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1 The Role of CD8 immune responses in HIV infection Tendai Mugwagwa Supervisor: Dr. Gareth Witten. (University of Cape Town) An essay submitted in partial fulfilment of the requirements for AIMS Diploma in Mathematical Sciences. May 2004. 1 2 Acknowledgements I would like to thank the following for their contribution in the success of this essay : Dr Gareth Witten (My supervisor), Dr Mike Pickles, Carl Scheffler and Prof Wesley Kotze. I am grateful to Prof Neil Turok, Prof Fritz Hahne and all my lecturers at AIMS in 2003-2004 for helping me through an exciting mathematical journey and the discovery of my passion for epidemiology. I would also like to thank the sponsors of AIMS and all its collaborators for making all this possible. Lastly I dedicate this essay to my family for their unconditional love and encouragement. Thank you Mom. 2 Abstract Mathematical models and experiments have shown the importance of CD8 responses in control of HIV infection. In this paper we review the theoretical and empirical evidence and how the two compliment each other as they shed more light on the progression of HIV infection. This paper also highlights the controversies pertaining the subject. We present some models that investigate the role of a CTL response and a CTL memory in control of HIV. These models show that a strong CTL response can control the viral load, however, in some cases the virus has been known to persist regardless of the immune response. We extend the basic immune response model to account for the escape of HIV from CTL responses via epitope mutations. We find that a broad and long lived CTL response efficiently controls the virus even in the event of mutations. On the other hand, a gradual switch from a slow replicating HIV strain to a faster replication kinetics, has also been suggested as a mechanism for disease progression. To explore the factors influencing this switch, we extend the CTL memory model to include macrophage cells, non-lytic CD8 responses and the evolution of HIV from a slow replicating strain(R5) towards a faster replicating strain (X4). We find that macrophages act as a reservoir for the virus hence promote viral persistence. However in the course of the infection, the success of a switch from the R5 strain to the X4 strain dependson the cytopathicity of the individual strains. The cytopathicity, evolution rate, infection rate and the strength of the immune response determine the time lapse before a switch occurs. We conlude that these factors determine the length of the asymptomatic period of HIV infection. Contents Abstract 2 1 Cellular immune response and disease progression 2 1.1 The stages of HIV infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Dynamics of HIV infection and the role of CD8 cells. . . . . . . . . . . . . . . 4 1.3 The role of CTLs in HIV infection . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Lytic and non-lytic CD8 responses . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Examples of mathematical models for the interaction between CD8 cells and HIV 18 2.1 The Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 The effect of Immune response on viral dynamics . . . . . . . . . . . . . . . . 20 2.3 CTL memory and viral dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Multiple epitopes and viral dynamics . . . . . . . . . . . . . . . . . . . . . . . 27 3 Gradual evolution of HIV as a mechanism for disease progression 32 3.1 Developments of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Conclusions 43 Bibliography 44 1 Chapter 1 Cellular immune response and disease progression HIV is a retrovirus which means that its genome is RNA and is translated into DNA during its life cycle. HIV attaches itself to target cell using a coreceptor (CCR5 or CXCR4). It then gains entry into the target cell and uses its machinery to complete its life cycle however destroys them in the process. Target cells include macrophages and T cells. A healthy human adult has about 1000 CD4 cells per micro litre of blood, but in an infected patient, the CD4 count can drop to lower levels. Currently, if a patient has a CD4 count of below 200 CD4 cells per micro litre, he or she is said to have AIDS. Like any other pathogen, invasion of the body by HIV stimulates an immune response. Although there is a wide range of immune responses, we will focus on T helper cells and CD8 cells, also known as cytotoxic T lymphocytes (CTLs). CD8 cells control the virus by either lysing the infected cell or inhibiting HIV replication and entry into target cells. It is unfortunate that the main target cell of HIV are CD4 cells because these play a major role in fighting viral infections [49]. In this chapter we review models and experimental evidence that has been presented to explain the relationship between CD8 cells and disease progression. 2 1.1 The stages of HIV infection 3 1.1 The stages of HIV infection To understand the development of AIDS from HIV infection, it is important to analyse the dynamics of HIV CD4 cells and CD8 cells throughout the period of infection. The pattern of disease progression is divided into three stages summarised in fig(1). Plasma concetration levels Primary phase Asymptomatic phase AIDS CD8+ cells CD4+ cells Virus load 2−10 weeks up to 15 years Figure 1.1: A qualitative diagram to show the time course of HIV infection in a typical infected adult.[36] • Primary Phase: During the first few weeks after infection with HIV, patients experience a period of increasing viral load and a decline in CD4 cells numbers. Flu like symptoms have been associated with this phase[49]. The end of this period coincides with the first signs of a CD8 immune response against HIV.[26, 37, 7] • Asymptomatic Phase: Although there are no visible symptoms present, the replication kinetics of the virus are extremely fast [20, 59, 38, 7]. However, there is little change in the viral load. The CD8 responses are thought to control the virus to low levels but the 3 1.2 Dynamics of HIV infection and the role of CD8 cells. 4 CD4 cell numbers continue to decline. The length of this phase may range from a few months to 15 or more years.[49, 7] • AIDS: This is the final stage of the disease. CD4 cells falls below 200 micro litres and is an overall weakness in the immune system allows opportunistic infections to frequently occur.[17, 7, 49]. Diseases from these infections eventually lead to death. 1.2 Dynamics of HIV infection and the role of CD8 cells. Mathematical models have provided insight into the dynamics of HIV during the different stages of infection. The models have been used to estimate how rapidly HIV replicates, the number of virus particles produced and cleared daily, and the lifespan of productively infected CD4 cells [49]. The basic model (discussed in greater detail in chapter 2) and a variant containing latently infected cells have been used to model the rise, fall and subsequent establishment of a viral load set point. However a combination of parameters may result in the variation of this set point. Muller et al[30] showed that small variations from patient to patient for several parameters resulted in large variations in the observed viral load set point. Although the basic model shows a fall in the viral load it does not include an explicit immune response. One then asks : Do CD8 cell control the virus in the primary phase of infection?. Phillips [40] showed that the fall of the viral load was due to a decline in the target cells(CD4 cells), a process called target cell limitation. There has been considerable controversy about the role of CD8 cell in viral control, in particular whether CD8 cells reduce the viral load. Simian immunodeficiency virus (SIV), infects monkeys (Macaques) in a way similar to HIV in human beings. Jin et al [22] used an antibody OKT8F to delete CD8 cells in 6 SIV infected macaques and 1 uninfected macaque. They observed a drastic CD8 level drop in all infected macaques and a subsequent increase in viral load. This seemed to support the idea that CD8 cells play a role in viral control. The infected macaques showed a subsequent drop in CD4 levels, however unexpectedly, the level also dropped in the uninfected macaque. This implies that SIV infection cannot solely account for the drop in CD4 cells. This was attributed this to 4 1.2 Dynamics of HIV infection and the role of CD8 cells. 5 the increase in viral load and activation induced apoptosis [22]. To investigate whether this drop was due to the deletion of CD8 cells, Jin et al [22] introduced a control antibody, P1.17, into uninfected macaques that did not delete the CD8 cells. CD4s dropped by 51%. This implied that antibodies could induce T cell activation hence increasing the target cells for the virus. The increase in viral load could thus have been due to an increase in target cells and not the absence of a CD8 response. Although some authors have doubted the role of CD8 cells in viral control([6, 14, 37]), there is strong evidence to show that HIV specific CD8 responses are generated and are inversely related to viral load [7, 2, 22, 44, 26, 37]. However there is no direct evidence that this immune response can modulate the natural history of HIV infection [37]. PLASMA CONCETRATION CTL RESPONSE VIRAL LOAD TIME Figure 1.2: A qualitative diagram to show the inverse relation between CTLs and viral load [29] CD8 responses can be divided into (i) The lytic response (Cytotoxic T Lymphocytes,CTLs) which make use of proteins in their cytoplasm such as peforin and granzymes for cell lysis. 5 1.3 The role of CTLs in HIV infection 6 This is also known as a direct killing response. (ii) Non-lytic responses (Chemokines) are soluble substances secreted by CD8 cells, for example is cytokines. These work by either inhibiting HIV replication or inhibiting viral entry into target cells. As we mentioned earlier, CD8 responses can change the viral set point, however it is not clear which of the two responses are involved. This marks the branching point in research on the role of CD8 responses in HIV infection. In the following subsections we will first examine key experiments and mathematical models developed for CD8 lytic responses (CTLs) and then those that include CD8 non-lytic responses. 1.3 The role of CTLs in HIV infection CD8 lytic responses are also known as CTLs (Cytotoxic T lymphocytes). The basic model for HIV infection was extended to include an explicit immune response [32, 56, 54, 50, 51]. These models have the typical form of an ecological food chain model [30]. CD4 cells are the prey, productively infected cells being their predators and the immune response as the top predator [32, 13, 30]. Nowak and Bangham [32] used such a model to hypothesise that viral control depended on CTL responsiveness and viral diversity. They suggested that a strong CTL response would decrease the viral set point however imposed a selection pressure on the virus resulting in increased viral diversity and escape from the immune response. A diverse virus resulted in an increase in viral load and thus disease progression. HIV has been shown to establish such persistence and escape from the immune responses over time [29]. The removal of CTLs has been shown to increase the lifespan of productively infected cells and thus increasing viral production[22], hence, CTL persistence is essential in viral control. On the other hand, Wodarz and Nowak [56] used a mathematical model to illustrated that apart from a high responsiveness, a long lived CTL response was efficient in viral control(fig 3). 6 1.3 The role of CTLs in HIV infection 7 2000 Viral load 1500 1000 500 0 0 0.005 0.01 0.015 0.02 r: CTL responsiveness Figure 1.3: A bifurcation diagram showing the effect of CTL responsiveness on the steady state viral load [32] They hypothesised that a long asymptomatic period was due to a long lived CTL response. This was supported by an observation that HIV infected individuals with an asymptomatic infection for 15 years had persistently low viral loads, a stable CD4 count and a strong proliferative response to HIV [23]. This result suggests that the quality of the immune response crucial in determining the course of disease progression. Although these models have demonstrated the key processes during the primary and asymptomatic phase of infection, they fail to represent the final stage, i.e the development of AIDS. After the primary phase of infection, these models reach a steady state where the CD4 cells, CTLs and viral loads are constant, but this has been shown not to be so[17, 7, 49]. We pose the question: should the rate of production of CD4 cells be a constant function as assumed in the models? It seems likely that the production of the cell may depend on other factors such as presence of antigen to stimulate thier production. Having established that a persistent CTL response is necessary to keep the viral set point low, we now look at factors affecting CTL persistence. 7 1.3 The role of CTLs in HIV infection 8 1.3.1 CTL exhaustion Before we look at factors that promote CTL persistence we will first look at why it fails. Wodarz et al [58] developed a mathematical model in which they found that for a lymphocyte infecting virus such as HIV, the ability to infect CD4s and a fast replication rate were the main factors causing of the decline in CD8 cells. They referred to this process as ‘CTL exhaustion’. A low initial CD4 count also led to the same result. They defined a threshold value depending on viral replication rate. The higher the replication rate, the more likely the CD8 count drop. They used this to describe the mechanism for disease progression as the gradual evolution of HIV towards higher replication rates. This was in agreement with the findings of Connor and Ho [10] that long-term non-progressors harbour only relatively slowly replicating HIV variants[58]. Van Den Boek et al[47] also showed that rapid replicating LCMV resulted in CTL exhaustion in mice. However, if there are variations is viral replication rate during disease progression, should β the replication rate in the models be a constant? Experimental evidence[47] which supports model by Wodarz et al[58] involved a non-cytopathic virus LCMV. HIV has been shown to have different viral strains at different infection stage. These viral strains have different cytopathicity levels [45, 48, 12]. Therefore should the death rate for infected cells in the models be also a constant? A lasting CTL response was suggested to depend on CD4 help [52]. If consider the latter question, a cytopathic HIV strain would impair CD4 help leading to a decrease in CTL response, a case not accounted for by Wodarz et al [58]. 1.3.2 CTL memory Now having established the causes of CTL exhaustion, we look at factors that promote CTL persistence, which is also know as CTL memory. Traditionally CTL memory had been associated with protection against re-infection. A persistent CTL response has been shown to be important for viral control. CTL memory was redefined to include the role it plays in clearance of primary infections [57]. CTLs were divided into two groups: • CTL precursors (CTLp): these are CTLs that have never seen the antigen and do not take part in the killing of target cells. They are also called the CTL memory cells. 8 1.3 The role of CTLs in HIV infection 9 • CTL effectors (CTLe): these differentiate from CTLps on an encounter with an antigen. These carry out the killing of target cells. Elevated numbers of CTLps have been observed long after viral clearance [21, 35, 19, 57]. This CTL memory is thought to protect the host form secondary infections. There are disagreements on the nature and protectiveness of CTL memory, in particular, the role of a persistent antigen or CD4 cells in maintaining memory. An experiment showed that initially CD4 deficient mice and wild type mice infected with LCMV, both had viral loads at undetectable levels. However, the virus reappears to high levels in the CD4 deficient mice. This observation was attributed to the failure in establishment of CTL memory. CD4 help was suggest to interfere with the generation and/or maintainance of CTL memory in primary infection [57]. Mathematical models were developed to test the hypothesis that CTL memory depended on CD4 help[57]. The models demonstrated that the impairment of CD4 help during primary infection resulted in CTL exhaustion. Although a CTL response would be produced, it would not last long. As mentioned earlier, CTL memory is favoured by a slow replicating virus and also requires a high initial value of CD4 count and a low viral load[58]. Long term non-progressors of HIV infection have been characterised by a relatively high level of CTL response despite low viral loads[18, 52]. Such patients have also shown strong CD4 proliferative responses[42, 52].However Wodarz et al[52] showed that specific CTL precursors (CTLp) in chronically HIV infected patients decayed rapidly after therapy[23]. With this result it is not clear whether the loss of CTLps was due to an impaired CD4 help or a reduction in antigen level by anti-viral therapy. Some researchers believe that CTL memory depends on CD4 help while others are in favour of an antigen-dependent CTL memory, however mathematical models have suggested that the two might work together[57]. CTL memory dependence on antigen and/or CD4 help The role of an antigen in CTL memory persistence was drawn from the traditional definition of CTL memory in which we required a persistent antigen. Kundig et al[28] showed that for an efficient immunisation, a large antigen dose was require to maintain CTL memory. However Wodarz et al[57] used a mathematical model to show that antigen-dependent CTL memory was not sufficient for successful viral control. During the asymptomatic period of HIV, the viral load 9 1.3 The role of CTLs in HIV infection 10 is low hence it may not fail to stimulate a CTL response. More recent studies insist on CD4 help as being nessecesary in the initial development of CTL memory however, the interactions between CD4 and CD8 remain unclear. McMicheal and Rowland-Jones [29] identified three functions of T help in relation to CD8 memory have. Function (i): Initiation of CD8 responses CD4 cells were said to initiate production of CD8 cells. Two pathways by which this occurs were defined. Traditionally, the role of CD4 cells was to produce a cytokine, interleukin(IL12). This cytokine trigger dentritic cells to produce CD8 cells. This is called the classical pathway [50]. On the other hand, some viruses can directly stimulate the dentritic cells. Whether HIV also does this remains to be clarified. Alternatively CD4 cells activate antigen-presenting cells APC which then trigger the production of CD8 cells. This is called the CD4-APC-CD8 pathway[50]. Wodarz and Jansen [50] used a mathematical model to investigate the role of each of these pathways in CD8 production. The model predicted that the ’classical pathway’ would be efficient at inducing CD8 cell expansion at high viral level in early infection stages. However, at lower viral levels the ’CD4-APC-CD8 pathway’ would be efficient in ensuring viral clearance. This result meant that the two pathways would work together for successful viral control throughout the infection period. The results also showed that CD4 help plays a role in the development of CTL memory but on the other hand suggesting that in the absence of CD4 help APCs alone can also establish a CTL memory. Experiments to distinguish the importance of the two pathways in view of different viral levels are still to be done. Function (ii): CD8 maturation Another function of CD4 is facilitating CD8 maturation. Animal models were used to show that the absence of T help leads to CD8 cell ’immaturity’[29]. Maturity here implying failure of CD8 cells to fully differentiate and carry out their functions. Kalams[23] showed that in the absence of CD8 help, CTL can be persistent but in a non-functional state. Low levels of peforin (a substance used by CTLs in cell lysis) were observed in HIV infected patients with a low CD4 count[29]. These results show that HIV specific CD8 cells may be less efficient in their lysing property than expected. We then pose the question: should the rate of CTL killing in model [54, 32, 58, 50] be a constant function? What could it be a function of? 10 1.3 The role of CTLs in HIV infection 11 Function (iii): Maintainance of CD8 memory The last function of CD4 cells was identified as the ability to maintain CTL memory. Kalams [23] presented data to show a positive correlation between T help and CD8 memory. However the relation was questioned when McMicheal and Rowland-jones [29] showing the long survival of CD8 memory in chronically infected patients despite the impaired CD4 help. This implied that naturally activated CD8 cells may survive better in the absence of CD4 help. Wodarz et al [52] use mathematical models to compare the role of CD4 help dependent and CD4 independent CTL responses. They found that a cooperation between the two responses was essential for viral control. During initial infection stages, viral load in high and this compromises the CD4 pool. The CD4-independent response dominates the immune responses. As it reduces the viral load, the CD4 cell pool recovers. On the other hand, as the availability of CD4 cells improves, the CD4 dependent response brings the viral load to lower levels. These low viral load levels cannot sufficiently stimulate a CD4 independent response. Wodarz et al[52] point out that help dependent CTL memory is broad hence viral escape would be difficult. They use this to accounts for the long asymptomatic period in HIV infected patients saying that CD4 help dependent CTL memory dominated this stage of HIV infection. However another mathematical model by Wodarz et al[56] showed that broad response can lead to immunodominance depending on the CTL lifespan and CTL responsiveness to various epitopes. Knowing the high mutation rate within epitopes, immunodominance could easily lead to loss of viral control. It would therefore be important to investigate these properties in HIV before concluding that a broad response would successfully control the virus. The mathematical model on CTL memory and multiple epitopes will be discussed in greater detail the chapter 2. From discussion thus far, we found that; • Progression of HIV infection is equivalent to progression towards the threshold for CTL exhaustion. • Impairment of CD4 help results in an inefficient CTL memory. • HIV can escape the immune responses via mutations. 11 1.3 The role of CTLs in HIV infection 12 In all three cases CTLs function to the benefit of the host. It has however been shown that CTL activity can be detrimental to the host [54]. This process is called CTL induced pathology. 1.3.3 CTL induced pathology A CTL response can cause severe immunopathology through destruction of the host cells. A classic example is an experiment in which a mouse infected with LCMV remains healthy in the absence of a CTL response [58, 31]. This result is due to the non-cytopathicity of LCMV . The presence of an efficient CTL response can successfully control the infection, however a less efficient CTL response can lead to severe immunopathological effect characterised by wasting of the mouse [31, 61, 57]. Such CTL mediated immunopathology has been suggested as a possible reason for the eventual development of AIDS. As mentioned earlier, the viral charectaristics differ depending on the host cell[53]. Since HIV infects a number of different cells such as macrophages and T cells, it becomes difficult to investigate the conditions for immunopathology in HIV experimentally. A mathematical model was used to examine the properties of CTL induced pathology an its implication for HIV infection [57]. The degree of CTL mediated pathology was defined as the total number of T cells found in the presence of a virus and CTL repsonse( the sum total of the infected and uninfected T cell). The model predicted that the degree of pathology was determined by the rate of viral replication relative to the CTL responsiveness. The faster the replication rate of the virus, the stronger the CTL response needed to be in order to be beneficial to the host and avoid pathology. Wodarz and Krakuer [54] then introduced the need for non-lytic CD8 responses in the model. They suggested that cytokines would reduce the replication rate by inhibiting viral replication. A reduced rate of replication can in-turn prevent cytopathology in an individual with an intermediate CTL response. In mice infected with a slow replicating Armstrong strain, the absence of a non-lytic response did not compromise viral control and no CTL induced pathology was observed. On the other hand, mice infected with a faster replicating Traub strain, the absence of a non-lytic response resulted in severe tissue damage and wasting of the host [46]. 12 1.4 Lytic and non-lytic CD8 responses 13 1.4 Lytic and non-lytic CD8 responses In the following section, we now look at the role of both lytic and non-lytic CD8 responses in viral control. Initially Researchers thought that non-lytic CD8 responses worked independently in control of viruses of different cytopathologies. Kagi et al[24] hypothesised that lytic CD8 responses were essential for the control of non cytopathic viruses whereas the no lytic CD8 responses were sufficient to deal with cytopathic viruses. However recent experiments have shown that non-lytic CD8 responses may also contribute to resolving non-Cytopathic viruses while lytic cell independently resolving some cytopathic infections. The importance of nonlytic responses was also discussed by Wodarz et al [58], who pointed out that HIV progression towards the CTL exhaustion threshold may not only be achieved by the virus evolving towards higher replication kinetics, but also by a loss of efficiency of those branches of the immune system that limit the overall replication kinetics of the virus. This branch being the non-lytic CD8 responses. A decrease in cytokine production may lead to increased viral kinetics driving the system towards CTL exhaustion. This view was supported by experiments in which some mice lacking inteferons (a kind of chemokines), where infected with a cytopathic virus, Vaccinia while others with a non-cytopathic virus LCMV [47]. In both cases no CTL activity was detected. This result was explained as follows, the absence of non-lytic responses to control the replication rate of the virus led to a failed CTL response possibly due to CTL exhaustion. Wodarz et al[53] then discarded the simple rule that lytic responses are required to deal with non-cytopathic viruses where as non-lytic responses were sufficient to deal with cytopathic viruses. They suggested that the relevance of the two responses in resolving viruses depended on the viral cytopathicity relative to its replication rate. They presented two cases; • Case 1) if viral cytopathicity lies below a certain threshold, a combination of lytic and non-lytic CD8 response is likely to resolve the infection. The non-lytic response would reduce the viral replication rate hence minimising immunopathology and enabling the lytic response to clear the infection. • Case 2) If cytopathicity lies above the threshold, both the lytic and non-lytic response can in principle, resolve the infection independently. 13 1.4 Lytic and non-lytic CD8 responses 14 The cytopathicity of HIV is not well understood. The virus has been characterised by different replication rates and levels of cytopathicity in different cell types for example in macrophages and T cells [53]. Experiments have been done with LCMV, a non cytopathic virus and VSV, a cytopathic virus to support Wodarz et al’s [53] hypothesis. In the experiments, peforin deficient mice infected with LCMV developed severe cytokine mediated aplastic anaemia and succumbed to the infection [4]. The lack of peforin meant the lack of a CTL response. It was also shown that with Human T cell leukaemia virus, a chronic state of activation of the immune system resulted in overproduction of cytokines and damage the host[4]. This showed that CTLs alone could not resolve the infection despite the non cytopathic nature of the virus. For VSV infected mice, they showed that animals lacking a CD4 and CTL response could not control the infection by use of non-lytic responses alone[4]. However it has been shown that in hepatitis B virus (HBV), viral control was achieved by an intricate balance between a lytic and non-lytic CD8 response [16]. Wodarz et al [53] then used mathematical models of lytic and non-lytic CD8 responses to conclude that reduction of replication rate the virus by non-lytic responses was always beneficial to the host. On the other hand, increasing the death rate of the virus by lytic responses could be both detrimental and beneficial to the host. Lytic responses were likely to be detrimental when the virus replicated at a fast rate. In the model, the relevance of a lytic and non-lytic CD8 response in viral control depended on viral cytopathicity. Since HIV has been characterised by different replication rates and levels of cytopathicity in different cell types[53], we therefore ask: is not important to include other target cells in the model such as macrophages. Macrophages are believed to form a harbour for the virus in which it can escape the CTL response [25]. 1.4.1 The role of latently infected cells An introduction of macrophages into the model also brings with it the issue of HIV tropisms as HIV entry into cells differs with different target cells. However, let us first look back at the models discussed thus far. Although they all provide insight into the events during primary infection and part of the asymptomatic period, none of these models shows the eventual development of AIDS. In addition, they do not explain variation of the length of the asymptomatic period which 14 1.4 Lytic and non-lytic CD8 responses 15 ranges between a few months and up to 15 years. Wodarz and Krakuer [54] explained that at the beginning of the asymptomatic period, the slow replicating HIV strain dominates the viral population while the replication kinetics of the virus increase during disease progression until the development of AIDS. In section (1.3.2) it was argued that impairment of T help cell in the primary stages of infection may lead to an inefficient CTL memory response. A weak CTL response and a fast replicating virus according their model, would lead to CTL induced pathology and contribute to the development of AIDS [54]. In this section we look at this change in viral strain bringing us back to HIV tropisms. HIV tropisms HIV can exhibit distinct cellular tropisms that have important implications for the viral pathogenesis and disease progression [12]. There two main tropisms named depending on the type of coreceptor they use and their target cells. The M-tropic HIV strain has been shown to dominate the primary and initial stages of the asymptomatic period of infection [12, 45, 60].Virus isolates in this period tend to use the CCR5 coreceptor hence it is also known as the R5 strain [5].It is characterised by : • a slow replication rate [45] • a relative acytopathicity • shows the non-synctium inducing(NSI) phenotype • infects macrophages and primary T cells [12, 11] • it can be inhibited by cytokines[51]. • associated with slow disease progression[9]. Later in the course of infection, HIV has been shown to evolve to the T-tropic strain which uses the CXCR4 coreceptor [12]. This viral strain is called the X4 strain [5]. Some individuals may retain thier use of the CCR5 coreceptor hence they show a dual tropism [12, 8]. Such individuals are said to have the R5X4 strain. However the X4 strain is characterised by : • a fast replication rate 15 1.4 Lytic and non-lytic CD8 responses 16 • a high degree of cell killing • shows the synctium inducing phenotype • infects T cells [51, 11] • associated with an accelerated disease progression [9]. Switching tropisms during the asymptomatic period The evolution of HIV towards a faster replicating strain has been hypothesised as the major driving force underlying the progression to AIDS [54, 45]. This is equivalent to a switch from the R5 strain to the X4 strain that marks the onset of AIDS. Wodarz et al [51] use mathematical models to show that at the beginning of HIV infection when the immune response is strong, a slow replicating virus such as the R5 strain can successfully infect macrophages. However, since lytic CTL responses are less effective in killing macrophages compared to T cells [54], macrophages create a buffer or refuge for the virus [25, 45]. This is thought to be the major contributor to viral persistence [45]. During the course of the infection the virus evolves towards to the faster replicating rates, at the same time escaping the immune system through epitope mutation [32, 7, 57]. This creates a condition favourable for the emergence of the X4 strain. During the primary infection stage, a slow replicating virus like the R5 strain allows for establishment of a persistent virus by infecting macrophages. This explains why individuals with defective CCR5 coreceptors,∆ CCR5 deletion, are said to be immune to HIV infection [50, 3]. 1.4.2 Why is the asymptomatic period long in slow disease progressors We know that HIV has a rapid viral turnover and hence mutation frequency is high. Surprisingly the asymptomatic period can be long in slow disease progressors[7]. This question has been a major challenge for researchers. Wodarz and Krakuer [54] proposed an explanation for the long time span in the asymptomatic phase. They found that increasing the replication rate of HIV results in evolution towards CCR5 tropism to escape CTL responses. However, the virus is then inhibited by non-lytic CD8 responses and pathology is prevented for longer periods of time. The X4 strain is also limited by lack of active T cell. However, opportunistic infections may 16 1.4 Lytic and non-lytic CD8 responses 17 cause high rate of T cell activation promoting the fast replicating virus to evolve towards the CXCR4 tropism. This proposal was supported by Giorgi et al[15] who found that short survival in HIV patients was associated with an increase in level of T cell lymphocyte activation. The non-lytic responses in a way slow down disease progression until there is enough activated T cells. Transition of the virus from R5 to X4 would mean that the virus becomes insensitive to inhibition by chemokines released by CD8 cells (see review by [29]). They also point out that efficient HIV escape from CTL killing would be via epitope mutations hence a change to X4 will actually be in favour of an escape from CTLs. Escape of HIV from CTLs through epitope mutation will be discussed in the Chapter (2). To summarise, during primary infection, increased viral loads cause an impairment of CD4 cells. This compromises the longetivity of CD8 memory and suppresses expansion of the X4 strain due to lack of targets. However a CTL response is stimulated by the high antigen levels, this brings the viral load to very low levels in the asymptomatic phase. The virus slowly evolves towards the fast replicating virus, conditions which favour the R5 strain. This does not mean the X4 strain is absent. A slow progressor is characterised by a low number of activated CD4 cells [15] and a strong CTL response. These two maintain a low viral load. However the low viral load cannot sustain an immune response due to reduced stimulation. Depending on the lifespan of CTLps at low antigen and CD4 levels the immune system gradually weakens. Eventually opportunistic infections invade the system causing an increase in activated CD4 cells, a condition suitable for the X4 strain. Since X4 has a higher replication rate it quickly dominates the system and escape the immune system via epitope mutations. The increased viral load or CTL induced pathology then leads to the development of AIDS [15, 54]. Alternatively, when the virus switches from R5 to X4, this is equivalent to an escape from chemokines the reduce the viral replication kinetics. A combination of the ability to infect CD4 cells and a fast replication rate will lead to CTL exhaustion and hence to the development of AIDS. 17 Chapter 2 Examples of mathematical models for the interaction between CD8 cells and HIV In a bid to understand the relationship between CD8 responses and HIV, mathematical model have been used to compliment experimental evidence. In this section discuss in greater detail some mathematical models mentioned in chapter (1). We restrict ourselves to models were developed to test the following hypothesis:(i) CTLs control the viral load steady state value,(ii) CD4 help determines CTL memory establishment, and (iii) HIV can escape a CTL response by way of mutations in epitopes. We start by looking at how the basic model is developed. We then look at its extensions and how they are used to explore the different hypothesis. 2.1 The Basic model HIV requires a host cell to reproduce itself. The basic model of viral dynamics has three variables; uninfected CD4 cells T , infected CD4 cells T ∗ and the free virus particles v. We assume that uninfected CD4 cells are supplied from precursor cells at a constant rate λ and the die at a rate dT T . Uninfected cells are infected by free virus particles at a rate βT V . Infected cell in turn die at a rate dT ∗ V . Considering that viruses can be cytopathic, we assume that infected CD4 cells have a high death rate compared uninfected CD4 cells hence dT ∗ > dT . Free virus particles are produced from infected CD4 cells at a rate kT ∗ and decline at a rate dv hence 18 2.1 The Basic model 19 the average lifespan of a viral particle is 1 .These dv assumptions lead to the following system of differential equations. The basic HIV model Ṫ = λ − dT T − βT V T˙∗ = βT V − dT ∗ T ∗ V̇ = kT ∗ − dv V (2.1) [32] The basic reproductive ratio for (2.1) is given by R0 = βλk dT ∗ dT dv (2.1) has two equilibrium points whose stability depends on R0 . If R0 < 1 the virus will not spread hence λ dT = 0 T1 = T∗1 (2.2) V1 = 0 If R0 > 1 the virus will spread and an infection is established T2 = λ dT R0 dT dv βk dT = (R0 − 1) β T∗2 = (R0 − 1) V2 (2.3) The value of R0 depends on the replication rate of the virus,β, and its cytopathicity dT ∗ . If the β > dT dT ∗ dv λk then infection succeeds and the virus persists. However the viral load may be controlled by unavailability of CD4 cells. This is known as target cell limitation. On the other hand if β < dT dT ∗ dv , λk the virus will die out and (2.1) converges to (2.2). Here the basic reproductive number is below unitary hence infection fails to establish. HIV depends on CD4 cells for reproduction. If the virus is highly cytopathic, it will deplete the CD4 cell pool before an infection can be established. If a < βkλ , dT dv there are enough CD4 cells to allow for a successful infection hence the (2.1) converges to (2.3). 19 2.2 The effect of Immune response on viral dynamics 20 2.2 The effect of Immune response on viral dynamics The introduction of an immune response into the basic model(2.1) affects the equilibrium points. We introduce a new variable C which is the magnitude of the CTL response. We assume that a CTL response proliferates in response to an antigen at a rate rCT ∗ . The parameter r denotes the CTL responsiveness, which is the growth rate of of CTLs after encountering an infected cell. In the absence of an antigen CTLs decay at a rate dc C. CTLs kill infected CD4 cells at a rate pCT ∗ . These assumptions lead to the following extension of the basic model: The basic immune response model Ṫ = λ − dT T − βT V T˙∗ = βT V − dT ∗ T ∗ − pT ∗ C V̇ = kT ∗ − dv V Ċ = rT ∗ C − dc C (2.4) [32] (2.4) has three equilibrium points given by (2.2), (2.3) and λrdv rdT dv + βdc k dc = r βk = rdv λβrk 1 ( − dT ∗ ) = p rdT dv + βdc k T3 = T∗3 V3 C3 (2.5) The equilibrium to which (2.4) will converge to depends on CTL responsiveness r.However the new reproductive ratio is given by R1 = 1 + βdc k rdv dv R1 is always greater than unitary hence elimination of the virus is not possible. If r > dc T2∗ the immune response is strong enough to control the infection hence the system (2.4) converges to (2.5). Comparing (2.3) and (2.5) we find; T3 > T2 , T3∗ < T2∗ and V3 < V2 . This shows that the activity of an immune response reduces the viral load and increases the equilibrium abundance of uninfected CD4 cells. The stronger the CTL responsiveness, the higher the equilibrium 20 2.2 The effect of Immune response on viral dynamics abundance of CD4 cells and the lower the viral load. If r < 21 dc T2∗ (2.4) converges to (2.3) assuming R0 > 1. The diagram below(fig 2.1) shows the changes in the CTL responsiveness affects the steady state value of the uninfected CD4 cells and the virus. It shows a positive correlation between CTL responsiveness and uninfected CD4 cells while there is a negative correlation between CTL responsiveness and the viral load. uninfected CD4 cells 50 40 30 20 10 0 0 0.005 0.01 0.005 0.015 0.02 0.01 0.015 0.02 0.015 0.015 0.015 0.02 0.02 uninfected Viral Viralload CD4 loadcells 50 2000 40 1500 1500 30 1000 1000 20 500 500 10 0 0 0 0 0 0 0.005 0.005 0.005 0.01 0.01 0.01 r: CTL responsiveness Figure 2.1: Bifurcation diagrams showing the effect of CTL responsiveness on uninfected CD4 cells and viral load The model predicts that a strong CTL response is required for viral control [32]. Studies have shown an inverse association between the frequency of HIV-specific CTLs and plasma viral RNA load [24, 43]. It was also shown that HIV specific CTL responses in slowly progressing individuals was more vigorous[34]. From this we can conclude that CTL responsiveness plays an major role in viral control. 21 2.3 CTL memory and viral dynamics 22 2.3 CTL memory and viral dynamics As discussed in chapter(1), there is substantial evidence that successful viral control requires a persistent CTL response [52]. Patients who do not progress to AIDS after 15 years or longer have been observed to have significantly higher levels of CTLs compared to typical HIV-infected patients [52]. However it remains unclear how this persistence is established. A favourable hypothesis has been that CTL memory depends on CD4 help [58]. Altfeld et al[2] found that maintainance of effective CTLs required virus specific T helper cells. Putting this into consideration, we split CTL response into two, CTL precursors (memory), W , and CTL effectors, C. We then assume the following: (i) CTL precursors proliferation depends on both infected and uninfected CD4 cells at a rate rT W T ∗. (ii) CTL precursors differentiate into CTL effectors in response to infected cells at a rate fWT∗ (iii) CTL precursors decay at a rate dw (iv) CTL precursors do not take part in killing infected cells (v) CTL effectors decay at a rate dc These assumption result in the mathematical model: Ṫ = λ − dT T − βT V T˙∗ = βT V − dT ∗ T ∗ − pT ∗ C V̇ = kT ∗ − dv V Ẇ = rT ∗ W T − f W T ∗ − dw W Ċ = f W T ∗ − dc C (2.6) [57] This system of differential equations has three stationary points given by (2.2), (2.3) and 22 2.3 CTL memory and viral dynamics T4 = where X = T∗4 = V4 = W4 = C4 = 23 p X 2 − 4dT rf λ 2dT r dw βk λr + f dT − dv dw rT4 − f kdw dv (rT4 − f ) dc (rT4 − f )( βk T − dT ∗ ) dv 4 X+ dw pf βk T dv 4 − dT ∗ p (2.7) Assuming that R0 > 1, stability of the equilibrium points to (2.6) depend on a number of factor including CTL responsiveness, CTL lifespan and viral replication rate. Equilibrium (2.3) describes a case where the CTL response goes to extinction a condition called CTL exhaustion. Equilibrium (2.7) on the other hand represent successful establishment of CTL memory. To determine which equilibrium (2.6) will converge to depends on on the following inequalities: X 2 < 4dT rf λ dT ∗ dv T4 < βk f T4 < r (2.8) (2.9) (2.10) If any of the inequalities is satisfied equilibrium (2.7) is unstable. However (2.3) is stable if β > dT dT ∗ dv . λk This implies that CTL exhaustion will allow viral persistence however target cells limitation comes into play. On the other hand, if all the inequalities are not satisfied, and β< dT dT ∗ dv , λk then equilibrium (2.3) becomes unstable and (2.6) converges to (2.7). This implies CTL memory is established because the virus has a relatively low infection rate which does not compromise the CD4 help. A CTL memory then reduces the viral load to lower levels. A strong CTL response can drive the virus to extinction. If the inequalities, (2.8),(2.9) and (2.10) are satisfied, and if β < dT dT ∗ dv , λk the outcome of infection depends on the initial conditions. If initially we have a high viral load and a low 23 2.3 CTL memory and viral dynamics 24 CD4 count, impairment of CD4 help will result in CTL exhaustion. However, if we initially have a low viral load and a high CD4 count, the CD4 help can contribute to the establishment of CTL memory and viral control. In summary, there are two threshold values β1 and β2 . If β < β1 , CD4 help impairment is minimal during the primary infection hence CTL memory is established. If β > β2 , the fast replicating virus impairs the CD4 help during primary infection and CTL memory cannot be established. If however β1 < β < β2 the establishment of a CTL memory depends on initial conditions. These findings are summarised in the table. CTL memory CTL exhaustion CTL exhaustion (CTL memory generation) -high initial virus load (CTL memory fails) -low initial CD4+ cell count -naive state of host CTL memory -low initial virus load -high initial CD4+ cell count -high initial CTLp numbers ↑ ↑ βT1 βT2 direction increasing rate of viral replication β → [58] The following diagrams (fig(2.2) and fig(2.3)) are results of computer simulations showing the outcome of a the system at different replication rates. 24 2.3 CTL memory and viral dynamics 100 25 16000 uninfected CD4 cells 98 virus 14000 96 12000 94 10000 92 8000 90 6000 88 4000 86 2000 84 0 0 100 200 300 400 200 180 160 140 120 100 80 60 40 20 0 500 600 700 0 100 200 300 400 500 600 700 600 700 4 CTLp CTLe 3.5 3 2.5 2 1.5 1 0.5 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 Figure 2.2: A simulation showing development of CTL memory at a low replication rate of Parameter values: β = 2.4 ∗ 10−6 dT = 0.01; dT ∗ = 0.24; dv = 2.4; k = 15000; dw = 0.05; p = 1; r = 0.005; λ = 1; f = 0.01; dc = 0.1. The low viral replication rate does not cause much damage to CD4 help, hence a CTL memory is successfully developed. This CTL memory implies a persistent CTL response which keeps the viral load at low values for a long time. 25 2.3 CTL memory and viral dynamics 100 90 80 70 60 50 40 30 20 10 0 26 90000 uninfected CD4 cells virus 80000 70000 60000 50000 40000 30000 20000 10000 0 0 100 200 300 400 500 600 700 600 0 100 200 300 400 500 600 700 600 700 25 CTLp CTLe 500 20 400 15 300 10 200 5 100 0 0 -100 -5 0 100 200 300 400 500 600 700 0 100 200 300 400 500 Figure 2.3: A simulation showing development of CTL exhaustion at a higher replication rate. Parameter values: β = 2.4∗10−5 ; dT = 0.01; dT ∗ = 0.24; dv = 2.4; k = 15000; dw = 0.05; p = 1; r = 0.005; λ = 1; f = 0.01; dc = 0.1. The high replications rate of the virus impairs CD4 help hence hampering the development of a CTL memory. In the absence of a CTL memory, the virus load increases but will then be limited by the availability of target cells(CD4s). 26 2.4 Multiple epitopes and viral dynamics 27 These findings are supported by studies that have shown that HIV-specific CTLs are hardly detectable at low CD4 T cell count [25]. During primary HIV infection, CTLs disappearances were observed, but these could not be attributed to escape mutants. This model suggest that this may have been a result of CTL exhaustion. In other experiments, the depletion of CTLs by antibodies, was shown to result in a rapid increase of the viral load in SIV-infected macaques [22, 43]. This also showed the role of CTLs in viral control. 2.4 Multiple epitopes and viral dynamics Despite the presence of CTLs, HIV has been known to persist. This has been attributed to the emergence of escape mutants. In this section we look at how escape mutants can arise and how they lead to loss of viral control even in the presents of a strong CTL response. We explain how the steady state during the asymptomatic phase is shifted in favour of the virus. Altfeld [2] observed that despite a strong antiviral response, most HIV infected individuals had poorly controlled viral loads and progress to AIDS quickly. The generation of viral escape mutants and development of high viral diversity over time was pointed out as the factor contributing. These mutations would occur within the viral epitopes. An epitope is a site on the viral surface. Immune response cells attach themselves onto these sites during infection. T cells have receptors which can recognises these epitopes. However these receptors are epitope specific. In the basic model we assume that the virus has one epitope which is recognised by the CTL response. It is however possible that a virus can have several of epitopes. If a virus has multiple epitopes, this implies that we also require a broader CTL responses for viral control. Therefore if we have 3 epitopes it means we also 3 different CTL responses for each of them. We modified the model [32] for viral dynamics by assuming: (i) The virus has 10 epitopes and 10 corresponding CTL responses Ci . (ii) Each CTL response Ci has a different responsiveness ri such that r1 < r2 < ... < r10 (iii) The CTL proliferation rates ri are positively correlated to their killing rate pi p1 < p2 < ... < p10 . (iv) The CTL responses decay at the same rate dc 27 2.4 Multiple epitopes and viral dynamics 28 The above assumptions gave rise to the followings system of differential equations. Ṫ = λ − dT T − βT V T˙∗ = βT V − dT ∗ T ∗ − T ∗ 10 X C i pi i=1 V̇ = kT ∗ − dv V Ċi = T ∗ ri Ci − dc Ci (2.11) From computer based simulations of the model, we found that the outcome of a infection depended on the lifespan of the CTL response 1 . dc If 1 dc was small, the less responsive CTL responses would be driven to extinction while the most responsive response remained. The competitively superior CTL clone reduces the viral load to levels too low to stimulate the weaker CTL clones [56]. This is known as immunodominance. On the other hand, if 1 dc was large or the CTL has a long lifespan, then the different CTL responses can coexisted and successfully controlled the virus. In this case the competitively superior CTL clone did not reduce the virus to low levels[56] and the weaker responses could persist . We also found that a long lived CTL response would more efficient in the overall control of the viral load as shown in fig(2.6). 28 2.4 Multiple epitopes and viral dynamics 29 -8 -8.5 log(CTL response) -9 -9.5 -10 -10.5 -11 -11.5 -12 0 50 100 150 200 250 300 time Figure 2.4: When the lifespan of the CTL response is small (dc = 0.01)the less responsive CTLs are driven to extinction while the strongest become dominant. The most responsive CTL clone is represented by the top curve. The rest are in descending order with the least responsive at the bottom. Parameter values dT = 0.01; dT = 0.24; β = 2.4 ∗ 10−5 ; dv = 2.4; k = 15000; λ = 1 -6 -6.5 log(CTL response) -7 -7.5 -8 -8.5 -9 -9.5 0 50 100 150 200 250 300 time Figure 2.5: When the lifespan of the CTL response is large (dc = 0.001)the CTLs coexist hence there is no immunodominance.The most responsive CTL clone is represented by the curve at the top. The rest are in descending order with the least responsive at the bottom. Parameter values the same as in fig(2.4) 29 2.4 Multiple epitopes and viral dynamics 30 35000 long lived CTL short lived CTL 30000 log(viral load) 25000 20000 15000 10000 5000 0 0 50 100 150 200 250 300 time Figure 2.6: A long lived CTL broad response controls the virus better than a short lived CTL response. Parameter values the same as in fig(2.4) except short lived CTL response: dc = 0.1 and k=1500,long lived CTL response: dc = 0.001 and k=1500 What is the effect of variations in epitopes Considering the fast replication rate of HIV, mutation are bound to frequently occur. CTLs provide a selection pressure for generation of escape mutants[2]. Studies have shown the positive selection of CTL escape variants within Nef during primary infection. This explains the disappearance of the inverse correlation between the viral load and CTL response just after primary infection. This model can explain this loss of control of the virus by CTLs as follows: if a mutation occurs in immunodominant epitope, the following can happen (i) a new CTL response may be induced together with the old ones. In this case immunodominance is maintained. (ii) a partial shift of immunodominance to the next epitope. (iii) the mutant may induce a new specific CTL response that is stronger than the original and also induce a partial shift to the next epitope. (iv) a total shift of immunodominance to the a weaker epitope[33] If a mutation does not induce a new CTL response, the virus is said to have escaped the immune response. The mutant epitope can then increase the viral load. The model predicts that if there is immunodominance, a variation if the most dominant CTL clone results in an increase 30 2.4 Multiple epitopes and viral dynamics 31 in the viral load. However if the CTL responses coexist a mutation in the same epitope will lead to a slight increase in the virus load but the virus can still be controlled. From this model we conclude that although CTLs reduce the viral load they also provides a selection pressure for emergence of viral mutants. Such mutation result in escape mutants. These mutants then create a window through which the virus can escape CTL responses leading to an increase in viral load. In general CTLs and target cell availability controls the viral load. A persistent CTL response depends on the availability of CD4 help. However at times the virus can persist in the presences of a strong immune response. This would have been a result of mutations within epitopes such that the immune system fails to recognise the virus. 31 Chapter 3 Gradual evolution of HIV as a mechanism for disease progression The gradually switching of HIV from a slow replicating strain to a faster one, has been suggested as a determinant of disease progression. We hypothesis that HIV makes use of different strains in adapting to its environment. Wodarz et al [51] developed a model which includes the different viral strains. However they did not include the effect of the non-lytic CD8 responses on the infected macrophages and the gradual evolution of the R5 strain towards the X4 strain during the disease progression. In this chapter we develop a model that seeks to investigate the parameters that affects the this gradual switch from one strain to the other. In so doing we also gain insight to the conditions affect the switching from one viral strain to the other. Our model was developed on the following assumptions 1). CTLs can reduce viral load [32, 43, 24] 2). CTL memory is required for successful viral control[50] 3). A lasting CTL memory depends on CD4 help[50] 4). Non-lytic CD8 responses also play a role in viral control [54] 5). HIV can infect macrophages and CD4 cells [51] 6). There are two main viral strains of HIV, R5 and X4.[5, 45] 32 3.1 Developments of the model 33 3.1 Developments of the model It has been shown that can have several coreceptors can be found on a single virus particle [12, 11, 5]. In our model we assume a single viral population with two coreceptors CCR5 and CXCR4. However the each virus particle can only use one coreceptor at a time. We therefore split the viral population into a population of the R5 strain ’R5 ’ and the X4 strain ’X4 ’. We also introduce two target cell populations, macrophages ’M’ and CD4 cells ’T’ [5, 45]. For simplicity we assume that the R5 strain infects macrophages while the X4 strain infects the CD4 cells [12, 11]. Since we assumed a single virus population, the infection rates for the two strains are assumed to be the same. Macrophages are produced at a constant rate µ, decay naturally at a rate dm∗ M and are infected by the R5 strain at a rate βMR5 . HIV can only replicate in active and dividing cells. However this is not a requirement for macrophage infection [51]. On the other hand CD4 cells are split into resting CD4 cells ’S’ and active CD4 cells ’T’ [51]. HIV has been shown to complete its life cycle in active cells [51]. We assume that the virus only infects active cell. Resting CD4 cells are produced at a constant rate λ. They are activated as a result of an interaction with antigens at a rate gS[T ∗ + M ∗ + N] where N is the background activation by other infections. This background activation is neseccary to keep the active CD4 population present at low virus levels, a condition required for the onset of AIDS [7]. Resting CD4 cells decay at a rate S and active CD4 cells decay at a rate dT T where ds < dT . Infected cells produce free virus particles into the plasma. In the model the R5 strain is produce at a rate KR5 M ∗ and the X4 strain at a rate KX4 T ∗ . Both viral strains decay at the same rate dv , however R5 evolves to X4 during the course of the disease [45, 12] at a constant rate ev R5 . Chemokines are believed to inhibit the replication and entry of the R5 strain [11]. We therefore modify the R5 strain infection term to βR5 M HC+1 where H is the efficiency of the the non-lytic response in inhibiting the viral replication and entry into macrophages. However, the CTL response has been shown to be inefficient towards the macrophages as compared to the CD4 cells. We do not include a CTL response for the infected macrophages but infected CD4 cells are killed by CTLs at a rate pT ∗ C. From the previous chapter we established that CTL memory dependented on CD4 help, we include a CTL memory term and CTL response term 33 3.1 Developments of the model 34 similar to that in (6). Both infected macrophages and infected CD4 cells activate CTLps at a rate rW T (T ∗ + M ∗ ). Going back to the characteristics of HIV tropisms [51] we came up with the following condition for our parameters; • KX4 > KR5 ; The replication rate of the X4 strain has been shown to be fast faster than the R5 strain. [51, 45, 48, 5] • dT ∗ > dm∗ ; The cytopathicity of the X4 strain has also been shown to be higher.[51, 45, 48, 5] • X4 = 0 and R5 6= 0 at time=0; The R5 strain has been shown to dominate the primary and part of the asymptomatic period of infection. On the other hand the X4 strain has been shown the appear in the later stages of infection [7, 12, 5]. We therefore had only a population of the R5 strain at the beginning of the simulations. The above assumption lead us to a system of 9 nonlinear differential equations given by: Ṡ = λ − ds S − gS(T ∗ + M ∗ + N) Ṫ = gS(T ∗ + M ∗ + N) − dT T − βT X4 T˙∗ = βT X4 − dT ∗ T ∗ − pT ∗ C βMR5 Ṁ = µ − dm M − HC + 1 βMR 5 Ṁ ∗ = − dm∗ M ∗ HC + 1 Ṙ5 = KR5 M ∗ − dv R5 − ev R5 Ẋ4 = KX4 T ∗ − dv X4 + ev R5 Ẇ = r(T ∗ + M ∗ )W T − f W (T ∗ + M ∗ ) − dw W Ċ = f W (T ∗ + M ∗ ) − dc C 34 (3.1) 3.2 Results and analysis 35 3.2 Results and analysis In this section we seek to investigate the factors the that affect the switching of the the two strains. These results are based on computer simulations. The general results of the simulations depended on the initial conditions. The first case is when T > f /r. In this case there are enough CD4 cells to establish a CTL memory. Here persistence of the viral population depends on the efficiency of the CTL response. For a weaker CTL response, the viral load persists but at low levels. However if the CTL response is strong then both viral strains are driven to extinction. The second case is when T < f /r. In this case there are a number of factors affect the dynamics of the simulation. The following observations were based on the assumption that this in equality was true. • CTL responsiveness Since T < f /r CTL memory fails to persist. However the CTL response can still be stimulated but does not persist for the entire period of infection. However this short lived presents has an effect on the out come of the system. Increasing the CTL responsiveness, r while maintaining the inequality T < f /c, the viral load is kept at a low level before the immune response fades. An increases in CTL responsiveness therefore increases the time taken for the X4 strain to appear and for a switch from the R5 strain to the X4 strain to occur. The rate at which the CTLps differentiate to CTLes f has an same effect as r. • Cytopathicity There was a negative correlation between the cytopathicity of both viral strains (dT ∗ and dm∗ ) and the steady state value their corresponding viral loads. However this was the case provided dm∗ < βλKR5 . dm (dv +ev ) The effect of each individual cytopathicity was independent of the other, for instance an increase in the cytopathicity of the R5 strain only decreased the the viral load of the R5 strain and not the X4 strain. There was a threshold for both dT ∗ and dm∗ which determined whether X4 < R5 or R5 < X4. Therefore if dT is above this threshold or dm∗ is below it, no switch would occur. If however dm∗ ≈ dm then the presence of a switch depends on the cytopathicity of the X4 strain(dT ∗ ). On the other 35 3.2 Results and analysis hand if dm∗ > βλKR5 , dm (dv +ev ) 36 then always R5 goes to extinction as the basic reproduction number is below unitary. In this case the R5 strain fails to establish an infection. In the absence of the R5 strain, the X4 strain also fails to emerge as it has to evolve for the R5 strain. In general increasing the cytopathicity reduced the number of infected cells hence the overall viral production. This then resulted in a reduced viral load. We also found that the cytopathicity of the R5 strain dm∗ , had a negative correlation with the time taken for a switch to occur. However the cytopathicity of the X4 strain had a positive correlation with this time. Results shown in Fig(3.1) and (3.2) (b) (a) 7e+08 R5 X4 6e+08 virions 5e+08 4e+08 3e+08 2e+08 1e+08 0 0 1000 2000 3000 4000 5000 Time(days) (b) (b) 7e+08 7e+08 R5 X4 virions 6e+08 6e+08 5e+08 5e+08 5e+08 4e+08 4e+08 4e+08 3e+08 3e+08 3e+08 2e+08 2e+08 2e+08 1e+08 1e+08 1e+08 0 0 0 0 0 0 1000 1000 1000 2000 2000 2000 Time(days) Time(days) 3000 3000 3000 4000 4000 4000 5000 5000 Figure 3.1: The effect of a variation in the cytopathicity of the X4 strain on the presence or absence of a switch. (a). dT ∗ = 0.01; Low cytopathicity of X4 maintains a high number of infected CD4 cells which can sustain the X4 population until a switch occurs. (b)dT ∗ = 0.1 the high cytopathicity of X4 reduces the number of infected CD4 cells present for viral production. This keeps the X4 viral load is kept at a low level and hence a switch fails. 36 3.2 Results and analysis 37 (a) 7e+08 6e+08 5e+08 4e+08 3e+08 2e+08 1e+08 0 R5 X4 0 1000 2000 3000 4000 5000 (a) (b) 7e+08 6e+08 6e+08 5e+08 5e+08 4e+08 4e+08 3e+08 3e+08 2e+08 2e+08 1e+08 0 R5 X4 0 1000 2000 3000 4000 5000 (b) (c) (c) 6e+08 3.5e+08 5e+08 4.5e+08 3e+08 5e+08 4e+08 3.5e+08 2.5e+08 3.5e+08 4e+08 3e+08 3e+08 2.5e+08 2e+08 3e+08 2.5e+08 2e+08 1.5e+08 2e+08 2e+08 1e+08 1.5e+08 5e+07 1e+08 1e+08 5e+07 0 5e+07 0 0 0 0 0 R5 X4 1000 1000 1000 2000 3000 2000 2000 Time(days) 3000 3000 4000 4000 4000 Figure 3.2: The effect of a variation in the cytopathicity of R5 on the time lapse for a switch to occur. (a). dm∗ = 0.005 (b). dm∗ = 0.01 and (c).dm∗ = 0.08. A lower cytopathicity of R5 increases the availability of infected macrophages for viral production. However since this does not affect the X4 viral load it therefore takes more time for the switch to occur as compared to when the cytopathicity is much high. Note that in (a) dm∗ < dT ∗ , (b).dm∗ = dT ∗ and (c)dm∗ > dT ∗ . 37 5000 5000 3.2 Results and analysis 38 • evolution rate The evolution rate determines the time taken for the X4 strain to emerge. If rate at which the R5 strain evolves to the X4 strain ev > dv then X4 > R5 throughout the simulation period. This implies that the X4 strain appears soon after initial infection and dominates the population throughout the period of infection. If ev < dv then the population is initially dominated by the R5 strain however there is a negative correlation between ev and the time taken for the X4 strain to emerge. There is also a similar relation with the time it takes for dominant population to switch from R5 to X4. The lower the value of ev the more time it takes for the switch to occur. On the other hand, ev had a positive correlation with the viral load of X4 while having a negative correlation with the R5 population. These findings are illustrated in fig(3.3). 7e+08 (1) R5 (1) X4 (2) R5 (2) X4 6e+08 5e+08 virions 4e+08 3e+08 2e+08 1e+08 0 0 1000 2000 3000 4000 5000 Time(days) Figure 3.3: An increase in the evolution rate from R5 to X4 reduces the time lapse for the emergence of the X4 strain and the time lapse for a switch to occur. However this increase also results in an increase of the X4 viral load and a subsequent decrease in the R5 viral load. (1). ev = 1 and (2).ev = 10−5 . note that in all cases ev < dv . 38 3.2 Results and analysis 39 • Infection rate There are two threshold value of β which affect the out come of the simulation. First if β < dm∗ dm (a+ev ) λKR5 then the the replication kinetics are too slow for an infection to occur. Despite the condition T < f /c, if β is low enough , a CTL memory can be established and drive the viral population to extinction. If the virus manages to persist, only the R5 strain is presence. The X4 strain fails to appear at all or weakly appears but is the driven to extinction. On the other hand if β > dm∗ dm (a+ev ) λKR5 but is lower than a second threshold β2 , persistence of the virus depends on the cytopathicity of the different strain relative to the efficiency of the CTL response. If cytopathicity is high for both strains, a weak CTL response may result in CTL induced pathology [54]. This pathology reduces the availability of target cells hence the viral strains are driven to extinction. However for a lower cytopathicity, both viral strains can coexist. In this case there is a negative correlation between the infection rate and the time it takes for the dominant strain to switch from R5 to X4. An increase in infectiousness reduces the time taken for the switch to occur. If the infection rate goes beyond β2 the system becomes limited by availability of target cells as CTL exhaustion and pathology come into play. In this case both viral strains are again driven to extinction. • Background activation Surprisingly the background activation term did not have an effect on the outcome of the simulation. One would have expected that in the absence of background activation, it would take more time for the switching to occur. Maybe this was because it was introduced right from the beginning of the simulation. On the other hand the activation term could be of a form other than the one we used. In similar models, background activation has been shown to affect the dynamics of the system [7]. Experiments have also shown that disease progression is associated with an increased T cell activation by infections other than HIV[15]. 39 3.3 Discussion 40 3.3 Discussion On the contrary to the model by Callaway et al [7], the outcome of our model depended on the initial conditions. If at the beginning the is a large pool of active CD4 cells (T > f /c), this is sufficient to stimulate and establish a strong and lasting immune response. The presents of such a strong immune response is enough to drive the R5 strain to extinction and the prevent the X4 strain from emerging. However, if the initial CD4 cell population is not large enough (T < f /c), then the virus can establish an infection but its outcome now depends on a number of other parameters. We restricted our analysis to the case when infection is established. As the virus persists, the CD4 cells gradually decline and so does the strength of the immune response (CD8 cells). When the CD8 cells go below a critical threshold the immune response becomes too weak to suppress the the R5 and X4 viral strains. At this point the X4 strain emerges and due to it fast replication rate [9, 27], its population grows exponentially. Depending on other parameters such as the cytopathicity of the X4 strain, there maybe a switch in the dominant viral strain from the R5 to the X4 strain. When the X4 strain first emerges there was a significant decline in the CD4 cell count. This has also been shown in experiments [27, 7]. However from the time when the X4 strain emerges the two strains coexist and this has also been supported by clinical observations in which both strains were shown to coexist late in infection [27, 7] We analysed the effect of six parameters on the outcome of the system. We choose these parameters because we thought they we important determinants of the properties of the system we were looking at. The properties were (1) the presents or absence of a switch from R5 to X4, (2) if there is a switch , the time that lapses before it occurs. We found that while some parameters affected either of the two properties, other parameters affect both the occurrence and the time lapse of the switch. The important feature of this model is the inclusion of the both the CD4 cells and macrophages and distinguishing of the virus tropisms. This model gives a better insight on how HIV adapts to its environment by making use of the different properties of the tropisms. During the primary stage, the immune response is generally stronger. A slow replicating 40 3.3 Discussion 41 virus and an escape from the lytic responses of CD8 cells gives the virus a better survival chance. Experiments [60, 48], have shown that the R5 strain dominates the initial population even if the X4 strains are transmitted as well. This has been shown to be the case if infection was via drug injection or blood transfusion [51]. As the immune system weakens, the virus slowly evolves towards a faster replicating strain. When the immune system goes below a critical threshold, the faster replicating strain then dominates the system paving way for accelerated disease progression. From this model we conclude that macrophages are essential for maintaining viral persistence in the presence of an immune response. We also conclude that the cytopathicity of the different HIV strains play an important role in progression of the disease by determining the success of HIV in switching from a slow replicating strain to a faster one. On the other hand, the rate of evolution, the strength of the immune response, infection rate and again the cytopathicity are major determinants of the time taken for the switch to a faster replicating strain to occur.The following is a table of parameter values used in the simulations of our model.[39] 41 3.3 Discussion 42 Parameter Value Description λ 600 day −1mm−3 Production rate of resting T cells µ 300 day −1mm−3 Production rate of macrophages g 0.000001 day −1 mm3 rate of T cell activation N 100 day −1mm−3 amount of antigen present for back ground activation β 2.4 ∗ 10−5 day −1 mm3 infection rate KR5 15000 day −1 number of new viral particles produced from infected macrophages KX4 20000day −1 number of new viral particles produced from infected CD4 cells ev varies (10−5 )∗∗ evolution rate p 1 day −1 mm−3 rate of T cell killing by CTLs H 20 efficiency of non-lytic response inhibition of entry into macrophages f 0.001 rate of differentiation from CD8 precursors to CD8 effectors r 5 ∗ 10−8 CD8 responsiveness ds 0.00001day −1 death rate of resting T cells dT 0.001day −1 death rate of active but uninfected T cells dT ∗ varies (0.01)∗∗ death rate of infected T cells dm 0.001 day −1 death rate of uninfected macrophages dm∗ varies (0.005)∗∗ death rate of infected macrophages dv 2.4 day −1 decay rate of free viral particles dc 0.005 decay rate of CD8 effectors dw 0.0001 decay rate of CD8 precursors ∗∗default value. 42 Chapter 4 Conclusions In this paper, we reviewed models and imperical evidence presented to explain the dynamics of HIV infections and progression to AIDS. The subject proved to be highly controversial as there remain a lot of unanswered questions. However from the review, assuming CD8 cells play some role in HIV infection, we established the following: (1). CD8 responses can reduce the viral set point value. (2). Persistent CD8 responses are required for successful viral control. (3). CD4 helper cells are efficient in establishing a persistent CD8 responses. (4). Lytic CD8 responses (CTLs) can be detrimental to the host hence a cooperation with non-lytic CD8 responses results in a more beneficial outcome. (5). HIV can escape CTL responses through variations with in the virus for instance mutations in epitopes. (6). HIV makes use of different viral tropisms to adapt to its environment. (7). Switching of viral tropics can be a mechanism for disease progression. We developed a model to investigate the factors that contribute to the switching of viral tropisms. We considered a case where the R5 strain dominated the viral population at the beginning of an infection while the X4 strains evolves from the R5 strain at a later stage. We found that a successful switch from the R5 strain to the X4 strain depended on the cytopathicity of the individual strains. However if the switch does occur, strength of the CD8 response, infection rate and the evolution rate had an effect on the time lapse before the switch occurs. We conclude that these factors may be important determinants of the 43 44 length of the asymptomatic period of infection. However our analysis of the model showed that background activation had no effect on switching of tropisms. This does not conform with experiments that have shown that disease progression is associated with an increased T cell activation[15]. We therefore suggest that the activation term be some other function other than a function of the resting T cells as suggested in our model. A possible candidate would be a function that peaks later during the asymptomatic period. This will account for the role of opportunistic infections which arise just before the onset of AIDS [49]. We used the ‘one at a time’ analysis technique whereby we varied the value of a single parameter and noted its individual effect. We suggest that a detailed analysis be carried out, for example an analysis of an interactions between different parameters. To extend this model we suggest the inclusion of some mechanism for viral escape from CD8 responses for instance variation within epitopes. As discussed earlier, HIV can evade the immune response through such variations [32]. Our model observations where based on the case where a CTL memory failed to establish since in the presence of a CTL memory the virus was always driven to extinction. This extension may give insight on the dynamics of the virus in the presence of an established CTL memory (in our model when T > f /r). 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