HIV and the Immune System - A Mathematical Model |
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by
Josef Lechner Amstetten, Lower Austria Abstract |
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The starting point of the model to be discussed is the increase of HI-viruses in the human body. This model gives an answer to the question why the space of time between HIVinfection and outbreak of AIDS differs to a great extent. Whereas some of the concerned die within months after infection, others ( but only a rather small number) are still alive without any troubles even 20 years later. After about 10 years half of all the infected suffer from AIDS. The following model - mainly developed by the Austrian mathematician M.Nowak - gives also insight into the phenomenon that our immune system is generally not able to root out HI-viruses completely as it is able to do with other types of viruses. A paradox result of this model is the evidence that HI-viruses are that „successfull„ because their replication is susceptible to mistakes. As the transcription of virus-RNA to DNA is regularly connected with random mistakes, the likelyhood for a completely correct transmission of the genetic information is only about 40 percent. Against every mutated
virus the immune system must create new specific killer cells, which can
only fight this special kind. The resistant cells act as specialists.
On the contrary all mutating viruses can destroy all kinds of resistant
cells against HIV or at least impair their function. The work as generalists. According to [LIPPA, 1997 bzw. REICHEL, 1999] the number vj(i) of the viruses of the kind i, j steps of time after infection can be described by means of the following (iterative) equation: |
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vj+1 ( i ) = vj ( i ) + vj ( i ) × ( R - P × aj ( i ) ) | |||||||||||||||||||||||||||||
In this context aj(i) , which affects the virus mutant i, is the number of immune cells per volume of blood. We call the rate of increase of the viruses R, supposing (in this simplified model) that R is equal ly big for all mutants (which is actually not the case!) The factor P describes the efficiency of the immune cells in their fight of resistance. It ....: 0 < P < 1, otherwise one immune cell would destroy more viruses than could exist at all. | |||||||||||||||||||||||||||||
For the developement of the resistant cells (T-helper cells, B-cells and antibodies respectively, killer-cells) an (iterative) equation of the following kind can be given. | |||||||||||||||||||||||||||||
aj+1 ( i ) = aj ( i ) + K × vj ( i ) - U × vj × aj ( i ) | |||||||||||||||||||||||||||||
Here the factor of proportionality K describes the increase of the kind i of the resistant cells, which is generated by the mutant i of the viruses. The factor U characterises the aggressiveness of the viruses. By analogy it is: 0<U<1. The term vj describes the total number of all viruses after j steps of time. |
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The following aspects show us why the simulation of the mathematical model is not easy: | |||||||||||||||||||||||||||||
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Tools which are available for math lessons have their specific strengths and weaknesses. No matter which tool is used, by using the medium adequately a simulation of this extensive model is practicable. | |||||||||||||||||||||||||||||
Realisation of HIV-simulation by means of TI-92 | |||||||||||||||||||||||||||||
Simulation 1 | |||||||||||||||||||||||||||||
Let‘s start with a trivial model with only one type of virus. In this case it‘s easy to perform a simulation with the CA-TR TI-92 and the TI-89 respectively. According to [LIPPA, 1997, NOWAK, 1992] the following parameters are relatively realistic: | |||||||||||||||||||||||||||||
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The sequence-mode
of TI-92 allows the direct input of the respective equations of the model.
The names of the equations are determined by u1,u2,u3, ... in this calculating
system. In our case u1(n) is the number of viruses of type 1 after n
steps of time, u2(n) the number of resistant cells |
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Now the development of the viruses and the resistant cells respectively can be observed and evaluated by means of tables or graphs with all the available tools of the calculator (e.g. TRACE). | |||||||||||||||||||||||||||||
Table |
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Graphic presentation of the development in time | |||||||||||||||||||||||||||||
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Graphic presentation of resistant cells against viruses | |||||||||||||||||||||||||||||
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Comment: One must switch Sequence-Mode / F7 to Custom. Choose u1 and u2for axes. | |||||||||||||||||||||||||||||
Simulation 2 | |||||||||||||||||||||||||||||
Now we want to work with two mutants, the second of which shall appear after 60 steps of time (which means after about 3.6 months). This can be reached by a simple branching out using the when-control structure. The respective inputs in the sequence-mode of the TI now read: | |||||||||||||||||||||||||||||
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Total number of viruses and resistant cells | |||||||||||||||||||||||||||||
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By ticking the respective (iterative) equations and choosing an appropriate graphing style (F6) the develpment of the individual types of viruses can be observed. | |||||||||||||||||||||||||||||
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As it can be seen above, the input of the simulation equations is already extensive with only two mutants. Therefore it seems to be advisable to use a suitable program for simulations with more mutants. | |||||||||||||||||||||||||||||
Simulation 3 | |||||||||||||||||||||||||||||
Now the number of mutants shall be increased to 11. The time for the appearance of the mutations shall be determined at random. This may lead to the following result. Here the breakout of AIDS happens after four years. | |||||||||||||||||||||||||||||
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How was this simulation
performed? The program allows the input of model parameters in the first step. |
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The item „Appearance of mutants„ allows the use to control the time of appearance of the individual mutants on purpose or to select it quickly or slowly by accident. In the case of ,quickly‘ all mutants appear within 500 steps of time (2.5 years), in the case of slowly within 4000 steps of time (20 years). These values can be easily changed. After this first step
the calculator is busy for some time - the respective equations are generated.
Finally |
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In a second step the appearace of mutants is controlled. If one has decided to choose one of the two random methods, one can see in the data-matrix-editor, when new mutants will appear. | |||||||||||||||||||||||||||||
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The values in column c2 describe, after how many steps of time the individual mutants appear. They can be changed at any time. If the user have chosen not a random method, the values must be supplied. |
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Two final remarks: | |||||||||||||||||||||||||||||
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Program
HIV: Source-text hiv() Prgm Ó(c) Josef Lechner, Mai 1999 Ólejos@nol.at Ó Sequence-Bereich löschen RclGDB leer Ó Lokale Variablen
Ó Definitionen "1"®mz:2®modus "0.1"®r:"0.002"®p "0.02"®k:"0.00004"®u Define vhsum(mz)=Func
While i£mz If i=1 Then vhsstr&"u1(n-1)"®vhsstr Else vhsstr&"+u"&string(2*i-1)&"(n-1)"®vhsstr EndIf i+1®i EndWhile vhsstr EndFunc Define ahsum(mz)=Func
While i£mz If i=1 Then ahsstr&"u2(n-1)"®ahsstr Else ahsstr&"+u"&string(2*i)&"(n-1)"®ahsstr EndIf i+1®i EndWhile ahsstr EndFunc "Define u"&string(i)&"(n)=when(n=(mdata[2])["& string((i+1)/2)&"],1,u"&string(i)&"(n-1)+u"& string(i)&"(n-1)*(r-p*u"&string(i+1)& "(n-1)))"®vstr(i) "Define u"&string(i+1)&"(n)=u"&string(i+1)& "(n-1)+k*u"&string(i)&"(n-1)-u*("&vhsum(mz)&")*u"&string(i+1)&"(n-1)"®astr (i,mz) "Define ui"&string(i)&"=0"®vistr(i) "Define ui"&string(i+1)&"=0"®aistr(i) "Define u"&string(2*mz+1)&"(n)="&vhsum(mz)®vsum(mz) "Define u"&string(2*mz+2)&"(n)="&ahsum(mz)®asum(mz) "Define ui"&string(2*mz+1)&"=0"®visum(mz) "Define ui"&string(2*mz+2)&"=0"®aisum(mz) Ó
Modell spezifizieren Ó
Iterationsgleichungen schreiben Ó
Auftreten von Mutanden steuern ![]() |
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Realisation of the HIV-Simulation by means of DERIVE | |||||||||||||||||||||||||||||
If you want to use the CAS DERIVE for a simulation, you have choose an other way. As every simulation of a dynamic system is an iterative process, you have to describe how to get from one condition to the next. Using DERIVE ITERATES is the proper command. Its efficiency is the basis for the following implementations. | |||||||||||||||||||||||||||||
Simulation 1 | |||||||||||||||||||||||||||||
Again we will start with the trivial model with only one type of virus. The following parameters are shall be remembered: R=0,1 / P=0,002 / K=0,02 / U=0,00004 | |||||||||||||||||||||||||||||
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After choosing the
model parameters the iterative equations are defined. NV(v,a) describes
the number of viruses after every step of iteration, VA(v,a) the number
of resistant cells after every step of iteration, |
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With the help of ITERATES the different values are calculated. The first list shows the two iteration equations, the second list the iteration variables, the third the initial values and the last parameter indicates how many iteration steps are wanted. Afterwards the calculation is started using the APPROX-command. | |||||||||||||||||||||||||||||
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If you also want to visualize the steps of time in the table - which is important for a later graph -, you can calculate the values in the time-column iteratively simply by adding a further item in the corresponding list. | |||||||||||||||||||||||||||||
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To be able to show the time-development of the system, you must create two tables: one table time-viruses, the other time-resistant cells. The output is restricted (Version 4.11) to the desired columns for plotting by using the command SUB SUB [column1, column2]. | |||||||||||||||||||||||||||||
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The result of the simulation can be reviewed in the graphic window |
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If you only plot the two original colums, you get the phase diagram of the system: | |||||||||||||||||||||||||||||
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Simulation 2 | |||||||||||||||||||||||||||||
If you want to work with more mutants, the shown method is only partly suitalbe, because you need at least the double number of iterative equations than there are mutants.So that the number of mutants does not play a role on the one hand and to create a clearly structured simulation it is recommended to express the current condition of the system in the form of a list (a vector) z. It shall contain the step of time, for each mutant and its type of resistant cell the respective equations and finally the total numbers of viruses and resistant cells. Additionally a second list (a second vector) mdata shall indicate the steps of time of the appearing mutants. This vector can be - if you want to - generated at random. | |||||||||||||||||||||||||||||
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The current system parameters are chosen and defined first. You also have to provide the number of mutants. The times of appearance are combined in the list mdata. | |||||||||||||||||||||||||||||
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The list z0 contains the starting situation (in this case a list only with zeros as items). | |||||||||||||||||||||||||||||
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The following two definitions calculate the total number of viruses and resistant cells: | |||||||||||||||||||||||||||||
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The following three
definitions form the core of the simulation. NV(z,i) calculates the number
of viruses of type i using the current condition z of the system. NA(z,i)
calculates the number of resistant cells against the virus mutant i also
using the current condition z. |
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Again the calculation is carried out by means of the ITERATES-command. This can be done in a surprisingly simple manner. The first function NZ(z) calculates the next using the current condition of the system, z contains the respective current condition, z0 the initial condition and the last parameter indicates the number of iterations. | |||||||||||||||||||||||||||||
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provides the following table: | |||||||||||||||||||||||||||||
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Finally a graph of any mutant and any type of resistant cell can be drawn be selecting only two columns. |
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Example 1: 3 Mutants | |||||||||||||||||||||||||||||
Choosing model parameters | |||||||||||||||||||||||||||||
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Calculating the model | |||||||||||||||||||||||||||||
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The development of the individual mutants and the immune cells respectively can be observed pretty well: | |||||||||||||||||||||||||||||
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Example 2 - 11 Mutants | |||||||||||||||||||||||||||||
Choosing model parameters: | |||||||||||||||||||||||||||||
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The same values for the time of appearance of the mutants as for the simulation with the help of the TI-92 have been chosen. This list should be compiled at random, which can be achieved in the following way: |
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The modul mutants creates a list wíth the starting times, when you provide the number of mutants and the time span within which they shall appear. Every evaluation leads to a new random list. | |||||||||||||||||||||||||||||
The two following definitions may help you to find the correct columns where you can find the total numbers of viruses and resistant cells respectively. The evaluation of the three ITERATES-lines allows to get graphs. | |||||||||||||||||||||||||||||
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Literature
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