A comparison of elasticities of viral levels to specific immune response mechanisms in human immunodeficiency virus infection
 Sarudzai P Showa^{1, 2}Email author,
 Farai Nyabadza^{2},
 Senelani D HoveMusekwa^{1} and
 Gesham Magombedze^{3}
https://doi.org/10.1186/175605007737
© Showa et al.; licensee BioMed Central Ltd. 2014
Received: 4 July 2014
Accepted: 2 October 2014
Published: 20 October 2014
Abstract
Background
The presence of an asymptomatic phase in an HIV infection indicates that the immune system can partially control the infection. Determining the immune mechanisms that contribute significantly to the partial control of the infection enhance the HIV infection intervention strategies and is important in vaccine development. Towards this goal, a discrete time HIV model, which incorporates the life cycle aspects of the virus, the antibody (humoral) response and the cellmediated immune response is formulated to determine immune system components that are most efficient in controlling viral levels. Ecological relationships are used to model the interplay between the immune system components and the HIV pathogen. Model simulations and transient elasticity analysis of the viral levels to immune response parameters are used to compare the different immune mechanisms.
Results
It is shown that cellmediated immune response is more effective in controlling the viral levels than the antibody response. Killing of infected cells is shown to be crucial in controlling the viral levels. Our results show a negative correlation between the antibody response and the viral levels in the early stages of the infection, but we predicted this immune mechanism to be positively correlated with the viral levels in the late stage of the infection. A result that suggests lack of relevance of antibody response with infection progression. On the contrary, we predicted the cellmediated immune response to be always negatively correlated with viral levels.
Conclusion
Neutralizing antibodies can only control the viral levels in the early days of the HIV infection whereas cellmediated immune response is beneficial during all the stages of the infection. This study predicts that vaccine design efforts should also focus on stimulating killer T cells that target infected cells.
Keywords
HIV immune responses Elasticity analysis Discrete time modelsBackground
The human immune system is a complex network of cells, chemicals and organs that keeps an individual healthy. There are three lines of defense that make up the human immune system. The three types of immunity are, passive immunity, innate/nonspecific immunity and adaptive/specific immunity. Passive immunity is borrowed from an external source and lasts for a short time. Innate immunity comprises of barriers such as the skin and the mucous membranes. There are two major branches of the specific immune responses which are the humoral immune response and the cellmediated response. Humoral immunity is mediated by B cells and cellmediated immunity involves the production of cytotoxic Tlymphocytes (CTLs), activated macrophages, activated natural killer (NK) cells and cytokines in response to an antigen and is mediated by Tlymphocytes.
Understanding the interactions of these immune system mechanisms during an HIV infection is of great importance in HIV treatment and vaccine development. However, there are no good animal models for this infection and mathematical models have been the basic tool used to understand these interactions [1–4]. All the models developed and the related HIV research pointed to the same conclusion that, although the immune response poses a tough challenge to HIV infection, the virus adopts several immune system escape mechanisms. The virus can escape the immune system through mutations [5–7], the formation of the viral latent reservoir [8, 9] and through the use of its proteins such as the Vif, Vpu, Tat[10] and Nef[11]. Treatment regimes were developed to augment the human immune system in the fight against HIV infection. However, these treatment regimes are not easily accessible in developing countries and hence the need to find a cure or vaccine for the infection.
In this study, we model the within host dynamics of the HIV infection using discrete time models because of their relative simplicity in computing transient elasticities of viral levels to immune system parameters compared to ordinary and partial differential equations (ODEs and PDEs). A discrete time model also allows the incorporation of all the life cycle aspects of the virus and yet remain relatively simple to analyse compared to ODEs and PDEs. The main advantage of transient analysis over asymptotic analysis is that it focuses on perturbations to the population structure rather than perturbation analysis on demographic rates only [12]. Transient elasticity analysis is also effective in analyzing the effects of changes of parameters and initial conditions on population levels in the short term.
Ecological modelling tools are employed to model the interplay between the immune system and the HIV pathogen. The human immune system is treated as an ecosystem and the components of the immune system are treated as species in that ecosystem. Pathogens, in this case, HIV, are then defined as exotic species, which can either invade the ecosystem or be driven to extinction. In an ecosystem, all organisms are connected by ecological relationships namely predation, competition, mutualism, commensalism, amensalism and parasitism. Such relationships are comparable to the human immune system. In the human immune system, examples of such relationships include CTLs hunting and killing infected cells, antibodies neutralizing viruses, one strain of HIV competing for CD4 ^{+} T cells, two or more HIV strains competing for CD4 ^{+} T cells, CD4 ^{+} T cells offering a catalytic effect to both the B cells and the CD8 ^{+} T cells and immune surveillance, where different components of the immune system interact to maintain homeostasis. Transient elasticity analysis is then used to compare the two arms of the specific immune response and hence inform on HIV vaccine or treatment development. Elasticity analysis can be defined as a method of evaluating how proportional changes in model parameters affect the population growth. Parameters with high absolute values of elasticities are predicted to give greatest changes in population levels when altered by a fixed quantity, thereby predicting good targets for the control. In this study, the immune response in which these parameters are found, is the immune response that is predicted to be most effective in controlling the viral levels.
Results
Model simulations
Numerical simulations are performed to show the dynamics of HIV infection under different immune response mechanisms. These simulations are performed in order to identify the specific immune system component that is most effective in controlling the HIV infection. Each immune response mechanism is considered separately by setting all the other immune response mechanisms to zero, except the ones under consideration. For instance, to consider the chemokine antiviral response which work by reducing the infectivity of the virus, the expressions labeled antibody, lytic and cytokine are set to 1 in the model which include specific immune responses (system of equations (7)(14)). This model (with chemokine response only) is called the chemokine response model in this study. The model with non zero immune system parameters is referred to as the combined model. The basic model is the resulting model after all the immune system parameters are set to zero (the system of equations (1)(6)). The function of the basic model is to act as a control, in the sense that it gives the viral, infected and uninfected cell levels, before the intervention of the immune system.
Parameter values
Parameter  Description  Value  Source 

β _{1}  Virus infectivity  0.000024 m l^{−1}d^{−1}  [13] 
β _{2}  Cellassociated  100−1000×  
Virus infectivity  0.000012 m l^{−1}d^{−1}  
h _{1}  Neutralizing efficiency  0.001 m l^{−1}d^{−1}  Est. 
h _{2}  CTLs killing efficiency  0.01 m l^{−1}d^{−1}  Est. 
h  Cytokine killing efficiency  0.0011 m l^{−1}d^{−1}  Est. 
μ _{ T }  Death rate of CD4 ^{+} T cells  0.02 d^{−1}  [18] 
ω  Saturation constant  0.01 m l^{−1}  Est. 
S _{ T }  Source term for CD 4^{+} T cells  10 cells d^{−1}  Est. 
μ _{ C }  CTL death rate  0.5 d^{−1}  Est. 
${\mu}_{{T}^{\ast}}$  Infected cell death rate  0.5 to 1 d^{−1}  [18] 
μ _{ B }  B cell death rate  0.5 d^{−1}  Est. 
θ _{1}  Transition probability  1/3  [19] 
θ _{2}  Transition Probability  0.06315  Computed 
θ _{3}  Transition Probability  0.43685  Computed 
ϕ  Viral production/cycle/provirus  1000  Est. 
ψ  Probability of proliferation of CTLs  0.01  Est. 
χ  Probability of proliferation of B cells  0.01  Est. 
a _{1}  Saturating constant  0.002 m l^{−1}  Est. 
a _{2}  Saturating constant  0.002 m l^{−1}  Est. 
The viral, infected and uninfected cell levels show an oscillatory behavior even if the time (t) is increased to 200. This is a common feature of the predatorprey models that we have used.
Elasticity analysis
It was observed that during the early days of the infection the viral level was most elastic to the probability of proliferation of B cells ψ, followed by h_{1}, the neutralizing efficiency of antibodies. Since these elasticity values are negative, it means that if the probability of proliferation of B cells and the neutralizing efficiency of antibodies are increased as soon as the virus is introduced, lower viral levels will be obtained. Since these two parameters are from the neutralizing antibody response, it means that the neutralizing antibody response is more effective in controlling the viral levels than any other specific immune component during the early days of the infection. However, as time progressed, the viral level was most elastic to the lytic antiviral response parameter h_{2}, followed by the probability of proliferation of CTLs, χ and was least elastic to h, the cytokine antiviral response parameter. This means that killing of infected cells is most effective in controlling the viral levels as soon as the infection become established, followed by the probability of proliferation of these CTLs. The elasticities of the viral level to the antibody response parameters changed from negative to positive as time progressed and these positive values are very close to zero. Thus we can predict that neutralizing antibodies are only efficient in reducing the viral levels in the early days of the infection and that increasing antibody response parameters once the infection become established will result in elevated viral levels. Elasticity analysis of the viral levels to immune response parameters predicted results that were also obtained by model simulations (see Figure 1). Elasticity analysis of the viral levels to other model parameters is given in Additional file 1.
Correlations of the viral levels to the immune system components predicted to be efficient in controlling the viral levels
At time 10, we observed a negative correlation between h_{1}, the neutralizing efficiency of antibodies and viral levels, and between the probability of proliferation of B cells and viral levels. At time point 100, it can be seen that as the values of h_{1} and ψ increase, the viral levels also increase. The scatter plots clearly show that the role of antibody (humoral) response depends on the stage of the infection.
Effects of transmission parameters on viral levels
We observed that the viral levels increase with increases in β_{1} and β_{2}. However, increasing β_{2} had a greater impact on viral levels when compared to increases in β_{1}, and thus we can conclude that the viral levels are more sensitive to β_{2} than β_{1}. When these two parameters were increased simultaneously, after an initial increase, the viral levels will start to decrease with increases in these two parameters. Theoretically, these results imply that the transmission parameters are bifurcation parameters. Increases above certain thresh hold values will impact negatively on viral levels.
Discussion
Discrete time HIV models that incorporate the life cycle aspects of the virus, the antibody (humoral) response and the Tcell response were formulated to determine immune system components that are most efficient in controlling viral levels. The interplay between the immune system components and the virus at the different stages of its life cycle was modelled using ecological relationships. The flexibility of the developed models allowed an indepth examination of the interactions between the specific immune system’s antiviral response and the HIV pathogen.
Neutralizing antibodies were predicted to be effective in controlling viral levels in the early days of the infection, a result observed in several experimental studies [21–25]. However, neutralizing antibodies were not able to control the viral levels once the infection become established as was observed in the studies [26, 27]. The reason for this could be the fact that antibodies (HIV predators) neutralize free virus (prey) thereby reducing the parasitoid (HIV) pressure on the host (CD4 ^{+} T cells). This will allow the host population to grow. Reduced levels of free virus means less stimuli and hence reduced levels of B cells. The virus population will then start to grow (due to increased levels of CD4 ^{+} T cells) with two advantages; first, more CD4 ^{+} T cells (host) and second, less B cells (predator). Increases in viral levels are at the expense of CD4 ^{+} T cell levels. When the B cells start to increase due to increased viral levels (prey), they do so with a disadvantage of low CD4 ^{+} T cell levels and hence their proliferation is impaired. It is also important to note that neutralizing antibodies target the early stage of the HIV life cycle, a stage at which viral levels were shown to be least sensitive to [28–30]. Currently there is no information besides the high viral mutation rates, that explain why neutralizing antibodies fail to control the infection. This result provide this understanding and prediction that could be critical in the knowledge required for the designing of effective antibody based vaccines.
The chemokine response model predicted viral levels which settled to the same levels as those of the basic model (a model without immune system intervention that was used as a control). The reason for this could be that, chemokines target the early stages of the HIV life cycle and hence are not effective in controlling viral levels. The lytic antiviral response was predicted to be most effective in controlling the viral levels amongst the three cellmediated immune responses considered in this study.
A model which combines the neutralizing antibody and cellmediated immune responses on HIV infection dynamics was also considered. This model demonstrates increased levels of uninfected CD4 ^{+} T cells and reduced viral and infected cell levels. This may have come as a result of the fact that the presence of neutralizing antibodies increases the CD4 ^{+} T cell levels and thereby improving the CTL antiviral response.
Transient elasticity analysis of the viral level to the immune system’s antiviral response parameters was performed. It was observed that during the early days of the infection the viral level was more elastic to neutralizing antibody response parameters than the cellmediated response parameters, thus increasing the neutralizing antibody response parameters at the sites of HIV entry may circumvent the infection better than the T cell response parameters. This result is logical given the fact that T cell based immune response is elicited by infected cells thus they will have to work when the infection become established. However, it should be noted that although the neutralizing antibody effect resulted in reduced viral levels in the early days of the acute phase, this was at the expense of increased infected cell levels. The elasticities of the viral levels to neutralizing antibody response parameters changed from negative to positive after a few time points. Thus their effect as the disease progresses will change from decreasing the viral levels to increasing the viral levels. From this study we can predict that cellmediated immune response cannot protect against infectious challenge but can control an established infection whereas the neutralizing antibody response can protect against challenge but can not control established infections.
The elasticities of the viral levels to the cellmediated immune response parameters are negative for all time points. This result agrees with experimental results that CTLs are effective at all stages of the infection from the acute phase [31–34] to the late days of the infection [35, 36]. We therefore predict that cellmediated immune response is more effective in controlling the viral levels than the neutralizing antibody response. This result is consistent with experimental results [37–40]. Amongst the four specific immune response components considered (neutralizing antibody, CTLs’ chemokines, cytokine and lytic responses), we found that the CTLs’ lytic antiviral response was most effective in reducing the viral levels. However, from the model structure, it seems that this conclusion is mainly because it is assumed that neutralizing antibodies and chemokine responses do not prevent celltocell spread of HIV at all. Nevertheless, if we were to assume that neutralizing antibodies and chemokines inhibit celltocell transmission thereby reducing the transmission parameter β_{2}, we will still find that these responses’ effect on viral levels will still be lower than that from the lytic response as these immune responses will be targeting the early stages of the viral life cycle, stages in which it was shown by several studies that the viral levels are least sensitive to [41–44].
Sensitivity analysis of the viral levels to other model parameters was done and details are in Additional file 1. It was shown that the viral levels are most elastic to ϕ followed by θ_{2} and least elastic to β_{1}. The implication for this result is that the viral levels are most sensitive to viral production per cell per unit time and least sensitive to the rate at which cells are infected by cell free virus.
Although our model provide predictions that can be used to inform on HIV vaccine and treatment development, it did not include all the specific immune responses such as the antibodydependentcellularcytotoxicity (ADCC) and antibodydependedcellmediatedvirusinhibition (ADCVI). Including these antibody effector mechanisms may bring more insights on the role of antibodies in an HIV infection. Moreover, most of HIV replication occurs in the lymph nodes and the lymphatic tissues and not in the blood. In this study replication in the blood was considered, with the hope that if replication in the blood is reduced then the livelihoods of infected persons will improve just as is the case of HIV treatment. Antiretroviral therapy can reduce the HIV viral levels to undetectable levels in the peripheral blood but its effect on the lymphoid tissue is very limited as predicted by failure by most drugs to penetrate the lymphoid tissue and stop the ongoing replication in these tissues [45]. However antiretroviral therapy improve the livelihoods of infected persons despite this loop hole.
HIVinfected cells continue to make viruses in lymphoid tissues even if an individual is under treatment and having undetectable viral levels in the blood [45]. If HIV replication in these tissues can be stopped then HIV can be cured [46]. Modelling HIV replication in the lymphoid tissue will help identify an immune response that is most efficient in controlling replication in this site where the bulk of HIV replication and pathogenesis occurs and this finding may have many potential implications for the future of HIV therapy and for attempts to try and cure HIV infection but is however impeded by lack of data in this compartment. In humans, blood is commonly monitored to measure disease progression and assess immune status and hence infection parameters in blood are readily available than in tissues.
Conclusion
The CTLs’ lytic antiviral response was predicted to be the most effective in reducing the viral levels during all stages of the infection. It was also observed that the killing of infected cells plays a very significant role in increasing the uninfected cell levels and decreasing the infected cell levels. We therefore predict that concentrating on antibodydependentcellmediating cytotoxicity may be more beneficial in the control of the HIV infection than focussing on neutralizing antibodies only. Another striking result that was observed was that, the effect of neutralizing antibodies on viral levels is depended on the stage of the infection.
Methods
In this section we develop our models: (i) the basic model considers the interaction between the virus and CD 4^{+} T cells (ii) the specific immune response model is an extension of the basic model that includes the specific immune responses. The models represent cell interaction in the blood compartment where perfect mixing of cells and the virus is assumed. Age structure of infected cells maybe added to the model to get a system of integrodifference equations in the case where age is a continuous variable or a discrete multistate model in the event where age is discrete. Segregating infected cells by age of infection may bring useful insights but at the expense of a complex model which might be difficult to analyze and may hence fail to meet the objectives of this study. We therefore ignored the age structure of infected cells and assume that infected cells are indistinguishable from each other.
The basic model
This model is referred to as the basic model in the rest of the manuscript. Equation (2) represents the amount of provirus in the first pseudo provirus stage. The proportion of the DNA that survive and grow from the D stage to the P stage is given by θ_{1} and the proportion of the provirus that survive and remain in the P stage is given by θ_{3}. Equation (3) represents the second pseudo proviral stage. The proportion of the provirus that survive and grow from the P to the V stage is given by θ_{2}, and θ_{3} is the proportion of the provirus that survive and remain in the Q stage. Equation (4) is the equation of the mature virus population, ϕ represents the number of virus particles produced per provirus per replication cycle. Viral production depends on the density of the infected CD4 ^{+} T cells. The parameter θ_{2} represents the transition probability from the provirus stage to the virus stage. The same parameters θ_{2} and θ_{3} were used in equations (2)(4) because the pseudo stages are assumed to be identical, so the transition and survival probabilities in these stages are the same. We have assumed that the viral level (density) at time t+1, does not depend on the viral level at time t, because plasma virus have a mean life span of 0.3 days [60] and the time step for the model is 0.5 days, meaning that no plasma virus is able to survive to the next time step.
Death of infected cells is modelled through the parameter ${\mu}_{T}^{\ast}$. Through this parameter, the intracellular steps associated with the dying cells are flashed out of the system when death of infected cells occurs. For this reason we keep track of the remaining infected cells by the term $\left(1{\mu}_{T}^{\ast}\right){T}_{t}^{\ast}$, which is then considered in the intracellular compartment. If one is to kill all infected cells, at time t, there will be zero populations at the intracellular stages at time t since the intracellular levels will be multiplied by zero to get the levels per ml.
Equation (5) models the number of CD4 ^{+} T cells at time t+1, S_{ T } is the constant supply from the thymus which is assumed to occur at the beginning of the time step and ν=1−μ_{ T }, where μ_{ T } is the proportion of uninfected cells that die per time step. Death is assumed to occur at the end of the time step so that 1−μ_{ T }, gives the proportion of uninfected cells that survive per time step. Uninfected cells must survive infection by infected cells and the virus for them to remain healthy. We have assumed that a CD4 ^{+} T cell can be infected by free virus particles or by cell associated virus through contacts of infected cells and uninfected cells in a random fashion (contacts are assumed to be randomly distributed). The proportion of cells that survive infection per time step is given by $exp\phantom{\rule{0.3em}{0ex}}\left({\beta}_{1}{V}_{t}{\beta}_{2}{T}_{t}^{\ast}\right)$. The last term in the equation represents the proliferation of CD4 ^{+} T cells and the expression was adopted from [61]. Proliferation is assumed to occur at the beginning of the time step.
Equation (6) is the equation for infected cells and ${\mu}_{{T}^{\ast}}$ is the probability that the infected cell dies naturally. Death is assumed to occur at the end of the time step, so that $1{\mu}_{{T}^{\ast}}$ is the proportion of infected cells that survive per time step. The first term on the right hand side of equation (6), gives the gain term due to infection of CD 4^{+} T cells. The expression $1exp\phantom{\rule{0.3em}{0ex}}\left(\phantom{\rule{0.3em}{0ex}}\left({\beta}_{1}{V}_{t}+{\beta}_{2}{T}_{t}^{\ast}\right)\right)$, represents the proportion of uninfected cells that become infected per time step.
Modelling the specific immune response
We have assumed a predator prey relationship where there is a hunt and kill relationship such as neutralizing antibodies eliminating/killing virus particles, CTLs hunting and killing the infected cells. In such relationships there is an exponential decay of the prey and an exponential increase in the predator therefore the primary reason for the exponential term selection. However we opted for a saturating type term in the case of an immune mechanism function depending on the amount (density/level) of specific cytokines or chemokines or a hybrid of saturation term in an exponential term in which case the decay depends on the density of the biological variables. For example, chemokines inhibit viral replication by blocking the critical interaction between coreceptors and the V3 domain of the viral envelope glycoprotein gp120 [66]. The infectivity of the virus is related to the number of envelope glycoproteins on its surface [67], so we assumed that the relationship is saturating because as soon as all the glycoproteins are interfered with, increasing the chemokines will not have any impact. The model is built on the assumption that the higher the levels of CTLs secreting these chemokines, the lower the infectivity of the virus. In the presence of the chemokine secreting CTLs (chemokine immune response), a virus infectivity function of the form $\frac{{\beta}_{1}}{1+\omega {C}_{t}}$, is assumed, where ω is a saturation constant. The function $\frac{1}{1+\omega {C}_{t}}\to 0$ as C_{ t }→∞ and $\frac{1}{1+\omega {C}_{t}}\to 1$ as C_{ t }→0.
The interaction of the HIV antigen and the antibodies is similar to predation with a modification that proliferation (population growth function) of B cells is dependent on helper T cells. We thus model the interaction between neutralizing antibodies and HIV using a predatorprey relationship. We further assume that the levels of circulating antibodies is proportional to the levels of B cells. The first term on the right hand side of equation (7) is multiplied by exp(−h_{1}B_{ t }), where h_{1} is the antibody neutralizing efficiency, to cater for the neutralizing effect of antibodies (predator effect), as B_{ t }→∞, exp(−h_{1}B_{ t })→0. It should also be noted that as B_{ t }→0, exp(−h_{1}B_{ t })→ 1. The expression ${\beta}_{2}{T}_{t}^{\ast}$ represents HIV genome entry through cell to cell contacts. Celltocell eliminates the rate limiting step of diffusion, thereby reducing the exposure time of viral particles to neutralizing antibodies and chemokines. There are conflicting reports with regards to the efficiency of antiviral responses such as neutralizing antibodies and chemokines in inhibiting celltocell transmission [68–72]. In this study, we assume that celltocell transmission is not affected by the extracellular environment, antibodies and the chemokine immune response in this case.
Cytokines have been reported to inhibit the viral life cycle at the transcription level [64, 65], by targeting early proviral gene expression [73]. We therefore assume that these cytokines affect the transition probability from the first pseudo provirus stage to the second pseudo provirus stage (Q). It is still unclear how the CD 8^{+} T cells specifically affect HIV RNA transcription at the molecular level [65], we assume that the survival probability of the P stage, θ_{2}, is reduced by a factor of exp(−h C_{ t }) in the presence of cytokine secreting CTLs. Here we again assume a predatorprey relationship between cytokines and HIV genes. The factor is depended on the levels of circulating CTLs such that C_{ t }→0, exp(−h C_{ t })→1 and C_{ t }→∞, exp(−h C_{ t })→0.
CTLs release proteins (perforin and granzymes) that kill infected cells [63]. The right hand side of equation (12) is multiplied by exp(−h_{2}C_{ t }) to cater for the killing of infected cells by CTLs. The killing of infected cells is dependent on the levels of circulating CTLs.
Equation (13) models the dynamics of CTLs at time t+1. The function f(T_{ t }), represents the proliferation term for CTLs. Proliferation of CTLs is dependent on CD4 ^{+} T cell density. The higher the density of CD4 ^{+} T cells, the higher the proliferation for CTLs. We propose a function of the form, $f\left({T}_{t}\right)=\frac{\chi {a}_{1}{T}_{t}}{1+{a}_{1}{T}_{t}}$, where χ is the probability that proliferation occurs. The interaction between infected cells and the CTLs is modelled using a predatorprey relationship. The proportion of infected cells that have been preyed on is represented by 1− exp(−h_{2}C_{ t }) so that $f\left({T}_{t}\right){T}_{t}^{\ast}(1exp({h}_{2}{C}_{t}\left)\right)$ gives the gain term of CTLs due to interactions/contacts with infected cells. The parameter μ_{ c } is the proportion of CTLs that die naturally. Death is also assumed to occur at the end of the time step.
The last equation models the dynamics of the antibodies. The function $g\left({T}_{t}\right)=\frac{\psi {a}_{2}{T}_{t}}{1+{a}_{2}{T}_{t}}$, represents proliferation of B cells and ψ is the probability that proliferation occurs. The proliferation term is dependent on the density of helper T cells. The interaction between antibodies and the virus is modelled using a predatorprey relationship. The proportion of virus that have been preyed on is given by 1− exp(−h_{1}B_{ t }) so that the gain term of antibodies is given by g(T_{ t })V_{ t }(1− exp(−h_{1}B_{ t })). The parameter μ_{ B }, gives the proportion of B cells that die naturally per time step. Death is assumed to occur at the end of the time step.
Model analysis
Boundedness of solutions
An important qualitative property of the discrete dynamical system (7)(14) relates to the boundedness and positivity of solutions, since negative solutions do not have biological meaning. We thus state the following proposition.
Proposition 1
Solutions of the system of equations (7)(14) remain nonnegative and are bounded whenever ν+a<1.
Proof
The system of equations (7)(14) is thus bounded whenever ν+a<1. □
Disease free equilibrium point
where $\widehat{T}=\frac{K(v+a1)+K\sqrt{{(v+a1)}^{2}+\frac{4{\mathit{\text{aS}}}_{T}}{K}}}{2a}.$ The following theorem is used to prove the stability of E_{0}.
Theorem 2

If all the eigenvalues of the Jacobian matrix of the system of equations (7)(14) evaluated at $\u0112$, ${J}_{\u0112}$, have moduli strictly less than 1, $\u0112$ is asymptotically stable.

If at least one eigenvalue has modulus greater than 1, then $\u0112$ is unstable.

If no eigenvalue of ${J}_{\u0112}$ is outside the unit circle but at least one is on the boundary (has a modulus of 1), then $\u0112$, maybe stable, asymptotically stable or unstable.
Proposition 3.
Proof.
The eigenvalues are $0,\phantom{\rule{1em}{0ex}}{\theta}_{3},\phantom{\rule{1em}{0ex}}\nu +aexp\left(\frac{\widehat{T}}{K}\right)a\frac{\widehat{T}}{K}exp\left(\frac{\widehat{T}}{K}\right),1{\mu}_{B},\phantom{\rule{1em}{0ex}}1{\mu}_{C}$ and $(1{\mu}_{{T}^{\ast}})+{\beta}_{2}K\widehat{T}$. All eigenvalues has moduli less 1 if $\nu +aexp\left(\frac{\widehat{T}}{K}\right)a\frac{\widehat{T}}{K}exp\left(\frac{\widehat{T}}{K}\right)<1$ and $\left\right(1{\mu}_{{T}^{\ast}})+{\beta}_{2}\widehat{T}<1$. Thus E_{0} is asymptotically stable if $\nu +aexp\left(\frac{\widehat{T}}{K}\right)a\frac{\widehat{T}}{K}exp\left(\frac{\widehat{T}}{K}\right)<1$ and $\left\right(1{\mu}_{{T}^{\ast}})+{\beta}_{2}\widehat{T}<1$. □
Endermic equilibrium point
Due to the complexity of the system of equations (7)(14), the conventional methods for finding the interior equilibrium point fails, however, we can be guaranteed that it exist by the theory of persistence and permanence of discrete dynamical systems. Persistence conditions ensure that no species will go extinct in a system of interacting species whilst permanence conditions guarantees that the size of each population is bounded and that each population settles above certain threshold values. The following are the definitions of persistence and permanence of the system of equations (7)(14).
Definition 4
 1.
The system of equations (7)(14) is strongly persistent at time τ if $\phantom{\rule{0.3em}{0ex}}{D}_{t},\phantom{\rule{1em}{0ex}}{P}_{t},\phantom{\rule{1em}{0ex}}{Q}_{t},\phantom{\rule{1em}{0ex}}{V}_{t},\phantom{\rule{1em}{0ex}}{T}_{t},{T}_{t}^{\ast}>0$ for each t=0,1,2,⋯τ.
 2.
The system of equations (7)(14) is weakly persistent at time τ if $\phantom{\rule{0.3em}{0ex}}{D}_{t},\phantom{\rule{1em}{0ex}}{P}_{t},\phantom{\rule{1em}{0ex}}{Q}_{t},\phantom{\rule{1em}{0ex}}{V}_{t},\phantom{\rule{1em}{0ex}}{T}_{t},{T}_{t}^{\ast}>0$ for each t=0,1,2,⋯τ−1 and ${D}_{\tau},\phantom{\rule{1em}{0ex}}{P}_{t},\phantom{\rule{1em}{0ex}}{Q}_{\tau},\phantom{\rule{1em}{0ex}}{V}_{\tau},\phantom{\rule{1em}{0ex}}{T}_{\tau}^{\ast}\ge 0$, D _{ τ }+P _{ t }+Q _{ τ }+V _{ τ },>0 and T _{ τ }>0.
 3.
The system of equations (7)(14) is strongly persistent if it is strongly persistent at time τ for τ=0,1,2⋯ and ${limsup}_{t\to \infty}{D}_{t},\phantom{\rule{1em}{0ex}}{P}_{t},\phantom{\rule{1em}{0ex}}{Q}_{t},\phantom{\rule{1em}{0ex}}{V}_{t},\phantom{\rule{1em}{0ex}}{T}_{t},{T}_{t}^{\ast}>0$.
 4.
The system of equations (7)(14) is uniformly persistent if it is strongly persistent at time τ for τ=0,1,2⋯ and there exist a positive constant ϖ such that $\phantom{\rule{0.3em}{0ex}}{liminf}_{t\to \infty}{D}_{t},\phantom{\rule{1em}{0ex}}{P}_{t},\phantom{\rule{1em}{0ex}}{Q}_{t},\phantom{\rule{1em}{0ex}}{V}_{t},\phantom{\rule{1em}{0ex}}{T}_{t},{T}_{t}^{\ast}>\varpi $.
 5.
The following theorem will be used to study persistence and permanence of the system of equations (7)(14).
Theorem 5.
 1.
f is dissipative,
 2.
f_{ Y } is isolated,
 3.
f_{ Y } is acyclic,
 4.
for each M _{ i }∈M, ${W}^{+}\left({M}_{i}\right)\bigcap \stackrel{\u0307}{X}=\varnothing $.
Theorem 6.
Proof.
where x=(D,P,Q,V,T,T^{∗},C,B).
componentwise. This contradicts the assumption that ${\text{lim}}_{t\to \infty}({D}_{t},{P}_{t},{Q}_{t},\phantom{\rule{0.3em}{0ex}}{V}_{t},{T}_{t}^{\ast},\phantom{\rule{0.3em}{0ex}}{C}_{t},{B}_{t})\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}(0,0,0,0,0,0,0)$. Hence ${W}^{s}\left(M\right)\bigcap \underset{+}{\overset{8}{R}}\setminus Y\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\varnothing $. Thus the system of equation (7)(14) is uniformly persistent. □
Proposition 7.
Proof.
The result follows from definition 1.5. □
Declarations
Acknowledgements
SPS would like to thank the Organization for Women in Science for the Developing World (OWSD) for the financial support, the Department of Mathematical Sciences, University of Stellenbosch and the National University of Science and Technology (NUST). FN acknowledges with thanks the support from the Department of Mathematical Sciences, University of Stellenbosch. SDH acknowledges with thanks the support from the National University of Science and Technology (NUST). GM acknowledges with thanks the support from National Institute for Mathematical and Biological Synthesis, (NIMBioS), NSF Award #0832858.
Authors’ Affiliations
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