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 requirementsfor AIMS Diploma in

Mathematical Sciences.

May 2004.

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2

Acknowledgements

I would like to thank the following for their contribution inthe 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 andthe discovery of my passion for

epidemiology. I would also like to thank the sponsors of AIMSand 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.

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Abstract

Mathematical models and experiments have shown the importance of CD8 responses in control

of HIV infection. In this paper we review the theoretical andempirical 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 memorymodel to include macrophage

cells, non-lytic CD8 responses and the evolution of HIV froma 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

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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 completeits 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.

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1.1 The stages of HIV infection 3

1.1 The stages of HIV infection

To understand the development of AIDS from HIV infection, itis important to analyse the

dynamics of HIV CD4 cells and CD8 cells throughout the periodof infection. The pattern of

disease progression is divided into three stages summarised in fig(1).

Primary phase Asymptomatic phase AIDS

CD8+ cells

CD4+ cells

Virus load

up to 15 years

Pla

sma

conc

etra

tion

leve

ls

2−10 weeks

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, patientsexperience

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

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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 inthe 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 doesnot 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 toa 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 theuninfected macaque. This implies

that SIV infection cannot solely account for the drop in CD4 cells. This was attributed this to

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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] introduceda control antibody, P1.17, into

uninfected macaques that did not delete the CD8 cells. CD4s dropped by51%. 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 increasein 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 aregenerated 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].

TIME

PLA

SM

A C

ON

CE

TR

AT

ION

CTL RESPONSE

VIRAL LOAD

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 peforinand granzymes for cell lysis.

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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 solu-

ble 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 re-

sponses 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 in-

fection. 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 de-

pended 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 thatapart from a high responsiveness,

a long lived CTL response was efficient in viral control(fig 3).

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1.3 The role of CTLs in HIV infection 7

0

500

1000

1500

2000

0 0.005 0.01 0.015 0.02

Vira

l loa

d

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 toa long lived CTL response.

This was supported by an observation that HIV infected individuals with an asymptomatic infec-

tion 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 loadsare 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.

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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 foundthat 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 defineda 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 agreementwith 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 replicationrate 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 virusLCMV. HIV has been shown to have

different viral strains at different infection stage. These viral strains have different cytopathicity

levels [45, 48, 12]. Thereforeshould the death rate for infected cells in the models be alsoa

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 lookat factors that promote CTL

persistence, which is also know as CTL memory. Traditionally CTL memory had been asso-

ciated with protection against re-infection. A persistentCTL 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.

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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 atundetectable 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 mod-

els 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 pre-

cursors (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 toan impaired CD4 help or a

reduction in antigen level by anti-viral therapy. Some researchers believe that CTL memory de-

pends 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

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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(IL-

12). This cytokine trigger dentritic cells to produce CD8 cells. This is called the classical path-

way [50]. On the other hand, some viruses can directly stimulate the dentritic cells. Whether

HIV also does this remains to be clarified. Alternatively CD4cells activate antigen-presenting

cells APCwhich 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 vi-

ral clearance. This result meant that the two pathways wouldwork 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. Experimentsto 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 maybe 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?

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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 tomaintain CTL memory. Kalams

[23] presented data to show a positive correlation between Thelp 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 theabsence of CD4 help.

Wodarz et al [52] use mathematical models to compare the roleof 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 com-

promises 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 viralload 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 viralescape 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 mathe-

matical model by Wodarz et al[56] showed that broad responsecan 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.

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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 iscalled 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 ofLCMV . The

presence of an efficient CTL response can successfully control the infection, however a less ef-

ficient 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 possi-

ble 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 investigatethe conditions for immunopathol-

ogy 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 respon-

siveness. 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 replicatingArmstrongstrain, 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 replicatingTraubstrain, the absence of a non-lytic response resulted

in severe tissue damage and wasting of the host [46].

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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 lyticand 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 etal[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 non-

lytic 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 efficiencyof 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 toincreased 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,Vac-

cinia while others with a non-cytopathic virusLCMV [47]. In both cases no CTL activity was

detected. This result was explained as follows, the absenceof non-lytic responses to control

the replication rate of the virus led to a failed CTL responsepossibly 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.

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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 differentcell types for example in macrophages

and T cells [53]. Experiments have been done withLCMV, a non cytopathic virus andVSV, a

cytopathic virus to support Wodarz et al’s [53] hypothesis.In the experiments, peforin deficient

mice infected withLCMVdeveloped 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]. Thisshowed that CTLs alone could not

resolve the infection despite the non cytopathic nature of the virus. ForVSVinfected 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 Bvirus (HBV), viral

control was achieved by an intricate balance between a lyticand 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 indifferent 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 mod-

els 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

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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 re-

sponse 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 evolveto 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

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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 hasbeen 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 theimmune 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 con-

tributor to viral persistence [45]. During the course of theinfection the virus evolves towards to

the faster replicating rates, at the same time escaping the immune system through epitope muta-

tion [32, 7, 57]. This creates a condition favourable for theemergence 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 mutationfrequency 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 lowviral 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 immunesystem 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 anescape from chemokines the

reduce the viral replication kinetics. A combination of theability to infect CD4 cells and a fast

replication rate will lead to CTL exhaustion and hence to thedevelopment 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 thissection 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) HIVcan 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 modelof viral dynamics has three vari-

ables; uninfected CD4 cellsT , infected CD4 cellsT ∗and the free virus particlesv. We assume

that uninfected CD4 cells are supplied from precursor cellsat a constant rateλ and the die at

a ratedT T . Uninfected cells are infected by free virus particles at a rateβTV . Infected cell

in turn die at a ratedT ∗V . Considering that viruses can be cytopathic, we assume thatinfected

CD4 cells have a high death rate compared uninfected CD4 cells hencedT ∗ > dT . Free virus

particles are produced from infected CD4 cells at a ratekT ∗ and decline at a ratedv hence

18

2.1 The Basic model 19

the average lifespan of a viral particle is1dv

.These assumptions lead to the following system of

differential equations.

The basic HIV model

T = λ − dT T − βTV

T ∗ = βTV − dT ∗T ∗

V = kT ∗ − dvV (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 onR0.

If R0 < 1 the virus will not spread hence

T1 =λ

dT

T∗1 = 0 (2.2)

V1 = 0

If R0 > 1 the virus will spread and an infection is established

T2 =λ

dTR0

T∗

2= (R0 − 1)

dT dv

βk

V2 = (R0 − 1)dT

β(2.3)

The value ofR0 depends on the replication rate of the virus,β, and its cytopathicitydT ∗. If

the β > dT dT∗dv

λkthen infection succeeds and the virus persists. However theviral load may

be controlled by unavailability of CD4 cells. This is known as target cell limitation. On the

other hand ifβ < dT dT∗dv

λk, the virus will die out and (2.1) converges to (2.2). Here thebasic

reproductive number is below unitary hence infection failsto 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. Ifa < 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 variableC which is the magnitude of the CTL response. We

assume that a CTL response proliferates in response to an antigen at a raterCT ∗. The param-

eterr 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 ratedcC. CTLs kill infected CD4

cells at a ratepCT ∗. These assumptions lead to the following extension of the basic model:

The basic immune response model

T = λ − dT T − βTV

T ∗ = βTV − dT ∗T ∗ − pT ∗C

V = kT ∗ − dvV

C = rT ∗C − dcC (2.4)

[32] (2.4) has three equilibrium points given by (2.2), (2.3) and

T3 =λrdv

rdT dv + βdck

T∗

3=

dc

r

V3 =βk

rdv

C3 =1

p(

λβrk

rdTdv + βdck− dT ∗) (2.5)

The equilibrium to which (2.4) will converge to depends on CTL responsivenessr.However the

new reproductive ratio is given by

R1 = 1 +βdck

rdvdv

R1 is always greater than unitary hence elimination of the virus is not possible. Ifr > dc

T ∗

2

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, T∗

3 < T ∗

2 andV3 < 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 21

abundance of CD4 cells and the lower the viral load. Ifr < dc

T ∗

2

(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.

0

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fect

ed C

D4

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l loa

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d

r: CTL responsiveness

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l loa

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l loa

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l loa

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r: CTL responsiveness

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l loa

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l loa

d

r: CTL responsiveness

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l loa

d

r: CTL responsiveness

0

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Vira

l loa

d

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 forviral control [32]. Studies

have shown an inverse association between the frequency of HIV-specific CTLs and plasma vi-

ral 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 progressto 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 howthis 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

rTWT ∗.

(ii) CTL precursors differentiate into CTL effectors in response to infected cells at a rate

fWT ∗

(iii) CTL precursors decay at a ratedw

(iv) CTL precursors do not take part in killing infected cells

(v) CTL effectors decay at a ratedc

These assumption result in the mathematical model:

T = λ − dT T − βTV

T ∗ = βTV − dT ∗T ∗ − pT ∗C

V = kT ∗ − dvV

W = rT ∗WT − fWT ∗ − dwW

C = fWT ∗ − dcC (2.6)

[57]

This system of differential equations has three stationarypoints given by (2.2), (2.3) and

22

2.3 CTL memory and viral dynamics 23

T4 =X +

√

X2 − 4dT rfλ

2dTr

where X = λr + fdT −dwβk

dv

T∗

4=

dw

rT4 − f

V4 =kdw

dv(rT4 − f)

W4 =dc(rT4 − f)(βk

dvT4 − dT ∗)

dwpf

C4 =

βk

dvT4 − dT ∗

p(2.7)

Assuming thatR0 > 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 dependson on the following inequalities:

X2 < 4dT rfλ (2.8)

T4 <dT ∗dv

βk(2.9)

T4 <f

r(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 relativelylow 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 valuesβ1 andβ2. If

β < β1, CD4 help impairment is minimal during the primary infection hence CTL memory is

established. Ifβ > β2, the fast replicating virus impairs the CD4 help during primary infection

and CTL memory cannot be established. If howeverβ1 < 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

↑ ↑

βT1βT2

direction increasing rate of viral replicationβ →

[58]

The following diagrams (fig(2.2) and fig(2.3)) are results ofcomputer simulations showing

the outcome of a the system at different replication rates.

24

2.3 CTL memory and viral dynamics 25

84

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CTLe

Figure 2.2: A simulation showing development of CTL memory at a low replication rate of

Parameter values:β = 2.4 ∗ 10−6dT = 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 26

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20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

-5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

CTLe

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, mostHIV 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 epitopeis a site on the viral surface. Im-

mune 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 responsesCi .

(ii) Each CTL responseCi has a different responsivenessri such thatr1 < r2 < ... < r10

(iii) The CTL proliferation ratesri are positively correlated to their killing ratepi p1 < p2 <

... < p10.

(iv) The CTL responses decay at the same ratedc

27

2.4 Multiple epitopes and viral dynamics 28

The above assumptions gave rise to the followings system of differential equations.

T = λ − dTT − βTV

T ∗ = βTV − dT ∗T ∗ − T ∗

10∑

i=1

Cipi

V = kT ∗ − dvV

Ci = T ∗riCi − dcCi (2.11)

From computer based simulations of the model, we found that the outcome of a infection

depended on the lifespan of the CTL response1dc

. If 1dc

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 1dc

was large or

the CTL has a long lifespan, then the different CTL responsescan 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 . Wealso 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

-12

-11.5

-11

-10.5

-10

-9.5

-9

-8.5

-8

0 50 100 150 200 250 300

log(

CT

L re

spon

se)

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 valuesdT = 0.01; dT = 0.24; β = 2.4 ∗ 10−5; dv = 2.4; k = 15000; λ = 1

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

0 50 100 150 200 250 300

log(

CT

L re

spon

se)

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

0

5000

10000

15000

20000

25000

30000

35000

0 50 100 150 200 250 300

log(

vira

l loa

d)

time

long lived CTLshort lived CTL

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 pos-

itive selection of CTL escape variants withinNefduring primary infection. This explains the

disappearance of the inverse correlation between the viralload and CTL response just after pri-

mary infection. This model can explain this loss of control of the virus by CTLs as follows: if

a mutation occurs in immunodominant epitope, the followingcan 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 attimes 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 thatHIV makes use of different strains

in adapting to its environment. Wodarz et al [51] developed amodel which includes the different

viral strains. However they did not include the effect of thenon-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 thatseeks 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 macrophageswhile 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 rateµ, decay naturally at

a ratedm∗M and are infected by the R5 strain at a rateβMR5. 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]. Weassume 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 rategS[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 rateS and active CD4 cells decay at a ratedT T whereds < dT .

Infected cells produce free virus particles into the plasma. In the model the R5 strain is

produce at a rateKR5M∗ and the X4 strain at a rateKX4

T ∗. Both viral strains decay at the

same ratedv, however R5 evolves to X4 during the course of the disease [45, 12] at a constant

rateevR5. Chemokines are believed to inhibit the replication and entry of the R5 strain [11].

We therefore modify the R5 strain infection term toβR5M

HC+1where H is the efficiency of the the

non-lytic response in inhibiting the viral replication andentry 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 ratepT ∗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

raterWT (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 andR5 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:

S = λ − dsS − gS(T ∗ + M∗ + N)

T = gS(T ∗ + M∗ + N) − dT T − βTX4

T ∗ = βTX4 − dT ∗T ∗ − pT ∗C

M = µ − dmM −βMR5

HC + 1

M∗ =βMR5

HC + 1− dm∗M∗

R5 = KR5M∗ − dvR5 − evR5

X4 = KX4T ∗ − dvX4 + evR5

W = r(T ∗ + M∗)WT − fW (T ∗ + M∗) − dwW

C = fW (T ∗ + M∗) − dcC (3.1)

34

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 whenT > 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, theviral load persists but at low

levels. However if the CTL response is strong then both viralstrains are driven to extinction.

The second case is whenT < 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

SinceT < 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 inequalityT < 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 theR5 strain to the X4 strain

to occur. The rate at which the CTLps differentiate to CTLesf has an same effect asr.

• Cytopathicity

There was a negative correlation between the cytopathicityof both viral strains (dT ∗ and

dm∗) and the steady state value their corresponding viral loads. However this was the case

provideddm∗ <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. Therewas a threshold for both

dT ∗ anddm∗ which determined whetherX4 < R5 or R5 < X4. Therefore ifdT is above

this threshold ordm∗ is below it, no switch would occur. If howeverdm∗ ≈ 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 36

hand if dm∗ >βλKR5

dm(dv+ev), 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 asit 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 straindm∗, had a negative correlation with the time taken for

a switch to occur. However the cytopathicity of the X4 strainhad a positive correlation

with this time. Results shown in Fig(3.1) and (3.2)

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

(b)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

(b)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

(b)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

Time(days)

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

Time(days)

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(b)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5X4

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(a)

R5X4

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 untila 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

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(a)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(a)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(a)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(a)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(a)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08 7e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(b)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(b)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(b)

R5

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(b)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(c)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(c)

R5X4

0 1e+08 2e+08 3e+08 4e+08 5e+08 6e+08

0 1000 2000 3000 4000 5000

(c)

R5X4

0 5e+07 1e+08

1.5e+08 2e+08

2.5e+08 3e+08

3.5e+08

0 1000 2000 3000 4000 5000

(c)

R5

0 5e+07 1e+08

1.5e+08 2e+08

2.5e+08 3e+08

3.5e+08

0 1000 2000 3000 4000 5000

(c)

R5

0 5e+07 1e+08

1.5e+08 2e+08

2.5e+08 3e+08

3.5e+08

0 1000 2000 3000 4000 5000

(c)

R5

0 5e+07 1e+08

1.5e+08 2e+08

2.5e+08 3e+08

3.5e+08 4e+08

4.5e+08 5e+08

0 1000 2000 3000 4000 5000

(c)

R5X4

0 5e+07 1e+08

1.5e+08 2e+08

2.5e+08 3e+08

3.5e+08 4e+08

4.5e+08 5e+08

0 1000 2000 3000 4000 5000

Time(days)

(c)

R5X4

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 moretime 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

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 strainev > dv thenX4 > 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. Ifev < dv then the population is

initially dominated by the R5 strain however there is a negative correlation betweenev

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 negativecorrelation with the R5

population. These findings are illustrated in fig(3.3).

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

0 1000 2000 3000 4000 5000

virio

ns

Time(days)

(1) R5(1) X4(2) R5(2) X4

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 casesev < dv.

38

3.2 Results and analysis 39

• Infection rate

There are two threshold value ofβ 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 conditionT < 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 ifβ > dm∗dm(a+ev)λKR5

but is lower than a second

thresholdβ2, 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 aredriven 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 takesfor 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 beyondβ2 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 aneffect 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 intro-

duced 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 outcomeof 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 X4strain 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 thoughtthey we important de-

terminants 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 parametersaffected 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 abetter insight on how HIV

adapts to its environment by making use of the different properties of the tropisms. Dur-

ing 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 sur-

vival 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 per-

sistence 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 deter-

mining 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 600day−1mm−3 Production rate of resting T cells

µ 300day−1mm−3 Production rate of macrophages

g 0.000001day−1mm3 rate of T cell activation

N 100day−1mm−3 amount of antigen present for back ground activation

β 2.4 ∗ 10−5 day−1mm3 infection rate

KR515000day−1 number of new viral particles produced from infected

macrophages

KX420000day−1 number of new viral particles produced from infected

CD4 cells

ev varies(10−5)∗∗ evolution rate

p 1 day−1mm−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 0.001day−1 death rate of uninfected macrophages

dm∗ varies(0.005)∗∗ death rate of infected macrophages

dv 2.4day−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 dynam-

ics of HIV infections and progression to AIDS. The subject proved to be highly controver-

sial as there remain a lot of unanswered questions. However from the review, assuming

CD8 cells play some role in HIV infection, we established thefollowing: (1). CD8 re-

sponses 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 dominatedthe 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 rest-

ing 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 opportunis-

tic 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 indi-

vidual 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]. Ourmodel observations where

based on the case where a CTL memory failed to establish sincein 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). Another problem faced with our model was the exhaustion oftarget cells. In

some cases the observed result could have been a due to targetcell limitation rather than

a variation in the parameters. For this reason we suggest a extension of the model which

maintains the availability of target cells for longer in theabsence of an immune response.

44

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