional lymphangiogenesis could be a promising approach for treating chronic inflammatory illnesses [15].

2.1.1. Lymph Node

As previously mentioned, the lymphatic system is a network of lymphoid organs and lymphatic lymph nodes (LNs) that play a crucial role in fluid homeostasis, lipid absorption, and immunity. Primary organs, such as bone marrow and the thymus, create immune cells called lymphocytes, while secondary organs, like LNs, spleen, tonsils, and specialized mucosal tissue, provide immune response sites. The lymph is composed of macromolecule-rich interstitial fluid, circulating antigen-presenting cells (APCs), and lymphocytes. Lymphatic transport is critical for maintaining fluid balance and allowing detection and immune response to harmful pathogens.

Lymph nodes (LNs) consist of stromal cells and macrophages, which actively combat infections and monitor inflammatory responses to foreign substances. Lymphatic fluid enters the lymph node by afferent lymphatic vessels and is deposited into subcapsular sinuses. The contents of the lymph follow several routes, while endothelial cells create a continuous barrier to block the entry of big particles into the LN cortex. Macrophages have the ability to capture and present antigens that are carried by lymph to immune cells that are entering the body. The LN exhibits a high degree of vascularity, as blood capillaries extend from the hilus and create a compact network. Lymphocytes, including B- and T-cells, enter the lymph nodes through specific blood arteries located in the paracortex and exit back into the bloodstream by efferent lymphatic capillaries [17].

2.1.2. Virology

Type I interferons (IFNs) play a critical role in the immune system’s response to different types of infections. They are a type of cytokines, which are molecular messengers, specifically proteins, that modulate the immune response by promoting intercellular communication. IFNs have a substantial impact on dendritic cell activation, maturation, migration, and survival. In addition, they directly augment the activity of natural killer cells and T/B cells, so bolstering the body’s immune response to viral infections.

Plasmacytoid dendritic cells (pDCs) are commonly known as the primary producers of interferon (IFN) because of their distinctive molecular mechanisms for detecting nucleic acids. These cells have a specific function in identifying viral genetic material and triggering the production of type I interferons as a component of the immune response against viruses. Macrophages, along with other cell types, play a substantial role in producing type I IFN, depending on the specific pathogen, its tissue preference, and the route of infection. Macrophages are essential components of the immune system, playing a crucial role in phagocytosis and the presentation of antigens. Type I interferons have the capacity to impact macrophage activity by improving their capability to ingest and eliminate infections. These cell types also participate in the synthesis of type I interferons (IFNs) in reaction to viral infections, contributing to the total immune response against viruses [18].

2.2. Lymph Flow

The general fluid flow equations are given by the mass conservation equation and the equations of motion. The mass balance/continuity equation is as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho v_x\right)}{\partial x}+\frac{\partial \left(\rho v_y\right)}{\partial y}+\frac{\partial \left(\rho v_z\right)}{\partial z}=0,}$$

(1)

The continuity Equation (1), expresses that the mass of a system does not change due to motion, with $\rho$ density of the fluid and $\mathit{v}=(v_x,v_y,v_z)$ fluid velocity. The Navier–Stokes equations are obtained from Cauchy’s equations:

$${\displaystyle a_i=F_i+\frac{1}{\rho }\frac{\partial T_{ij}}{\partial x_j}}$$

(2)

where $\mathit{T}$ is the Cauchy stress tensor. For Newtonian fluids, the form of the stress tensor is

$${\displaystyle T_{ij}=(-p+\lambda \theta ){\delta }_{ij}+2\mu D_{ij},\mathrm{where}\theta ={\mathrm{D}}_{ii},{\mathrm{D}}_{ij}=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)}$$

(3)

and, therefore, ${\displaystyle a_i=F_i+\frac{1}{\rho }\left(-\frac{\partial p}{\partial x_i}+(\lambda +\mu )\frac{\partial \theta }{\partial x_i}+\mu \Delta v_i\right)}$ [19].

Lymph flow dynamics corresponding to (1) and (2) are

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial A}{\partial t}}+{\displaystyle \frac{\partial \left(Au\right)}{\partial x}}=0\hfill \\ \hfill & \rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}\right)=-{\displaystyle \frac{\partial p}{\partial x}}+\mu {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-R\left(u\right)\hfill \end{array}$$

(4)

where $A(x,t)$ is the cross-sectional area of the vessel, $u(x,t)$ is the lymph flow velocity, $p(x,t)$ is the pressure within the vessel, x is the axial position along the vessel, t is the time, $\rho$ is the density of lymph, $\mu$ is the viscosity of lymph, and $R\left(u\right)$ is the resistance term accounting for friction and vessel properties [19,20].

The dynamics of the vessel wall intervene with the relationship between the pressure and the cross-sectional area: $p-P_{\mathrm{ext}}=K\left(\sqrt{{\displaystyle \frac{A}{A_0}}}-1\right)$, where $P_{\mathrm{ext}}$ is the external pressure acting on the vessel, K is the vessel wall stiffness coefficient, and $A_0$ is the resting cross-sectional area of the vessel.

The valve dynamics in a lymph flow are expressed by a unidirectional flow ensured by the lymphatic valves. The flow rate across a valve, Q, can be modeled as follows:

$$Q=\left\{\begin{array}{cc}{C}_v\left(P_2-P_1\right),\hfill & \mathrm{if}{P}_2>P_1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where $P_2$ is the upstream pressure, $P_1$ is the downstream pressure, and $C_v$ expresses the conductance of the valve. The flow rate equation expresses the relation between the volumetric flow rate to velocity and the cross-sectional area: $Q=Au$.

Lymph flow is influenced by external forces such as muscle contractions, respiration, and arterial pulsations. The external force applied to the vessel can be modeled as follows: $F_{\mathrm{ext}}=\alpha cos\left(\omega t\right)$, with $\alpha$ the amplitude of the external force and $\omega$ the frequency of the force.

The energy conservation law for the lymph flow dynamics that includes contributions from flow energy, dissipation due to viscosity, and work performed by external forces is represented as follows:

$${\displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}\rho u^2+\frac{p}{\rho }\right)+\mathrm{Dissipation}=\mathrm{Work}\mathrm{from}\mathrm{external}\mathrm{forces}.}$$

• Hagen–Poiseuille movement

Unidirectional flow takes place in the presence of a pressure gradient, alongside a velocity distribution and a stress distribution (wall shear stress). The pressure is associated with the tension in the capillaries.

Initially, the flow is similar to that of blood through arteries, capillaries, and veins, in shape.

$${\displaystyle \Delta p=\frac{8\eta LQ}{\pi R^4},\tau =\frac{4\eta Q}{\pi R^3}}$$

(5)

where $\Delta p=$ the pressure difference between two ends of one tube, $\eta =$ viscosity, $L=$ length of tube, $Q=$ flow, $R=$ radius of tube section, $A=$ surface of tube section, and $\tau =$ wall shear stress [21].

Although this approach accurately explains the movement of blood in arteries and veins, it needs adjustments to account for the movement of lymph in lymphatic channels, which is influenced by valves and contractions of the vessel walls, Figure 4.

In [22], the Hagen–Poisseuille equation is also used, but in the context of graph/subgraph theory to compute the direction of lymph flow. In addition, the balance of the flow through vertices due to the mass conservation leads to a symmetric weighted Laplacian matrix.

Pulsatile flow occurs throughout the circulatory system, regulating pressure through water exchange between the arterial, venous, and lymphatic systems. Fluid transport relies on pulsatile mechanisms, with external pressure playing a crucial role, and cells arranged to ensure unidirectional lymph movement.

One key feature is how initial lymphatic capillaries regulate lymph entry into the lymphatic system via processes that maintain pressure homeostasis. Primary valves, or flap valves, are one-way gates formed by overlapping endothelial cells. With the help of the anchoring filaments in the extracellular matrix, these valves open when the interstitial hydrostatic pressure exceeds the pressure within the lymphatic tube, enabling fluid, cells and chemicals to enter. When the pressure within the vessel exceeds the interstitial pressure, the valves close, preventing backflow. This procedure ensures that lymph flows in one direction and that tissues are thoroughly drained. Figure 1B, from [17] contains additional information as well as a visual representation.

Inside the lymphatic capillary (LC), the lymph flow is modeled as a porous medium using the Darcy–Brinkman equation, while outside the LC, the Navier–Stokes equation is applied. Equation (4)₂ becomes

$${\displaystyle \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho }\frac{\partial p}{\partial x}=-8\pi \eta \frac{u}{A}.}$$

The second equation explains the fluid flow in relation to pressure forces in a capillary, forces that control fluid passage across capillary walls. The capillary filtration coefficient (K) measures how quickly fluid goes through the capillary wall, which reflects its permeability. Capillary hydrostatic pressure ($P_2$) forces fluid out of the capillary and into the interstitial space, while interstitial hydrostatic pressure ($P_1$) pulls fluid back into the capillary. The reflection coefficient ($\sigma$) measures how well proteins are maintained within the capillary—larger values mean better retention. This parameter takes up values between 0 and 1, where $\sigma =0$ for a perfectly permeable membrane, and $\sigma =1$ for a membrane that is perfectly selective. Oncotic pressures also play an important role. Capillary oncotic pressure, created by plasma proteins such as albumin, pushes water back into the capillary, whereas interstitial oncotic pressure, caused by proteins in the interstitial space, pushes water out. These forces work together to determine the optimal balance of filtration and reabsorption.

The viscosity of lymph is primarily influenced by its protein composition, which includes albumin, globulins, fibrinogen, and lipoproteins, as well as immune-related molecules such as cytokines and antibodies [23]. Muscle contractions, interstitial fluid pressure, the integrity of lymphatic vessels, and outside stimuli like massage or greater capillary permeability are some of the things that cause lymph to flow. A rise in lymph viscosity, which can be caused by more protein or inflammation chemicals, can make lymphatic drainage much slower. This can change the flow of immune cells and the balance of fluids [24]. This paper goes on to show that when someone has hepatitis, their body makes more cytokines, which cause inflammation. Higher cytokine levels increase capillary permeability, allowing more proteins to leak into the interstitial space. The dynamics of oncotic pressure are changed by lowering the oncotic pressure in the plasma and raising the oncotic pressure between the molecules. This causes more fluid to build up in the tissues, which raises the interstitial fluid pressure. Along with greater capillary permeability and osmotic imbalances, these changes cause lymphatic flow to increase at first as a way to balance things out. But inflammation that lasts for a long time and high protein levels in lymph can make it thicker, which slows lymphatic flow and makes tissues swell.

2.3. Associated Models

In the following section, we use three different systems of equations in order to generate a better overall picture of how the mechanism of an infection works, being able to go from an overview into a more detailed version of the infection simulation. We start with a system containing three equations, where the only variables involved are the viral load, infected cells, and target cells. Then, for the second system, the cells are divided into dendritic activated and non-activated cells, healthy and infected hepatocytes, and cytotoxic T lymphocytes, providing a better understanding of the cells’ interaction. The final system that is analyzed, the most complex one, takes into account the dynamics of the virus in the spleen, liver, and blood, as well as the infected and uninfected macrophages and dendritic cells.

The first model of the HCV infection was proposed by Neumann et al. [25], where the dynamics of the infection were analyzed through a system of three differential equations representing the target cells, infected cells, and viral load, as seen in Figure 5.

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d{y}_1}{dt}=s-d{y}_1\left(t\right)-(1-\eta )\beta y_3\left(t\right)y_1\left(t\right)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_2}{dt}=(1-\eta )\beta y_3\left(t\right)y_1\left(t\right)-\delta y_2\left(t\right)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_3}{dt}=(1-\epsilon )p{y}_2\left(t\right)-c{y}_3\left(t\right)}\hfill \end{array}$$

(6)

This closed system takes into account the production rate of target cells (s) and their death rate (d). The cells become infected with the constant infection rate ($\beta$) and die with the constant rate ($\delta$). Hepatitis C virions are produced by infected cells at an average rate of p virions per cell per day and are cleared with clearance rate constant c. The possible effects of IFN in this model are to reduce either the production of virions from infected cells by a fraction (1 − $\epsilon$) or the de novo rate of infection by a fraction (1 − $\eta$).

The classical HCV model is as follows [26]:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d{y}_1}{dt}=r{y}_1\left(t\right)\left(1-\frac{y_1\left(t\right)+y_2\left(t\right)}{k}\right)-{\beta }_1{y}_1(t)y_2(t)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_2}{dt}={\beta }_1{y}_1(t)y_2(t)-d_1{y}_1(t)-{\beta }_2(t)y_2(t)y_5(t)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_3}{dt}=\lambda -{\delta }_1{y}_3\left(t\right)-\omega y_3\left(t\right)y_2\left(t\right),}\hfill \\ \hfill & {\displaystyle \frac{d{y}_4}{dt}=\omega y_3(t)y_2(t)-{\delta }_2{y}_4(t)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_5}{dt}=\nu y_4(t)y_5(t)-{\beta }_3{y}_2(t)y_5(t)-\mu y_5(t).}\hfill \end{array}$$

(7)

The aim of this study focuses on replicating the initial response of the immune system with a particular emphasis on the macrophages and the plasmacytoid dendritic cells. This early response is triggered by the antiviral interferon $\alpha$, whose quantities we want to monitor. The model we based our study on focuses on the compartmental model of the interferon response when it comes to contact with the hepatitis C virus (HCV), considering each of the variables as populations that interact with each other. The variables analyzed are the dynamics of the interferon, $y_1=I\left(t\right)$, the dynamics of virus in the spleen $y_6=VS\left(t\right)$, blood $y_7=VB\left(t\right)$, and liver $y_8=VL\left(t\right)$, and the infected/uninfected plasmacytoid dendritic cells and macrophages, as can be seen in Figure 6. This results in a non-linear dynamic system, whose evolution will be observed, while considering the delays with which the virus sets into the infected individual [27].

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{d{y}_1}{dt}}\hfill & \hfill =& r_1·y_2\left(t-{\tau }_1\right)+r_2·y_3\left(t-{\tau }_2\right)-d_2{y}_1\hfill \\ \hfill & {\displaystyle \frac{d{y}_2}{dt}}\hfill & \hfill =& s_1·y_6\left(t\right)·y_4\left(t\right)-d_3·y_2\left(t\right)\hfill \\ \hfill & {\displaystyle \frac{d{y}_3}{dt}}\hfill & \hfill =& s_2·y_6\left(t\right)·y_5\left(t\right)-d_4·y_3\left(t\right)\hfill \\ \hfill & {\displaystyle \frac{d{y}_4}{dt}}\hfill & \hfill =& -s_1·y_6\left(t\right)·y_4\left(t\right)+d_5\left(c_1-y_4\left(t\right)\right)\hfill \\ \hfill & {\displaystyle \frac{d{y}_5}{dt}}\hfill & \hfill =& -s_2·y_6·y_5+d_6\left(c_2-y_5\left(t\right)\right)\hfill \\ \hfill & {\displaystyle \frac{d{y}_6}{dt}}\hfill & \hfill =& {\displaystyle \frac{r_3}{{\displaystyle 1+\frac{y_1}{t_1}}}·y_2\left(t-{\tau }_3\right)+\frac{r_4}{{\displaystyle 1+\frac{y_1}{t_2}}}·y_3\left(t-{\tau }_4\right)-\left[s_1·y_4\left(t\right)+s_2·y_5\left(t\right)\right]y_6\left(t\right)}\hfill \\ \hfill & \hfill & \hfill -& (d_1+m_1)·y_6\left(t\right)+m_2·y_7\left(t\right)·{\displaystyle \frac{b_4}{b_3}}\hfill \\ \hfill & {\displaystyle \frac{d{y}_7}{dt}}\hfill & \hfill =& {\displaystyle m_1·y_6\left(t\right)·\frac{b_3}{b_4}+m_3·y_8\left(t\right)\frac{b_5}{b_4}-\left(m_2+m_4+m_5\right)·y_7\left(t\right)}\hfill \\ \hfill & {\displaystyle \frac{d{y}_8}{dt}}\hfill & \hfill =& b_1{y}_8\left(t\right)·\left({\displaystyle 1-\frac{y_8\left(t\right)}{b_2}}\right)-m_3·y_7\left(t\right){\displaystyle \frac{b_4}{b_5}}\hfill \end{array}$$

(8)

Considering the flow of the lymph tip fluid to the node, the values of the intervening functions become initial conditions for the functions that describe the model in the node, which are being used in solving the system (8).

3. Main Study and Results

3.1. Parametric Analysis

A qualitative study on the system solution also involves the analysis of the parameters involved in the model, trying to see which parameter, through its small variation, influences the results so that a state of recovery can be obtained more quickly. For the first model, (6), for different values of the parameter $ϵ$ using numerical integration as Runge–Kutta order four and parameters described in Appendix A, the results are depicted in Figure 7 and Figure 8.

The graph (Figure 7) depicts the evolution of infected cells over 48 days for various antiviral efficacy ($\epsilon$) levels. High efficacy ($\epsilon =0.99$) inhibits infection, resulting in a reduction in infected cells. Lower efficacy ($\epsilon =0.5$) allows for rapid infection increase before stabilization, whereas moderate efficacy ($\epsilon =0.75$) slows the infection but maintains a high peak of infected cells. Higher antiviral efficacy leads to better infection control and fewer infected cells over time. The right graph depicts the early-stage dynamics of infected cells in logarithmic scale over the first two days for various antiviral efficacy levels. Initially, all curves show a rapid increase, indicating early infection spread. However, differences in growth rates become apparent as time progresses. Despite minor variations, all cases stabilize at similar levels within this short period, suggesting that early infection dynamics are less influenced by antiviral efficacy, while long-term outcomes depend more on $\epsilon$. The evolution of a virus is, in time, double scaled for both the short and long term, as shown in Figure 7a,b. Figure 7b suggests the need for a short-term study that is longer than two days, as the infected cells do not stabilize in value.

Figure 8a represents the logarithmic evolution of target cells T, which are hepatocytes, over time for different antiviral efficacies. The high efficacy ($\epsilon =0.99$) hepatocytes steadily increase, indicating effective viral suppression and minimal cell loss. Lower efficacy, shown by $\epsilon =0.5$, allows an initial rise followed by a decline and stabilization at the lowest level out of these three values, suggesting a lower control over the infection of hepatocytes. Intermediate efficacy, simulated with an $\epsilon$ of 0.75, shows a similar trend but with a less pronounced higher minimum, reflecting partial control of infection. Higher antiviral efficacy better preserves hepatocyte levels over time.

Figure 8b depicts the logarithmic progression of viral load, V, over time for various antiviral efficacies ($\epsilon$). The maximum efficacy ($\epsilon$ = 0.99) indicates that the viral load drops quickly and stays low, indicating successful viral suppression. Moderate efficacy ($\epsilon =0.5$) allows for an early reduction followed by a rise, which stabilizes at a high viral load, indicating infection control. Intermediate efficacy ($\epsilon =0.75$) follows a similar pattern, but with a slower increase and slightly lower stability level. Higher antiviral efficacy is required to reduce and maintain low virus levels.

The simulations in Figure 9 and Figure 10 are based on the 5 equation system, the classical HCV model (7) described above.

According to Figure 9a, the function $y_2\left(t\right)$, representing infected hepatocytes, shows an exponential drop that is unaffected by the initial number of infected target cells. The evolution of cytotoxic T lymphocytes in Figure 9b is simulated for two days with different initial values of $y_2\left(t\right)$. It can be observed that the highest initial value of infected hepatocytes, with the value of 0.70, leads to a lower activation of cytotoxic T lymphocytes, likely due to increased immune system strain or resource depletion. As a result, the cytotoxic T lymphocyte levels are slightly reduced at a higher initial infection value compared to lower values of 0.60 and 0.70.

This graphic depicts $y_4\left(t\right)$, the activated dendritic cells, over a period of 2 days. It can be observed that the activation process has a peak after half a day, at around 14 h. The peak form and the rate at which dendritic cells are activated are quite similar at different values of $y_2$ (infected hepatocytes), indicating that the activation rate has a stable way of developing; however, more dendritic cells activate at a higher $y_2\left(t\right)$. The activation rate is faster in the first 12 h, reaches a peak, and then decreases with a lower velocity. The immune response activates quickly, adapts to the size of infection, and is consistent. Because the null value of activated dendritic cells is reached after 5 days for all three infection sizes, suggesting that after this period of time, no further plasmacytoid dendritic cells are activated, the graph was not extended up to 48 days of observations. Doing so would have decreased the accuracy with which we can study the evolution of the dendritic cells and would lead to an uncertainty in reading their values.

3.2. Analysis and Graph Interpretations

This chapter explores the complexities of interpreting the resulting graphs created using Python. By applying the robust tools provided by Python, we can generate visual depictions of data that offer meaningful insights into the way the immune system works. Analyzing these graphs is essential for revealing patterns of the immune response.

• Hepatits infection, model (8)

In the first graph depicted below, Figure 11a, an increase in the concentration of type I interferons is observed 18 h after infection with the virus in mice. In humans, the process may be prolonged due to the substantial increase in body weight and tissue surface area. This results in a higher time of lymph movement through the whole body, requiring a longer period of time to gather the virus and transport it to the lymph node where the lymphocytes eliminate it. This graph underlines the initiation of the immune response, particularly how long it takes for the interferons to be released and subsequently stimulate the B-cells and T-cells, as well as enhance the superior quality of macrophages and plasmacytoid dendritic cells.

The following graph, Figure 11b, illustrates the secretion of macrophages during the initial stage of infection, with a subsequent decrease in production due to their prompt immunological response. Macrophages play a crucial role in controlling the spread of the virus. They accomplish this by engulfing and breaking down both viruses and bacteria through a process called phagocytosis. Additionally, this type of cell emits anti-inflammatory cytokines later in the immune response to help reduce inflammation.

A linear secretion pattern of plasmacytoid dendritic cells is illustrated in the following graph, Figure 12, with a significant increase in quantity during the initial 18 h. The quantity of infected pDCs stabilizes thereafter, which can be correlated to their primary function as ‘sensors’ for viruses and bacteria. As the viral load decreases, the infected pDCs remain constant. It is anticipated that their secretion will then decrease as the infection is combated until complete elimination.

Upon analyzing the graphs collectively, it can be concluded that infected macrophages and pDCs, 18 h after first exposure to the virus, release interferons that augment their ability to eradicate the virus. Additionally, these interferons activate crucial lymphocytes (B- and T-cells) that effectively eliminate the parasites. These graphs serve as mathematical evidence demonstrating that during the period of infection of pDCs and macrophages with the virus, they secrete the necessary interferons. Both cell types exhibit a linear increase in the first 18 h, while the concentration of interferons remains null. The active immune response, characterized by an increase in the IFN level, occurs only after this incubation and ‘virus reconnaissance’ period. Another key point is that plasmacytoid dendritic cells gain significance as the infection advances, because they persistently present antigens to T-cells in the lymph nodes. Meanwhile, macrophages play a crucial role in containing the virus and initiating the active immune response through processes such as phagocytosis and the release of other cytokines, particularly anti-inflammatory proteins that restrict the virus spread.

• Variation of infection rates for the model (8)

For the subsequent numerical simulations, the focus was on analyzing how the number of monocytes changes in response to variations in parameters that could feasibly be modified in real-world scenarios. The infection rate, which represents the efficiency with which a virus infects target cells, is influenced by a combination of biological, environmental, and therapeutic factors [28].

To investigate this, the parameters $s_1$ and $s_2$ were varied, representing the infection rates of macrophages and plasmacytoid dendritic cells (pDCs), respectively. Variances of ${10}^{-5}$ and ${10}^{-7}$ were selected, as these values are in proximity to the typical range of ${10}^{-6}$ under normal conditions.

In the first graph (Figure 11a), the relationship between interferon concentration and monocyte infection rate is depicted. When monocytes such as macrophages and pDCs become infected, they release interferons as part of the immune response to combat viruses or bacteria. Consequently, the graph representing higher infection rates (represented by the red line) illustrates that interferon concentration exceeds 2500 pg/mL after 30 h. In contrast, the graph for lower infection rates (represented by the blue line) shows a concentration below this threshold at the same time point. These results suggest a direct proportionality between infection rates and interferon concentrations, as a higher number of infected monocytes release greater amounts of interferons.

Figure 12 focuses on the changes in pDC concentrations under higher infection rates. Small differences are detected when having different infection rates. The orange graph, which is the one with a higher plasmacytoid dendritic cells infection rate has a much more abrupt surge in infected cell concentration compared to the blue graph with the lower infection rate. Both graphs seem to indicate a stabilization around the same value after slightly more than 30 h. However, no noticeable differences are observed in these graphs, which is likely due to the macrophages initiating the immune response as the first line of defense upon encountering the virus [29]. Macrophages are widely distributed across various tissues, including the lungs [30], whereas pDCs are predominantly located in lymphoid organs and lymph nodes [31].

This distinction is further supported by Figure 11, which displays the concentration of infected macrophages over time. The graphs reveal that with higher infection rates, represented by the orange graph, a slight increase in the number of infected macrophages is evident after 25 h. Additionally, the macrophages start at a value greater than zero because of the population of infected macrophages already in the system, since they are the first immune cells to react to any already existing pathogen. For the first 18 h of the infection, both graphs show a linear rise for both infection rates (the orange and blue lines), implying that macrophage growth follows the same pattern at this early stage regardless of what the infection rate is. This highlights the role of macrophages in stabilizing the virus and initiating the immune attack [32], which is subsequently carried out by other immune cells, including pDCs.

4. Conclusions

This study provides valuable insights into the complex interaction between lymphatic flow and the immune system, specifically in the context of viral diseases like hepatitis. This research improves our understanding of the immune response by using mathematical models to analyze the movements of lymph, immune cells, and interferons. The results emphasize the intricate nature of lymphatic circulation and the crucial functions of macrophages, plasmacytoid dendritic cells (pDCs), and interferons in orchestrating a synchronized immune response against viral infections.

Interferons, although they play a crucial role in impeding virus replication and stimulating immune cells, encounter obstacles in current therapies, including a restricted efficacy and undesirable side effects. The study used an interdisciplinary strategy that combines virology with lymphatic dynamics to gain a more comprehensive understanding of how the immune system functions.

It is clear that the interferon concentration increases rapidly in order to be able to fight the infection and is in agreement with the results from [27]. However, for the evolution of the infected macrophages, the steep incline, followed by the fast decline, suggests that the infection is being treated accordingly. This seems to not be the case with the Figure 1 from [27], where the infected macrophages first have a small decrease before starting to grow in order to reach their peak, meaning that other internal factors may act upon the fight with the infection.

The proper functioning of the vascular system ensures the balance of tissue fluid levels, therefore preventing the development of edema and maintaining overall homeostasis. Hence, the coordinated initiation and cessation of activity in these cells, along with the functioning of lymphatic valves, are essential for the efficient functioning of the lymphatic system and overall health. The mathematical models presented in this paper are universally applicable for different types of populations, whether at a cellular level or at a patient level. The parameters used define the population and reflect its characteristics.

The research highlights the need for additional studies on the dynamics of lymph flow, given the complex structure of lymph nodes, arteries, and capillaries. Due to its intricacy, it is essential to dedicate time and conduct thorough research in order to enhance models and therapy procedures. This study is crucial for advancing medical progress and enhancing the effectiveness of interferon therapies and the control of viral infections.