Abstract: The behaviour of lymphoid cells in the absence of viruses has already been published in the year 2013. This study is a continuation of recent attempts to understand, via mathematical modeling, the behavior of lymphoid cells in the absence and in the presence of viruses. In this study, which is the behaviour of lymphoid cells in the presence of viruses will be treated in three respects. Firstly, the innate immune response stage, secondly, the overlap of innate and adaptive immune responses stage and finally, the adaptive immune response stage of viral infections. The adaptive immune response stage considers the viremia and cell-mediated immune responses stage. The steady states and the stability for these differential models are deduced. Each of the models permit the existence of two types of stationary states. There is the state of no infection, with no virus cells while the other is the state of co-existence where a virus cell persists against the background of immune response. The state of no infection is asymptotically stable and a state of infection is unstable. It is found from the study that the state of no infection represents the preparedness of the immune state prior to the infection. Numerical simulation analysis suggests that the cells (NK, Tc, Th and B) grow exponentially as a result of proliferation and saturation because of the contacts between them and reach therefore reach plateau as time (t) increases. These immune cells are able to reduce viral load to the barest minimum if not reducing it to zero.
INTRODUCTION
Infectious diseases are the second leading cause of death among humans worldwide, but number one cause of death in developing countries (Heffernan et al., 2009). The practical importance of understanding the dynamics and evolution of infectious diseases and specifically viral infection is steadily increasing in the contemporary world (Iwasa et al., 2007). This study is rich with epidemiological models, which have greatly added to our understanding of outbreaks, epidemics and pandemics of diverse pathogens. Generally, diseases transmitted by viral agents, such as influenza, measles, rubella (German measles) and chicken pox, confer immunity against reinfection, while diseases transmitted by some bacteria, such as tuberculosis, meningitis and gonorrhea, confer only partial immunity against reinfection (Brauer, 1984).
Resistance is better known as immunity. The immune system is the body’s defense mechanism against infectious diseases. The immune system includes the organs, tissues, cells and molecules responsible for immunity. There are two types of immunity. The first is innate (natural or native or nonspecific) which refers to the basic resistance to disease that an individual is born with. The second one is the specific or acquired or adaptive immunity which requires the activity of a functional immune system involving cells called lymphocytes and their products (Nirvanagrewal, 2012).
Innate defense mechanisms provide the first line of host defense against invading pathogens until an acquired immune response develops. The interaction between the pathogen and the components of innate immunity triggers generation of an adaptive immune response that usually consists of pathogen-specific cytotoxic T cells (CTLs) and antibody molecules produced by B cells (Heffernan et al., 2009).
Acquired or adaptive or specific immunity reflects the presence of a functional immune system that is capable of specifically recognizing and selectively eliminating foreign microorganisms. T cells, B cells and natural killer cells play an important role in immunity against viral infections. T cells have been divided into two major subsets that are functionally and genetically different.
T helper cells (CD4+ T cells) which function to mediate responses by the secretions of lymphokines that stimulate other cells involved in the immune responses. The second subset are cytotoxic T cells (CD8+ cells) and these cells are directly involved in the killing of certain tumor cells, virus infected cells, transplant cells and sometimes eukaryotic parasites. The CD8+ T cells are also important in down regulation of immune responses. In this case, they are referred to as T suppressor cells. Natural killer cells are similar to the CD8+ T cells. They function as effector cells that directly kill certain tumor cells and virus infected cells. When the innate immune response fails to control the infection, the adaptive immune response is activated and this involves the production of antibodies and primed T cells. The CD8+ cytotoxic T lymphocytes (CTLs) kill target cells infected with viruses or bacteria. The CD4+ T-helper (Th1) cells provide help for B cells to develop into antibody secreting plasma cells following stimulation by foreign antigens such as bacteria antigen and tumor cells. Antibodies are specialized proteins that specifically recognize and bind to specific antigens that call their stimulation. Antibody production and binding to foreign antigen is often critical as a means of signaling other cells to engulf, kill or remove that substance from the body. Antibody production and secretion of cytokines that play a role in immune-regulatory functions or have a direct effect on invading pathogens (Germain, 1994). Human immune response to viral infections is caused by a variety of cells in the innate and adaptive mechanisms (Baron, 1996). Many different population growth models have been used for modeling disease progress curves. The logistic model has many real world applications in biology, ecology, statistics, neural networks, reaction models (chemistry), Fermi distribution (physics) and in medicine. Logistic growth model has been assumed in many epidemic models where population growth is limited (Ackleh and Allen, 2003). The main point about the logistic model is that it is a particularly convenient form to take when seeking for qualitative dynamic behaviour in populations (Murray, 2001).
Mathematical modeling using differential equations and dynamical systems have been used in the studies of immune response to various infections, most notably that of the HIV. The question now is how does the human body develop immunity or immune response to these infectious diseases such as viruses? The mathematical biologists Anderson and May (1992) proposed a theory and a mathematical model to explain this phenomenon. Bittner and Wahl (2000) studied the immune response against conserved and variable viral epitopes. The main immune cell studied was cytotoxic T lymphocytes. Wodarz (2004) reported on mathematical models which have investigated the importance of lytic and non-lytic immune responses for the control of viral infections. Lytic immune responses fight the virus by killing infected cells, while non-lytic immune responses fight the virus by inhibiting viral replication. All these researchers have not dealt with the behavior of these lymphoid cells which fight the viruses. Wodarz et al. (2007) published a study on the dynamics of killer T cell inflation in viral infections in which authors analyzed the impact of innate and adaptive immune responses. According to the study of these authors, a potentially contributor to cytotoxic T lymphocytes inflation is a competition between the specific cytotoxic T lymphocytes response and an innate Natural Killer (NK) cell response. Hancioglu et al. (2007), presented a simplified dynamical model of immune response to uncomplicated Influenza A Virus (IAV) infection which focuses on the control of the infection by the innate and adaptive immunity. Long et al. (2008), also worked on a Mathematical Modeling of Cytotoxi Lymphocyte-Mediated Immune Response to Hepatitis B Virus Infection in which the Human Immunodeficiency Virus (HIV) infection was successfully to simulate the interaction between HIV and cytotoxic lymphocyte mediated immune response and also considered the indicator of the liver cell damage between Hepatitis B and the cytotoxic mediated immune response and the indicator of the liver cell damage. Wiah et al. (2011) presented a mathematical model of immune response to Hepatitis B Virus (HBV) infection which focuses on the control of the infection by the interferons, innate and adaptive immunity. Nakata (2011) published study on the global dynamics of a cell mediated immunity in viral infection models with distributed delays which admitted three possible equilibria states. Pawelek (2012), also published a study on the mathematical modeling of virus infections and immune responses in which HIV infection was considered. The author of this study examined the relative roles of target cell availability and innate and adaptive immune responses in controlling the viruses. In addition, the study of Pawelek (2012), provided a quantitative understanding of the biological factors which could explain the viral and interferon kinetics during a typical influenza virus infection. Ben-Shachar and Koelle (2014) also published a study on Minimal within-host dengue models which highlights the specific roles of the immune response in primary and secondary dengue infections. Tian and Wang (2015) published a study on the stability and analysis for viral infections which focused on humoral immunity.
These authors dealt with specific viral infections and therefore do not seem to continue with the work of Anderson and May (1992). This current study seeks to modify the work of Anderson and May (1992) which consisted of the behaviour of two effector cells (T and B lymphocytes) in the presence of viruses. Anderson and May (1992) considered the adaptive stage of virus clearance. This study seeks to extend it by including the dynamics of two effector cells (Natural Killer cells and T helper cells) and also to consider the innate immune response stage where these NK cells provide the first line of defense to viral infections. The study also seeks to consider an overlap of the innate and adaptive immune responses of these effector cells to the viral infection and finally consider the adaptive stage which also has two sub-divisions thus viremia stage and cell-mediated immune responses.
MATERIALS AND METHODS
The model is developed in four stages as the innate immune response stage, the overlap of innate and adaptive immune responses stage, the viremia stage of viral clearance and finally the cell-mediated adaptive immune response stage. The model of the study contains five variables and these are Natural killer cells (N), Cytotoxic T cells (Tc) T helper cells (Th) and B lymphocytes (B). The assumptions of the mathematical model as well as certain parameter values are borrowed from the dynamics and data provided in the study of Anderson and May (1992). The constants reproduction rates, the self-reproduction rate, death rates and the rate at which the immune cells and the viruses interact to saturate are considered in the models, the equilibrium points and their stability for the system of the extended differential equations are analyzed and the stability of the linearized equations is determined. Time histories of these systems of differential equations are also used to analyze the systems. Phase portrait is drawn to show the interaction between the T cells and the virus cells. Parameter values which had been estimated by Anderson and May (1992) are extended based on the assumptions made for the construction of the models and these are used in the study. The different phases used in the model are presented.
The model describes the behaviour of lymphoid cells in the presence of viruses. This model described three main stages. The first stage is the innate immune response stage of viral infection, the second is the overlap of innate and adaptive immune responses stage of viral infection and lastly the adaptive immune response stage which was considered by Anderson and May (1992), but with the difference that two sub-stages are considered (viremia and cell-mediated immune adaptive immune responses to viral infections). The innate immune response stage of viral infection model involves one type of lymphoid population (Natural killer cells) denoted by N and the virus population denoted by V. The overlap of innate and the adaptive immune responses stage of viral infection model also involves three types of lymphocyte populations denoted by Tc, B and N respectively. The final model which was considered by Anderson and May (1992) and which is the final stage is also considered with the inclusion of two sub-stages.
Innate immune response: Innate immune response is essential for the early detection of invading viruses which help to trigger the activation of adaptive immune responses. The rate of interaction between the virus cells (V(t)) and the natural killer cells (N(t)) occurs on the 1st day of the viral infection as they try to provide the first line of defense when the human body is infected with virus. The rate of interaction between these two cells has the following key properties:
| • | New lymphoid cells of (NK cells) are produced by the bone marrow at a constant rate of An |
| • | NK cells die at a per capita rate of μn |
| • | NK cells kill virus cells in proportion to the number of contacts between them |
| • | The virus cells have an intrinsic growth rate when they enter a living being |
Model of innate immune response: These assumptions lead to the system of differential Eq. 1 and 2:
N = An-μnN+γNV |
(1) |
V = rV-kVN |
(2) |
where, An corresponds to new lymphoid cell of NK cells produced. The μnN corresponds to the rate at which NK cells die. The γNV corresponds to the rate of growth of NK cells due to interactions with the virus cells. The term rV represents the intrinsic growth of the virus cells while kNV corresponds to the rate at which the virus cells die due to the interactions with NK cells.
Existence of steady states in the innate immune response stage: In the presence of virus cells in the human body, we considered three main stages. These were the innate immune response stage, the overlap of innate and adaptive immune responses stage and finally the adaptive immune response stage.
By system of Eq. 1 and 2, the steady states are obtained:
An-μnN+γNV = 0 |
(3) |
rV-pVN = 0 |
(4) |
where, N*, V* is a trivial steady state solution. The Jacobian matrix of system of Eq. 1 and 2 is as follows:
| (5) |
Overlap of innate and adaptive immune responses stage: This stage is where the T cytotoxic, T helper and B cells have just started to fight the virus cells. In this case, all the four lymphoid population types will still be in the human body before the N cells become inactive. The interaction with the virus cells has the following key properties:
| • | T cytotoxic cells directly kill virus cells in proportion to the number of contacts between them and they proliferate because of these contacts |
| • | The T helper cells also activate the T cytotoxic cells to kill the virus cells |
| • | T Helper cells do not directly interact with the virus cells, but they continue to regulate the B cells to produce antibodies to kill the virus cells and the cells proliferate as well |
Model of overlap of the innate and adaptive immune responses stage: The key properties of (N, B, Tc and V) cells lead to the system of Eq. 6-9 of differential equations:
| (6) |
| (7) |
| (8) |
| (9) |
where, A in Eq. 6 represents the constant production rate of B, μbB is the self-reproduction of B cells at μb:
represents the proliferation and saturation nature of the growth of the two lymphocytes (ThB) as they interact. The term KVB in Eq. 7 represents growth rate of B lymphocytes as a result of the interaction with the virus cells. The term λTcV in Eq. 8 represents the growth rate of T as a result of the interaction between the T cytotoxic cells and the virus cells. The term rV in Eq. 9 represents the intrinsic growth rate of the virus cells.
Existence of steady states of the overlap of innate and adaptive immune responses: By system of Eq. 6-9, we obtained steady states by putting derivatives to zero. We obtained (N*, B*, T*, V*) as a trivial steady state solution. Similarly, we obtained the Jacobian matrix as:
| (10) |
Viremia stage of adaptive immune response: B cells, T cytotoxic cells and T helper cells have been recruited to take full control in fighting the viral infection. This is because Natural killer cells are overwhelmed and have become inactive by the virus cells. Here, T helper cells activate B cells to produce more antibodies to fight the virus. The T helper cells stimulate the T cytotoxic cells to maturity to kill the virus cells directly. The interaction between the lymphocytes and the viral cells has the following key properties:
| • | Virus cells have an intrinsic growth rate as they are in a living being |
| • | T helper cells activate the T cytotoxic cells to maturity to kill virus cells in proportion to the number of contacts between them and they proliferate because of these contacts |
| • | T helper cells do not directly interact with the virus, but they continue to regulate the growth of B cells to produce more antibodies to kill the virus cells |
Model of the viremia stage in the adaptive immune response stage: The rate of interaction of the cells (B, Tc, Th and V) results in the following system of Eq. 11-14 of differential equations:
| (11) |
| (12) |
| (13) |
| (14) |
where, Ab represents the constant production rate of B, μbTb is the self-reproduction of B cells at:
represents the proliferation and saturation nature of the growth of the two lymphocytes (ThB) as they interact. The term KVB in Eq. 11 represents growth rate of B lymphocytes as a result of the interaction with the virus cells. The term λTcV in Eq. 12 represents the growth rate of Tc as a result of the interaction between the T cytotoxic cells and the virus cells. The term rV in Eq. 14 represents the intrinsic growth rate of the virus cells.
Existence of steady states in the viremia stage of adaptive immune response: By system of Eq. 11-14 we obtained steady states by putting derivatives to zero. We have the corresponding Jacobian matrix as:
| (15) |
Cell-mediated adaptive immune response stage: Cell-mediated immune response is very important in the fight against viral infections. This is especially with infections that involve oncogenic viruses ( that is: viruses that spread directly from cell to contiguous cell). Antibody in such situations cannot reach the virus but rather virally induced antigen on the surface of the infected cell can be recognized by different effector cells such as cytotoxic T cells. Cell-mediated immune response involves T cytotoxic cells (CD+ 8 T cells) and T helper cells (CD+ 4 T cells).
The interaction between the lymphocytes and the viral cells has the following key properties:
| • | Virus cells have an intrinsic growth rate as they are in a living being |
| • | Activated T helper cells produce a number of cytokines that defend against viruses directly |
| • | The T cytotoxic cells have the ability of producing cytokines to directly attack the virus cells |
| • | The interaction between T helper and T cytotoxic cells lead to proliferation and saturation of the two effector cells |
| • | The T helper and T cytotoxic cells are the main components of cell-mediated antiviral defense |
The interaction of these cells results in the following system of differential equations:
| (16) |
| (17) |
| (18) |
where, Ac represents the constant production rate of Tc, μcTc represents the self-reproduction of T cytotoxic cells at:
represents the proliferation and saturation nature (plateau) of the growth of the two lymphocytes (TcTh) as they interact. The term λTcV in Eq. 16 represents growth rate of Tc lymphocytes as a result of the interaction with the virus cells. The term rV in Eq. 18 represents the intrinsic growth rate of the virus cells. It should be noted that the dynamics of Tc and Th cells are similar which reflects in Eq. 16-17:
Jacobian matrix of cell-mediated adaptive immune response stage:
| (19) |
| Table 1: | Parameters values supplied by Anderson and May (1992) |
Statement of the problem: At the dawn of the 21st century humankind is faced with new, more resilient diseases including HIV/AIDS and hepatitis B that come along with a death toll from preventable infectious diseases that remain high due to poor sanitation and malnutrition among other conditions in many parts of the world. A good understanding of the dynamics of viral infections, the various stages of viral infections and the interaction of immune cells at these specific phases are important to help our natural countermeasures which could be augmented with modern medicinal techniques and immunotherapies (Table 1).
Computational procedure
MATLAB: Ordinary differential equation solver ‘ode45’ is used to evaluate the numerical solutions of the systems of differential equations. The MATLAB software is installed on a Laptop with the specification as Toshiba (Brand) with 4.4 rating of Windows experience index with a processor of Intel (R) Pentium (R) CPU B 960 at 2.20 GHz, memory installed is 2.00 GB and 32 bit operating system and has windows 7 ultimate edition installed. The MATLAB software version is R2009a. The data are substituted into Eq. 5, 10, 15 and 19 to get 2 by 2, 4 by 4, 4 by 4 and 3 by 3 Jacobian matrices for innate immune response stage, overlap of innate and adaptive immune responses, viremia stage of adaptive immune response and cell-mediated immune response stage respectively. MATLAB codes were written to find the equilibrium points and their corresponding eigenvalues. The eigenvalues were then classified into either a stable point or unstable point which describes the behaviour of these eigenvalues.
RESULTS
This study is an extension of the model by Anderson and May (1992) on human immune response to virus infectious diseases and it has yielded results that are in consistent with the model by these previous authors with the additional results which come along with the addition of other phases of immune response which had been captured by Anderson and May (1992).
| Table 2: | Parameters values in the model estimated by the authors |
The various results of the study are presented.
Linearization stability analysis: Although it is usually not easy to determine the stability of an equilibrium point of a system of differential equations, the determination of the asymptotic stability is usually quite easy. The method involves linearization of the equations about the equilibrium point and the determination of the stability of the linearized equations. The numerical calculation of eigenvalues of matrices can easily be carried out with many mathematical software packages (e.g., MATLAB, MAPLE, MATHEMATICA). The linearization method examines the behaviour of the system close to equilibrium point. The stability of the equilibrium point can be determined by finding the eigenvalues of the system.
Equilibrium points in the innate immune response stage: We obtained the equilibrium points of systems of Eq. 3 and 4 by substituting the parameter values of Table 2.
1-1.25N+0.05 NV = 0 |
(20) |
0.0V-1.25VN = 0 |
(21) |
The equilibrium points are determined by Universal Mathematics Equation Solver.
There exists two equilibrium points and these are in Eq. 22:
| (22) |
Substituting the values of the parameters of Table 2 in Eq. 23, we have:
| (23) |
At (N*, V*) = (0.8,0) we obtained:
| (24) |
| Fig. 1: | Interaction between the natural killer cells and the viral cells at the early stages of viral infection |
| Table 3: | Classification of the equilibrium points in the innate immune response stage |
The corresponding eigenvalues were found to be:
λ1 = -1.2500 and λ2 = -0.9000 |
(25) |
At (N*, V*) = (10, -255) we obtained in Eq. 26:
| (26) |
The corresponding eigenvalues are:
λ1 = -12.5894 and λ2 = 0.0894 |
(27) |
It is observed from Table 3 that one of the equilibrium points is stable while the other one is unstable. This gives an assurance that even at the early stages of viral infections, the human system is somehow stable if only the natural killer cells are functioning well.
Figure 1 represents the interaction between the natural killer cells and the virus cells. The y-axis represents the N cells whiles the x-axis represents the virus cells. We observed that all the arrows converge to a particular point (0.8, 0). This confirms that there is stability when the virus cells enter the human body for the 1st week. The natural killer cells are able to suppress viral abundance to a very low level in the host within this 1st week.
Figure 2 represents the interaction between the N cells and the virus cells in the innate immune response stage. In this figure, we plot N cells and virus cells against time. The curve shows both cells decrease with time but the rate of decrease of the virus cells is greater than the rate of decrease of the N cells. The virus cells decrease drastically with time. We observed that the virus cells die faster than the natural killer cells as time increases. As a result of the interaction of the two cells, both cells decrease asymptotically as time increases. Besides, the virus cells approach a constant endemic value of about 0.001 whereas the N cells approach a fixed value of 0.8.
Equilibrium points of the overlap of innate and adaptive immune responses stage: We obtained the equilibrium points of systems Eq. 6-9 by substituting the parameter values of Table 2:
| (28) |
| (29) |
| (30) |
| (31) |
MATLAB is then used to find the equilibrium points as:
| (32) |
We realized that two of these equilibrium points involve complex numbers and therefore are neglected. By substituting parameter values, we obtained:
| (33) |
At (N*, B*, T*, V*) = (1, 0.8, 1, 0) we obtained:
| (34) |
| Fig. 2: | Growth nature of the Natural killer cells (N) and the Viral cells (V) at the innate immune response stage |
The corresponding eigenvalues were found to be:
λ1 = -2.2230, λ2 = -2.3808, λ3 = -1.2500 and λ4 = -1.1900 |
(35) |
At (N*, B*, V*) = (5, 0.8, 51, 0) we obtained:
| (36) |
The corresponding eigenvalues were found to be:
λ1 = -6.1217, λ2 = -2.5280, λ3 = -2.3579 and λ4 = -0.4197 |
(37) |
At (N*, B*, T*, V*) = (20, 0.8, 20, 0) we obtained:
| (38) |
The corresponding eigenvalues are found to be:
λ1 = 3.0239, λ2 = 2.3983, λ3 = 1.2500 and λ4 = -24.9400 |
(39) |
At (N*, B*, T*, V*) = (0.0781, 0.07686, 0.8126, -235.1599), we obtained:
| (40) |
The corresponding eigenvalues are found to be:
λ1 = 1.4035, λ2 = 2.8973, λ3 = 12.740 and λ4 = -13.0002 |
(41) |
It is observed from Table 4 that there is greater percentage of stability when (NK, Tc and B cells) get recruited to fight the infection. The behaviour of two of the equilibrium points are asymptotically stable while the other two are not stable.
Figure 3 represents the overlap of innate and adaptive immune response stage of viral infection. In this figure, we plot N cells, T cytotoxic cells, B cells and virus cells against time. We observed that B and T cells increase to a certain peak value of 20 and approach this constant endemic value as time increases. However, the N killer cells and the virus cells decrease to certain constants values of about 0.8 and 0.0001, respectively. This confirms the fact that the N cells get overwhelmed and become inactive after some time of viral infection.
Equilibrium points in the viremia stage of adaptive immune response: We obtained the equilibrium points of Eq. 11-14 by substituting the parameter values of Table 2.
| (42) |
| (43) |
| Fig. 3: | Growth nature of the viral cells and the immune cells (Natural killer cells, T cells and B lymphocytes) at the overlap of innate and adaptive immune responses stage |
| (44) |
| (45) |
MATLAB is then used to find the equilibrium points as:
| (46) |
And four other complex equilibrium points which have been displayed in the Appendix.
We evaluated the equilibrium points by substituting the parameter values of Table 3-4:
| (47) |
At (B*, Tc*, Th*, V*) = (-10.0792, 35.5400, -0.0735, 4.9196) we obtained:
| (48) |
The corresponding eigenvalues were found as:
λ1 = -46.6113, λ2 = -3.6669, λ3 = 13.6031 and λ4 = 9.7602 |
(49) |
At (B*, Tc*, Th*, V*) = (-1.4616, 0.4864, -1.6365, 1.4487), we obtained:
| (50) |
The eigenvalues were found as:
λ1 = 0.6865, λ2 = 0.3353, λ3 = -1.6458 and λ4 = -1.10064 |
(51) |
At (B*, Tc*, Th*, V*) = (0.0810, 0.7538, 0.0841, -9.0231), we obtained:
| (52) |
The corresponding eigenvalues were found as:
λ1 = -12.2183, λ2 = -10.1715, λ3 = -2.9879 and λ4 = -1.2974 |
(53) |
At (B*, Tc*, Th*, V*) = (0.5379, 0.8650, 0.5479, -0.6609), we obtained:
| (54) |
The corresponding eigenvalues were as below:
λ1 = -3.1767, λ2 = -1.4653+0.8241i, λ3 = -1.4653-0.824i and λ4 = -1.0953 |
(55) |
Table 5 displays the classification of equilibrium points of the viremia stage of adaptive immune response. Two of the equilibrium points are unstable while the other two equilibrium points are stable.
Figure 4 presents the viremia stage of adaptive immune response which involves the interaction between B cells, T cytotoxic cells, T helper cells and the virus cells. In this figure, we plot B cells, T cytotoxic cells, T helper cells and virus cells against time in days.
| Table 4: | Classification of the equilibrium points in the overlap of innate and adaptive immune responses |
| Table 5: | Classification of the equilibrium points in the viremia stage of immune response |
| Fig. 4: | Growth nature of the viral cells V and the immune cells (T helper, T cytotoxic and B lymphocytes) at the viremia stage of adaptive immune response |
We observe that B cells, T cytotoxic cells and T helper cells rise to peak values of 3.0, 1.5 and 1.2, respectively. There is a very sharp increase in B as compared with T cytotoxic and T helper cells. On the other hand, virus cells decrease drastically towards zero. This indicates that B cells are more active in the viremia stage of immune response to viral infections.
Cell-mediated stage of adaptive immune response: By substituting parameter values into Eq. 16-18, we obtained the Jacobian matrix hence finding the corresponding equilibrium points using MATLAB:
| (56) |
The equilibrium points are as follows:
| (57) |
We evaluate the equilibrium points by substituting the parameter values of Table 2 into Eq. 19:
| (58) |
At (B*, Tc*, Th*, V*) = (1, 1, 0) we obtained:
| (59) |
The corresponding eigenvalues were found to be:
λ1 = -2.0050, λ2 = -2.5010 and λ3 = -1.100
At (B*, Tc*, Th*, V*) = (5, 5, 0), we obtained:
| (60) |
The corresponding eigenvalues were found to be:
λ1 = -1.2844, λ2 = -2.4996 and λ3 = -29.90
At (B*, Tc*, Th*, V*) = (0.3221, 0.2583, -2.252), we obtained:
| (61) |
The corresponding eigenvalues were found to be:
λ1 = -2.1272, λ2 = -2.50 and λ3 = -479.90
At (B*, Tc*, Th*, V*) = (0.3221, 0.2583, -2.252), we obtained:
| (62) |
| Fig. 5: | Growth nature of the viral cells V and the immune cells (T cytotoxic and T helper cells) |
| Table 6: | Classification of the equilibrium points in the cell-mediated stage of adaptive immune response |
The eigenvalues were found to be:
λ1 = 0.1296, λ2 = -3.9391 and λ3 = -3.1447
At (B*, Tc*, Th*, V*) = (-0.3646, -0.2285, 4.7646), we obtained:
| (63) |
The eigenvalues were found to be:
λ1 = 0.1296, λ2 = -3.9391 and λ3 = -3.1447
Table 6 displayed the equilibrium points in the cell mediated stage of the adaptive immune response. This is the last stage of immune cells’ destruction of viral infections. The behavior of the equilibrium points show stability more than the other stages provided there are no deficiencies with the two main cells (Th and Tc) that are the main components of the last phase of viral destruction.
Figure 5 the cell-mediated stage of adaptive immune response stage of viral infection. This involves the interaction between T cytotoxic cells and T helper cells, respectively (CD+ 8 cells and CD+ 4 cells). We plot T cytotoxic, T helper and Virus cells against time in days. We observed that virus cells approaches zero with time. There is a sharp increase in the growth of T helper and T cytotoxic cells as time increases. The growth rate of T cytotoxic cells is very fast as compared to T helper cells. We observed that as T cytotoxic approaches a certain endemic value of 200, T helper cells approach a constant endemic value of 70 within the same duration of growth. We observe that at this last phase of immune response to viral infections, the viral load approaches zero. This shows that T cytotoxic cells are more active than the T helper cells.
RESULTS
This study is an extension of Anderson and May (1992) model on human immune response to virus infectious diseases and it has yielded results that are in consistent with these workers except that this current study included the dynamics of a third lymphoid cell (Natural Killer Cells) which provide a first line of defence and if a therapy is designed for it, viral infections could be compacted faster. In the presence of viruses Anderson and May (1992) considered only one phase of the adaptive immune response and had three different behaviour of the equilibrium states which are two unstable states at:
λ1 = -1.25, λ2 = -0.75397 and λ3 = 0.09 and
λ1 = -1.25 λ2 = 0.05 and λ3 = 0.05
respectively and one asymptotically equilibrium state as λ1 = -1.25, λ2 = -0.6785 and λ3 = -0.1. The last stage of immune response which was considered by Anderson and May (1992), which yielded three steady states gives five behaviour of steady states in this current study and these are three asymptotically stable states and two unstable states.
The current study realizes two behaviour of equilibrium points at the innate immune response stage (the first line of defense) that was not considered by Anderson and May (1992) and these are.
λ1 = -1.25 and λ2 = -0.90-asymptotically stable and λ1 = -12.5894 and λ2 = 0.0894-saddle point. Four behaviour of equilibrium points are realized at the overlap stage of immune response and these are: λ1 = -2.2230 λ2 = -2.3808, λ3 = -1.2500 and λ4 = -1.1900-asymptotically stable: λ1 = -6.1217, λ2 = -2.5280, λ3 = -2.3579 and λ4 = -0.4197: λ1 = -3.0239, λ2 = -2.3983, λ3 = -1.250 and λ4 = -24.94-asymptotically stable: λ1 = -1.4035 λ2 = -2.8973, λ3 = -12.740 and λ4 = -13.00-saddle point and finally-saddle point. These behaviour indicate the systematic stability that the human system maintains at the various stages of immune response to viral infections. The viremia stage gives only one stable state of equilibrium since NK cells become inactive and T helper cells are also not active. The only stable state is:
λ1 = -3.177 λ2 = -1.465+0.824i,
λ3 = -1.465-0.824i and λ4 = -1.095
which is also a stable sink. All the rest of the equilibrium points give unstable states and these are:
λ1 = -46.6113, λ2 = -3.6669,
λ3 = 13.6031 and λ4 = 9.7602
λ1 = 0.6865, λ2 = 0.3353,
λ3 = 1.6458 and λ4 = -1.1064
λ1 = -12.218, λ2 = -10.172,
λ3 = -2.988 and λ4 = -1.287
The study also considered the dynamics of the two types of T cells thus T cytotoxic and T helper cells that have different dynamics. The inclusion of these cells have helped to explain the viral infection in a wider perspective. We realized that there is stability in the human body when the virus cells enter the body. All the models considered give an equilibrium state that is asymptotically stable. At this point, the lymphoid cells are said to be in the immune state and any further infections result in a rapid re-equilibration. The models, therefore, display all of the major macroscopic characteristics of the human immune response to viral infections.
DISCUSSION
This results is in consistent with the results of Anderson and May (1992) which predicts two different types of study states thus a stable state and an unstable state even though their model is on two lymphocyte population types and this current study also considers four lymphocyte types. The unique aspect of this study is the inclusion of other two types of effector cells thus Natural Killer cells and T helper cells (NK cells and CD+ 4 cells). The current study had considered the immune response to viral infection in a wider perspective and each of the models at the specific stages of immune response also predicts two different types of steady states thus a stable state and an unstable state. The model of Bittner and Wahl (2000) predicts the existence of four different types of study states thus a state of no infection, infection being controlled by immune cells, all mutants held in check by T lymphocytes and responses to both conserved and variable epitopes control the infection. The results of Bittner and Wahl (2000) are in consistent with the results of both the current study and the study of Anderson and May (1992). The main immune cell studied in the model of Wodarz (2004) is cytotoxic T lymphocytes and the results show that T lymphocytes induce pathology in the clearance of viral cells and also suppress viremia to a certain degree but there is no obvious correlation between pathology and viral load. The study of this author is on Hepatitis C Virus infection. The results of this current study is in consistent with the results of Wodarz (2004), since the numerical simulation analysis shows the suppression of the viral load at all the stages of viral clearance. The simulation and sensitivity analysis of the model of Hancioglu et al. (2007) shows that the diseases fall into either of these three categories: asymptomatic diseases, typical diseases and severe diseases which represent various viral loads and the analysis of the adaptive immune response showed that whenever there is sufficient antibody response with enough specificity, the health of the host will restore, irrespective of the intensity of the innate response. The results of Hancioglu et al. (2007) are in consistent with this current study since the simulation analysis of the final stage of the adaptive immune response of the current study showed enough stability in the host. The categories with their corresponding viral loads also corresponds to the continuous reduction of viral load at the various stages of this current study. The results of the study of Wiah et al. (2011) showed the existence of disease free and endemic equilibrium states. The analysis of the adaptive immune response stage of this same study showed that with sufficient antibody response with enough specificity, the dynamics is able to restore the health of the host irrespective of the intensity of the innate response. The results of Nakata (2011) showed three possible equilibria and these are an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infections. This results is also in consistent with the results of the current study which are infection equilibrium (unstable state) and infection free equilibrium (asymptotically stable equilibrium state). The model of Tian and Wang (2015), established two threshold parameters which represents the infection-free equilibrium and the endemic equilibrium respectively. This results is also in the consistent with the results of the current study.
This study has extended the work of Anderson and May (1992) which considered only the adaptive immune response stage of viral infection and also dealt with two main effector cells thus (T cytotoxic and B cells) to four main stages with four main effector cells thus (Tc, Th, B and NK cells) thus innate, overlap of innate, viremia and cell-mediated immune responses stage. The system of three differential equations by Anderson and May (1992) of the adaptive immune response stage has been extended to a system of four differential equations yielding three asymptotical stabilities as compared to the one asymptotical stability in the adaptive immune response stage in the work of Anderson and May (1992). By numerical simulation analysis, the immune system is seen to be very effective in the fight of viral infections if the specific immune cells (Tc, Th, B and NK cells) function effectively. By stability analysis, there is the state of infection free steady state(s) at each stage of the stages of immune response to viruses that corresponds to the asymptotical stability which represents the ability of the immune cells (Tc, Th, B and NK cells) to fight viruses without the activation by drugs if there are no deformities of these cells. In summary, all the equilibrium points fall into either an asymptotical stable state or unstable state as compared to the study of Bittner and Wahl (2000) which shows four main stability states. The state of endemic steady state where the equilibrium points are unstable represents the immune-deficiency as a results of one or more of these cells not functioning properly or such cells may be absent altogether. The simulation analysis shows that at all the stages of immune response to viral infections, the cells of the immune system considered increase as viral load decreases which agrees with the three categories of viral loads of the study of Hancioglu et al. (2007). In summary, the stability analysis in the current study is in consistent with the results of Anderson and May (1992), Bittner and Wahl (2000), Wodarz (2004), Hancioglu et al. (2007), Wiah et al. (2011), Nakata (2011) and Tian and Wang (2015).
CONCLUSION
It can be concluded that the characteristics and growth nature of the immune cells are different thus it is recommended that further research be carried out on variable sensitivity analysis and also bifurcation analysis to determine the specific immune cell which is paramount to fight viral infection and at what specific stage it will function. It is also recommended that further research be carried on a specific viral infection in the near future which could predict an immunotherapy for such a viral infection.
| Appendix: | MATLAB Codes for numerical solutions and simulations |