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MBW:Dynamics of Drug-Resistant HIV Mutant

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Article review by Jessica Moton, March 2010.

Article: Anti-viral Drug Treatment: Dynamics of Resistance in Free Virus and Infected Cell Populations, M. A. Nowak, S. Bonhoeffer, G. M. Shaw, and R. M. May. J. Theoretical Biology, 184:203-217, 1997.


Mathematics Used

Continuous-time ordinary differential equations (ODEs) are used in this model. Parameters describing different behaviors are incorporated into the ODEs, and the ODEs are solved either analytically or numerically with different parameter conditions in order to simulate the effect of parameters on the system.

Type of Model

A continuous-time population dynamics model is used, similar to a different model describing cell-mediated immune response to tumor growth.

Biological System Studied

The HIV virus is an extensively studied virus due to its widespread impact on human populations. This study, in particular, investigates the effect of point-mutations in HIV viruses which allow them to continuously adapt to treatment.


Basic virus dynamics models have been essential in understanding quantitative issues of HIV replication. However, several parts of the viral life cycle remain elusive. The viral load (i.e. the abundance of free virus in plasma) has been shown to be an effective measure for the progression of human immunodeficiency virus type 1 (HIV-1). However, it is not enough to understand the virus as having one particular genome through which its reproduction can be prevented. Via point mutations or other complex evolutionary changes, the virus mutates to a state that is resistant to the forms of treatment that are available today: reverse transcriptase inhibitors, protease inhibitors, and newly developed entry inhibitors. Therefore, to contribute to the overall understanding of the progress of HIV-1, Nowak et al have developed analytic solutions for the emergence of resistant virus under single drug therapy.

The authors uniquely define their model to measure and identify the resistant mutant virus in different compartments of the virus populations:

  • free plasma virus
  • cells infected with actively replicating virus
  • cells that harbor latent virus
  • cells infected with defective virus

to better understand how the decline in the wildtype strain during treatment impacts the ability for the mutant to thrive. For an HIV model without drug-resistance, see MBW:HIV_Dynamics.

Biological Phenomenon

This study examines the effects of point mutations within the HIV-1 viral genome on the viral persistence and viral load of infected individuals. A point mutation is a single nucleotide base change in the genetic code of the virus (in this case RNA). Point mutations are divided into three main categories: insertion mutations, deletion mutations, and substitution mutations. Insertion and deletion mutations consist of the addition or subtraction of a single nucleotide and can alter the reading frame. Substitution mutations consist of the replacement of a single nucleotide with a different nucleotide. In general, mutations are either missense (protein is still created, with altered amino acid composition), nonsense (protein is prematurely terminated due to mutation to stop codon), or silent (change in base is masked by codon degeneracy. For this study, missense and nonsense mutations are the most important because the slight change in substitution rate that occurs with silent mutations is dwarfed by the impact of insertion and deletion mutations.

The dynamics of the emergence of drug resistant HIV-1 mutations rely on both the frequency of point mutations and the strength of the drug regimen that the patient is undergoing. At the time of publication, there were only two classes of HIV-1 inhibiting drugs. Some mathematical work has been done on the dynamics of these drugs, such as ritonavir. Protease inhibitors work by binding the HIV-1 protease and disrupting the proper protein cleavage necessary to form infectious virions. The inhibition is mediated by an irreversible binding event between the protease and the protease inhibitor. The second class of inhibitors that was available at the time was reverse transcriptase inhibitors. Since HIV-1 is a retrovirus it stores its genetic information in RNA rather than DNA. Once an HIV virion enters a cell and sheds its coat proteins, it must be transcribed back to DNA so that it can be incorporated into the cellular DNA. First generation reverse transcriptase inhibitors (like AZT are merely nucleotide analogues which are incorporated into the transcriped DNA strand, and act to terminate the growing DNA strand. For more information on the process of HIV treatment see When to Initiate HIV Therapy.


Untreated HIV disease is characterized by a gradual deterioration of immune function as CD4 cells are killed during the course of infection. HIV leads to AIDS in which immunosuppression leaves patients to suffer from infections of the lungs, intestinal tract, brain, eyes, and other organs, as well as debilitating weight loss, diarrhea, neurological conditions, cancer, and certain types of lymphomas.

HIV is a retrovirus, which means that it cannot replicate on its own. Instead, it replicates by infecting other cells. The virus particle is a small amount of genetic material surrounded by one or more protective shells. If it gains entry to a host (CD4) cell, it hijacks the cell's machinery for its own replication. A diagram of an HIV virion is shown below:


The spread of the virus occurs in phases, and requires that the basic reproductive ratio of the virus, R0, be greater than one. During the primary phase of the infection (before the immune system has had time to recognize and attack the virus), the number of uninfected target cells in reduced by the disease until each virion is expected to give rise to exactly one new virion. During the second phase, the killer T cells attack and kill infected cells. If the infection persists, then each virion continues to produce exactly one new virion, and the infection persists as long as the reproductive ratio of the virus is greater than one.

When a virion invades a CD4 cell, one of three types of infected cells results:

  • actively virus-producing cells
  • latent cells
  • defective cells

In the latent cells the virus is able to hide for a period of time, disallowing the immune system from clearing the virus before any drastic decrease in CD4 cells occurs. This delay is further investigated by this mathematical model.

There are two types of anti-viral treatments that are intended to stop the virus from reproducing:

  • reverse transcriptase inhibitors
  • protease inhibitors

When a patient is treated with wither type of drug, there is a rapid decrease in plasma virus and an increase in CD4 cells.

Treatment under a single ant-HIV drug almost always leads to an emergence of resistant virus.[1] With neverapine (NVP), high levels of resistance occure as a result of point mutations.[2] McLean & Nowak (1992) postulate that the increase in available target cells (as a result of the anti-HIV drug) allows the mutation to reproduce and take the place of the wildtype virus in terms of abundance.[3]

Drug-resistant strains evolve during therapy, or exist in the virus population before the onset of therapy. They are maintained by a mutation-selection balance and are selected to grow when the drugs are applied. If drug-resistant strains evolve during therapy, then the dosage of treatment should be increased so that the residual replication of the sensitive virus during treatment is minimized. On the other hand, if resistant strains exist before therapy, the effect of the drug on wild-type virus does not matter in the long run. A potent drug will lead to a fast decline of wild-type virus and a fast rise of resistant mutants. A weak drug will lead to a slow decline of wild-type virus and a slow rise of resistant mutants.[4]

Model Characteristics

The mode developed by Nowak et al for simulating HIV replication in the presence and absence of a single inhibiting therapy utilizes ordinary differential equations (ODE's) to interrogate the relationship between the various populations, rates, and cells present in the model system. They introduce three systems of equations; basic, basic including treatment, and basic including treatment and cellular diversity. The basic system can be solved analytically, and the two other systems are solved using approximations due to the complexity of the number systems and the system parameters. From a cellular perspective the important modeling parameters that are included are:

  • number of infected cells
  • number of uninfected cells
  • number of wildtype virions
  • number of mutant virions
  • number of latently infected cells
  • number of defective mutant virions

The basic model tracks the number of infected cells and unifected cells in the absence of any treatment and compares that to the basic model with treatment (where resistant mutations arise). The final model they investigate allows for emergence of resistance as well as the production of damaged or replication incompetent virus. The cell types allowed in the final model are uninfected, latent infected, active infected, and defective infected. The authors use these parameters to predict patterns of viral load, viral half life, resistance emergence and the limiting ratio of infected to uninfected cells. In addition to developing the mathematical modeling, the authors compare their simulations to actual data from patient groups.

Mathematical Model

The authors define three systems of ODEs that model the interaction inside the body between the virus and the immune system: the basic model, the basic model with treatment, and the basic model taking into account the possible types of infected cell types.

They first develop the basic model, which they describe as the simplest possible host cell dynamics. Three variables are defined:

  • x: uninfected cells
  • y: infected cells
  • v: virus particles

They also assume that uninfected cells are produced at a constant rate, λ, from a pool of precursor cells and die at rate dx. Virus reacts with uninfected cells to produce infected cells. This happens at rate βvx. Infected cells die at rate ay. Virus is produced from infected cells at rate ky and dies at rate uv. This gives rise to the following system of differential equations:


If the basic reproductive ratio of the virus,


is larger than one, then the system converges to the equilibrium


In order to understand the rapid development of resistant virus to all known drugs, an expansion to the basic model is created. Three additional variables are defined:

  • ym: cells infected by mutant virus
  • vm: mutant virus particles
  • ε: probability of mutation from wildtype to resistant mutant during reverse transcription of viral RNA into proviral DNA. (Approximately 10-3 – 10-5 if wildtype and mutant differ by a single point mutation.)

giving rise to the following mutation-selection process:


If we assume that the drug completely inhibits the replication of wildtype virus (β = 0), but does not affect the mutant. Then (8) becomes:


The basic model (1) is also expanded to include the large proportion of infected cells that do not produce new virus particles because they harbor latent or defective virus. Three new variables are defined:

  • y1 = infected cells that contain actively replicating virus
  • y2 = infected cells that contain latent virus
  • y3 = infected cells that contain defective virus

to obtain:


in which qi describes the probability that upon infection a cell will enter type i; ∑ qi = 1. Thus q1 is the probability that the cell will immediately enter active viral replication; y1 cells will produce virus at rate k. The parameter q2 is the probability that the cell will become latently infected with the virus and produce virus at a much slower rate c. The parameter q3 specifies the probability that infection of a cell produces a defective provirus that will not produce any offspring virus. The decay rates of actively producing cells, latently infected cells and defectively infected cells are a1, a2, a3, respectively.

Provided the basic reproductive ratio of the wildtype,


is larger than one, the system converges to the equilibrium




The following system depicts the full dynamics, and is an expansion of (10) via the addition of the different cell types’ dynamics to the basic model that accounts for virus mutations.


Simulation Results from Paper

In order to provide analytic approximations for the rate of emergence of resistant virus, the authors analyze the dynamics of the rise of resistant mutant virus populations, and their infected cells, following the sustained administration of drug after t = 0 (equation (36) with (β = 0) for t > 0).

Below is a computer simulation of the extended model (36) describing the dynamics of drug-sensitive wildtype virus and drug-resistant mutant virus in the free virus population, v, and infected cell populations harboring replication active virus, y1, latent virus, y2, and defective virus, y3. The declining wildtype virus is shown by a continuous line, whereas the broken line indicates the rising mutant virus. Panel (e) shows the relative frequency of mutant virus emerging in the actively infected cell population (broken line), followed by the free virus population (broken line long dashes) and finally followed by the defectively infected cell population (broken line with 2 short dashes 1 long dash). Panel (f) shows the dynamics of the uninfected cell population, x.



Experimental Results

In order to evaluate the effectiveness of the model, the authors compared model results with data on the rise of drug-resistant virus in three HIV-1 infected patients treated with neverapine (NVP) – a reverse transcriptase inhibitor. Plasma virus load, CD4 cell counts, and infected peripheral blood mononuclear cells (PBMC) were measured sequentially after the start of therapy. In all three compartments (plasma virus, infectious PBMC, and total infected PBMC) they quantify the proportions of NVP-sensitive wildtype virus and NVP-resistant mutant virus at day 0, 14, 28, 42 and 140 after initiating therapy.

In reference to the following figure for patient 1625,

  • Plasma virus denotes free virus particles in 1 μl plasma.
  • Infectious cells denotes PBMC harboring replication competent HIV (per 1 ml blood)
  • Provirus denotes number of PBMC with HIV proviral DNA (per 1 ml blood)
  • The continuous line is drug-sensitive wildtype virus
  • The broken line is drug-resistant mutant virus
  • Mutual frequency show the rise of NVP-resistant mutant in the free virus population (continuous line), in the infectious PBMC population (broken line short dashes), and in the total infected PBMC population (broken line long dashes)



The authors discovered that there are two phases for the rise in resistant mutant under drug therapy.

  • Phase I: Due to the equilibrium set by the mutation-selection balance, the resistant mutant initially falls, and the uninfected cell count is too low to maintain the mutant. As the uninfected cell population rises, the resistant mutant increases. This first phase ends when the resistant mutant becomes of order unity, which for small f is approximately at t = 2/(Rmd).
  • Phase II: The resistant mutant continues to grow exponentially on a 1/a timescale, causing the uninfected cell population to fall. The resistant mutant will peak when the uninfected cell population is 1/Rm. At this point, the uninfected cell population is still decreasing, so the resistant mutant will now fall (still on the timescale of 1/a at first), until the uninfected cell population ceases to decline. This occurs when the number of uninfected cells is 1/1+ ym. At this point, the resistant mutant will be enough below unity (and continuing to fall) that the uninfected cell population climbs again. Eventually the resistant mutant will climb back up to Rmx > 1, whereafter the uninfected cell population will rise and the cycle completes.

There are slowly damped oscillations eventually leading to the equilibrium x = 1/Rm and ym = Rm – 1.


As seen in the clinical experimental results, the NVP-resistant virus rises rapidly just as the model predicted. Additionally, this experiment allows us see that the virus rises in the free plasma virus population followed with a small delay in infected cells harboring replication competent virus and with considerable delay in infected cells harboring HIV provirus. Patient 625 is particularly interesting, because at day 28 plasma virus contains 100% resistant virus. (The other two patients contain 76% and 92% at this time.)

Subsequent Research

Since the publication of the Nowak paper, significant additional research has been conducted on mathematical modeling of HIV. In addition to new mathematical techniques, new information concerning the life cycle, replication parameters, replication kinetics, and immune system-virus interactions has become available. One of the most productive areas of research has been the expansion of the basic ODE approach utilized by Nowak to PDE's and delay differentials. Nelson and Perelson 2002 explore the effects of adding delays to the basic model employed by Nowak as well as others. Of particular interest is the additional information their model produces about drug efficacy. Utilizing delay differentials they are able to model the CD4 T-cell population in light of partially effective reverse transcriptase inhibition therapy. This work produces better agreement with patient viral loads and data and their relationship with the loss of CD4 T-cells due to viral infection.

The estimate of the \delta parameter (CD4 T-cell death) can drastically influence treatment strategies because it is closely linked with disease progression. Thus, research which further elucidates this relationship could have important ramifications in the medical as well as socio-political dynamics of treatment.

Ogunlaran et al, 2016 further refine the model by introducing control variables and an objective function that incorporates efficiency of drug treatments, costs of treatment, viral dynamics, and cell population model. The goal of their model is to maximize the objective function, which results in a minimum proportion of infected cells and minimum cost of drug treatment. Similar to the model presented here, Ogunlaran et al. use a system of continuous differential equations, but also incorporate two control variables which are bounded Lebesgue integrable functions simulating antiviral therapy. Based on their model, they used a Gauss-Siedel-like implicit finite difference method to simulate the populations of healthy T-cells, infected T-cells, and viruses and were able to develop functions describing optimal treatment levels over time for the different drugs.


The results that the authors present detail the exact challenge that scientists must confront when treating HIV patients. The mutation-selection balance between wildtype and mutant strains is an important force that has prevented the success of ant-viral treatment thus far. Ideally, the authors would like their kinetic picture of HIV infection incorporated into the dynamics of disease progression. Their results can help doctors anticipate mutant viral loads, thereby allowing them to create a longitudinal treatment plan.

  1. B. Larder, G. Darby, D Richman (1989). HIV with reduced sensitivity to zidovudine (AZT) isolated during prolonged therapy. Science 243, 1731-1734.
  2. D. Richman (1994). Drug resistance in viruses. Trends Microbiol. 2, 401-407.
  3. A. McLean, M. Nowak (1992). Competition between zidovudine sensitive and resistant strains of HIV. AIDS 6, 71-79.
  4. Bonhoeffer S, Nowak M A (1997) Proc R Soc London Ser B 264:631–637.