Using Mathematical Modelling For Autoimmunity Biology Essay

Mathematicss is used a batch in the universe that mean common people are blissfully incognizant of. Everyone goes through the instruction system larning mathematics in the early old ages as numeration from 1 through to 100, and so subsequently on, in their teens, as trigonometry and random equations. From the instruction we gain as a child, it ‘s difficult to conceive of mathematics to be utile in many countries of employment. This is untrue. Mathematics is everyplace, working behind the scenes. One of these unidentified countries is Medicine.

Mathematic is used extensively to assist derive progresss in medical research ; from the Pharmacology of new drug mixtures to the apprehension of the mechanics of complex diseases. Mathematicians are working with medical research workers every twenty-four hours to understand how different triggers affect the human organic structure, making some signifier of mathematical equation or system to demo this, and so to make a mathematical system to foretell how inauspicious effects can be counteracted.

Immunology is a complex country of Medicine and Biology. Rather than the molecular and cellular survey, it deals with complex non-linear biological systems. Although there are an increasing figure of efforts to utilize and develop computational package, these frequently do n’t see the nature of the interactions between the assorted cellular and molecular constituents nor do they normally give much biological penetration into how the system works. Mathematical modeling is used to foretell tumour growing and malignant neoplastic disease spread, where as another subdivision of mathematics, statistics are used to construe informations collected from clinical tests.

As the rubric suggests, this paper will be concentrating on the mathematical modeling used to help research into Autoimmune Diseases. First, there will be a reasonably brief overview of what autoimmunity is, followed by the basic modeling. The modeling will get down with the basic additive theoretical accounts, where a brief penetration will be given to their belongingss. Towards the terminal of the paper, non-linear theoretical accounts will be presented with a much deeper penetration to the workings of the theoretical account.

Biological Background

So, what is an Autoimmune Disease? An Autoimmune disease is when the organic structure ‘s immune system becomes hyperactive or ‘confused ‘ and starts to bring forth a response against the individual ‘s healthy cells. In kernel, assailing and damaging them. There are two different responses the immune system can take ; it can bring forth antigens to do direct harm to a individual organ or tissue doing a ‘localised ‘ autoimmune disease, or produces a response against multiple tissues and variety meats ensuing in a ‘systemic ‘ autoimmune disease.

Basic Biological Footings

Here is the definition of the basic biological footings used throughout this paper.

Target Cell – the organic structure ‘s healthy cells

Damaged Cell – a cell that has been damaged, i.e. by a virus or the organic structure ‘s immune response.

Antigen – a molecule which when released into the organic structure activates the immune response, bring forthing antibodies to assail what it recognises as a possible harmful encroacher.

Antibody – used by the immune system to recognize and destruct possible encroachers.

Key Effector Cell – A cell that performs the organic structure ‘s reaction to a stimulation

APC’s – The cell that process the antigens to let T cells to recognize them

T Cells – A set of cells belonging to a group of immune cells ( white blood cells ) called lymph cells

“ in vivo ” – where experimentation has been done in unrecorded stray cells as opposed to a whole being.

Barbarous Cycle of Autoimmunity

Before go oning, the rhythm of autoimmunity demands to be explained, in brief.

( 6 )

( 5 ) Damaged cell


Target cell

Key effecter cell




Lymph Vessel

Armored personnel carrier

( 7 )

( 1 )

Fig 1. A Cycle of Autoimmunity

Dynamic Properties of autoimmune disease theoretical accounts: Tolerance, flare-up, quiescence

( Iwami et al. , 2007 )

( 4 )

( 8 )

( 3 )

( 2 )

When autoimmune disease is initiated by some event, an bing cell becomes a cardinal effecter cell ( 1 ) . Then, this cardinal effecter cell onslaughts and amendss a healthy cell ( 2 ) . Here, the protein of the damaged cell ( antigen ) ( 3 ) is captured by an APC ( 4 ) and the protein is shown as a self-antigen at the lymph vas ( 5 ) . Then the immune cells which are specific to the protein are induced ( 6 ) , and these specific immune cells once more attack and damage mark cells, ( 7 ) get downing the rhythm once more ( 8 ) .


Autoimmune diseases by and large do n’t merely hold remarkable symptoms, or needfully a specific trial for diagnosing, therefore it can be difficult for medical practioners to supply a diagnosing. Besides many symptoms can overlap from one disease to another, as the organic structure reacts in similar ways ; it can be difficult for them to set up which disease the patient is enduring from.

There can be many possible triggers to trip the disease activity, and besides to find the badness. Many common life constituents ; like the conditions, emphasis etc. ; can do a wholly non-linear reaction to a sick persons immune response.

Therefore, because of the many varying symptoms and triggers, there ranges a big figure of different drug and physical therapies. Through medical literature there exists plentifulness of documents on the topic of autoimmune disease and it ‘s complications, yet there is limited theoretical work in mathematical literature. However, over the old ages, mathematical theoretical accounts have been proposed in relation to T cell inoculation ( Iwami et al. , 2008 ) , the behavior of the immune web and the activation of immune cell activation, for illustration. For the latter, there is a paper ( Wodarz and Jansen, 2003 ) look intoing the ratio of cross-presentation to direct-presentation of antigen showing cells presuming their figure is variable. However these theoretical accounts are reasonably specific to certain autoimmune tendencies.

Current apprehension of Autoimmunity is that tissue hurt causes T cell reactions and the activation of some immune cells. Consequently, this leads us to presume that patient ‘s autoimmune symptoms are based on the population size of healthy ( mark ) cells. Hence, the smaller the population size, the more terrible the symptoms. In the undermentioned theoretical account we will presume that the figure of antigen showing cells ( henceforth known as APC ‘s ) is changeless. More specifically, the Dendritic Cells ( DC ‘s ) , a type of APC, are known to transport out about all antigen presentations to T cells and the figure of these are said to be changeless in vivo.

Throughout the balance of this paper, we will look into a simple theoretical account for general autoimmune disease.

Basic theoretical account

In the undermentioned theoretical account we will be demoing the effects of ;

The immune system cells, C, which of course die at the rate I?

The healthy ( mark ) cells, T, which of course die at the rate I? and are

produced at a rate of I»

The damaged cells, D, which of course die at the rate I± ;

where I± E? I?

The immune system cells, C, damage the mark cells, T, at a rate proportional to their population, I?TC. Here, I? represents the efficaciousness of the procedure.

From the above background, a basic dynamical theoretical account is obtained, which is a system of non-linear differential equations. In order to simplify the theoretical account we assume that damaged cells already exist and disregard the kineticss of the cardinal effecter cells and APC ‘s ; these could hold been caused by an earlier viral or bacterial infection, or tissue hurt. We combine the kineticss of immune cells and mark cells and obtain a theoretical account as follows:

Target cell growing – force of harm by immune cells

Force of harm by immune cells – natural rate of decease of damaged cells

Personal immune response – natural rate of decease of immune cells

The immune system is a batch more complicated than our simple mathematical theoretical account above based on the mark cell growing map and the personal immune response map, but this will be equal for the minute. includes the rates that immune cells find and win in assailing mark cells. The effects of mark cell growing and the personal immune response will be investigated.

Target Cell Growth Function

The above mathematical theoretical account is based on a common theoretical account used to understand HIV infection. Reasonable maps, to stand for mark cell growing in worlds have been investigated. ( Martin A. Nowark and Robert M. May, 2000 ; Martin A. Nowark et al. , 1996 ; Alan S. Perelson and Patrick W. Nelson, 1998 )

The first is a simple equation which is merely, the rate at which the new mark cells are produced, minus ( rate of decease of mark cells ten population size of mark cells ) which was investigated by Martin A. Nowak et Al. ( 2000 ; 1996 )

The 2nd equation is somewhat more complex, ( Alan S. Perelson and Patrick W. Nelson, 1998 ) , with the add-on of an excess logistical term to take into history natural mark cell proliferation. Here, represents the maximal proliferation rate and L ; the mark cell population denseness at which proliferation shuts off.

It is noted that the above equation is density dependant. However, an alternate equation has been investigated by Liancheng Wang and Michael Y. Li ( 2006 ) , in order to include denseness dependance, and is every bit follows:

However, for mathematical simpleness and without altering the qualitative behavior of the system, we will presume that the first equation of is the functional signifier including denseness dependance.

Personal Immune Response Function

If the personal immune response map is defined as, the relationship between the immune cell incentive and the symptoms of autoimmune disease can be investigated. Immune response maps can change from individual to individual or they may depend on a patient ‘s status or the sort of immune cells. Immune cell incentive is considered to be a sensible map shown as:

Fig 2. Personal Immune Response

Dynamic Properties of autoimmune disease theoretical accounts: Tolerance, flare-up, quiescence,

Iwami et Al. ( 2007 )

The above diagram shows APCs do non bring forth immune cells if merely a few antigens exist, but so immune cells are bit by bit induced when comparatively many antigens exist.

Henceforth, the paper will look into the behavior of two appraisals of the personal immune response maps ;


And Non-Linear:


Linear Personal Response Function

In this subdivision, the additive personal response map and it ‘s representation of autoimmune disease symptoms will be investigated, i.e. , where represents the mean magnitude of activation of Immune Response by APCs per damaged cell.

Here, the figure of immune cells induced by APC ‘s is relative to the figure of damaged cells, so D can be considered as the proliferation rate of immune cells by APCs.

Linear Target Cell Growth Function

To get down, we select the simplest equation for the population kineticss of mark cells ; . This gives the undermentioned simple theoretical account of autoimmune disease kineticss:


The above theoretical account is the same as a basic HIV theoretical account studied by several research workers ( Nowak and May, 2000 ; Leenheer and Smith, 2003 ; and others ) , but merely the deductions for autoimmune disease will be explored in this paper. Numeric solutions were found for this system of differential equations with parametric quantities ( Iwami et. Al, 2007 )

Now, if is little so the damaged cells, D, vanish and the immune cells, C, are non activated therefore the mark cells, T, do non consume. However, if is big so the figure of damaged cells, D, increases triping the immune cells, C, and consuming the figure of mark cells, T. Hence, the larger is, the more likely a patient is to develop an autoimmune disease.

This system has two equilibria:

The tolerance equilibrium, , where,

The chronic infection equilibrium,

Next, the basic generative figure is defined as, which shows how many freshly damaged cells produced from one damaged cell in an person who does non hold an autoimmune disease presently. It has been shown that is globally asymptotically stable, if, by De Leenheer and Smith ( 2003 ) . If, under certain conditions is globally asymptotically stable.

The statement of continues here. The chronic infection equilibrium co-ordinates, from direct computation, are given by:

Taking the derived function of and, we get ;

As additions there is a transferal from tolerance to autoimmune disease activity from the consequence of. From ague symptoms the disease transportations to a chronic stage. Here, increasing deteriorates the symptoms shown by the consequences of, and. Namely, increasing activates the immune cells, and reduces the figure of mark cells. Hence, , which represents the mean magnitude of activation of the immune response, affects the badness of the autoimmune disease onslaught and it ‘s symptoms. It is a individual ‘s reactivity to a trigger or intervention.

Density-Dependant Target Cell growing Function

In the undermentioned subdivision the density-dependant map is used to obtain a similar theoretical account, which is besides a HIV theoretical account that has been antecedently researched ( Wang and Li, 2006 ; Nevo et al. , 2003 ; and others ) , one time more merely the autoimmune disease deductions are considered.

Again, ( Iwami et al. , 2007 ) , numerical solutions of this system where found with parametric quantities. This leads the theoretical account to look as ;

Iwami et Al ( 2007 ) demonstrated the following values of ;

As, mark cells reproduce themselves. The different values of determine the province of the disease ;

gives a tolerance of the immune response ensuing in no autoimmune disease,

gives a slow patterned advance of the disease ensuing in mild symptoms for the patient: The mark cells lessening bit by bit, nevertheless, in the chronic stage, there is still a comparatively high degree of mark cells.

gives repeated outbursts of an autoimmune disease.

gives a more terrible reaction than ( two ) . There is a speedy patterned advance of the disease ensuing in the rapid lessening of mark cells, and later, a low degree of mark cells in the chronic stage.

These consequences show that the values of in ( I ) and ( two ) are similar to the old consequences with the additive mark cell growing map ; hence, the higher the value of the more terrible the disease symptoms for the patient. The consequence for ( three ) behaves interestingly. It shows a periodic form for the disease symptoms, which relate to repeated outbursts of the autoimmune diseases.

The ground for this flare-up form in ( three ) can be explained as the followers ; if is comparatively little and the figure of mark cells is little, so the figure of mark cells increases because of the nature of ( i.e. the logistical term: ) . Then there is an associated addition in damaged cells and immune cells. The immune cells attack mark cells can do a lessening in the figure of mark cells and the rhythm repetitions.

In ( four ) , mark cells can non be increased any more due to the logistic nature of, if is big. Therefore, the consequence of density-dependant mark cell growing dramatically changes the symptoms of autoimmune disease.

Mathematically, this system is more interesting than the old one, as a positive equilibrium point could be unstable. This system has been investigated by De Leenheer and Smith ( 2003 ) and besides by Iwami et Al. ( 2007 ) . It has two equilibria:

The tolerance equilibrium, ,


The chronic infection equilibrium,

The basic generative figure is defined as

De Leenheer and Smith ( 2003 ) show that if, so is globally asymptotically stable and if so is merely globally asymptotically stable under certain conditions but can besides be unstable under certain conditions.

Non-Linear Personal Response

In the subsequent subdivision, the non-linear personal immune response map, , and it ‘s relationship with the symptoms of autoimmune disease is to be investigated. In ecological population kineticss, this map is called the “ functional response ” , or more accurately a Holling type III or sigmoid functional response.

Here, the parametric quantities of the Functional Response are defined ;

= maximal proliferation rate of immune cells caused by Armored personnel carriers

= figure of damaged cells at which the proliferation of immune cells is half of the upper limit

Therefore, the map can be regarded as the proliferation rate of immune cells by APCs.

This non-linear response map intensely changes the construction of the equation systems. Namely, both the tolerance and chronic infection equilibriums can be at the same time stable under certain conditions. Specifically, is ever stable.

Linear Target Cell Growth Function

As with the additive personal response map, we start with the additive mark cell growing map. Using this, the theoretical account alterations to the followers ;

Now, with ( i.e. the mark cell ‘s natural rate of decease is higher than the immune cell ‘s natural rate of decease ) , the deductions for autoimmune disease suggested by the theoretical account in this subdivision are considered. Numeric solutions are investigated with parametric quantities: , ( Iwami et. Al, 2007 )

Substituting the parametric quantity values into the templet of the theoretical account we get ;

Then utilizing the initial conditions:

gives the undermentioned consequences ;

gives a representation of quiescence of an autoimmune disease. In the beginning, the mark cells ( T ) remain at a reasonably steady big degree, while the immune cells ( C ) remain steady at a low degree. At some point, without some evolutionary event ( i.e. without altering parametric quantities ) , the figure of mark cells all of a sudden falls to a reasonably little degree while the sum immune cells increases. This consequence suggests that for an autoimmune disease to go evident, evolutionary events are non indispensable.

gives a representation of tolerance within the immune system, though this is different to the old consequences as the parametric quantities stay the same, merely altering the initial values. Equally long as the initial value of damaged cells is little, this system will ever demo a tolerance.

Here is given a stableness analysis for this system. This theoretical account has three equilibria:

, where

Hereafter, a elaborate account of the stableness of these equilibria is given. To happen the stableness of these equilibria, the eigen-values of a Jacobian matrix for each of these equilibria co-ordinates are investigated.

The expression for the Jacobian matrix for this differential equation system is ;

Hence, the Jacobian matrix of the first equilibrium point is as follows ;

Consequently, the eigen-values of this matrix are and. And so, is ever stable.

Traveling onto the Jacobian matrix for, the co-ordinates of which are found to be ;

The characteristic equation of this matrix is found as ;

In order to do this equation simpler, the coefficients for the indeterminate of the multinomial, s, are denoted by,

All eigen-values have negative existent parts if ;

is uncluttering true, so the concluding two conditions need to be proven.

By utilizing the co-ordinates for, is the same as


Then if and merely if

which implies that,

which is interpreted as for. However, if it exists, is ever unstable.

Finally, spread outing the plants of Iwami et Al. ( 2007 ) the last status is investigated:

So, if, so and is ever stable when it exists.

Henceforth is the statement for. Using the co-ordinates, we get:

Therefore, this is similar to the old systems with respects to the displacement of the badness of symptoms, where m is big. On the other manus, the fact this system is bi-stable gives the possibility for a displacement to disease tolerance with no affair how big is. If the figure of damaged cells is little, this system will ever demo tolerance.

Density-Dependant Target Cell Growth Function

In this subdivision, the 2nd representation of g from Section 3 is chosen, i.e.

altering the basic theoretical account to the signifier:

From the paper by Iwami et Al. ( 2007 ) , numerical solutions were found with the parametric quantities Hence, the theoretical account becomes ;

To transport out some simulations of this theoretical account, different sets of initial values are used, for different values of:

MathworksA© Matlab is used in order to derive elaborate consequences.

[ Matlab Codec in Appendix A ]

( I )

Time = T

Target Cells, T,

Damaged Cells, D,

Immune Cells, C,









































( two )


Target Cells, T,

Damaged Cells, D,

Immune Cells, C,









































( I )


Target Cells, T,

Damaged Cells, D,

Immune Cells, C,









































( two )


Target Cells, T,

Damaged Cells, D,

Immune Cells, C,









































Using the above tabular arraies, and diagrams, it is easy to see the forms caused by the discrepancies in initial values and in add-on the fluctuation of. A. ( I ) shows a quiescence stage of the disease followed by repeated ‘flare-ups ‘ . The degree of immune cells starts of at a systematically low rate during the quiescence stage, but as this ends, all of a sudden there is a important bead in the steady high rate of mark cells, with a so big addition of the figure of immune cells. However, this form of autoimmunity does non so switch the symptoms to chronic stage, but alternatively, shows the perennial outbursts.

A. ( two ) shows a tolerance of autoimmune disease. It can be explained as if merely a few antigens exist the APC ‘s do non bring on the activation of immune cells, and no autoimmune disease develops.

B. ( I ) shows a quiescence stage to get down but, where the figure of immune cells, with utmost awkwardness, additions. When this coatings, the mark cells all of a sudden lessening and the immune cells all of a sudden increase. As the figure of antigens additions, and the mark cells lessening, the APC ‘s induce the activation of the immune cells, doing the displacement from tolerance to disease activity, where this theoretical account shows the symptoms settling into a chronic province. B. ( two ) once more shows a consistent tolerance of autoimmune disease.

The above figures show two things ;

is connected to the mechanism of perennial outbursts of autoimmune disease.

is connected to the mechanism of quiescence and onslaught of autoimmune disease.

Now is given a stableness analysis for this theoretical account. From reexamining the theoretical account, it is evident that a boundary equilibrium exists, i.e.

Next, the eigen-values of the Jacobian matrix for are investigated. Hence the Jacobian matrix is as follows ;

The eigen-values of the above matrix are. Inserting the equation for, and simplifying, . Hence, all eigen-values are negative and so is ever stable.

The being of interior equilibria still needs to be shown. Using the equations for gives ;

Now replacing the equation for D into C ‘ gives ;

Furthermore, utilizing the equation for, a representation for C is obtained, as follows:

Substituting this equation for C back into the old equation, gives ;

The positive roots of this equation are the T component.

The undermentioned graphs from “ Dynamical belongingss of autoimmune disease theoretical accounts, by Iwami et Al ( 2007 ) ” give a ocular representation of

and, and


Fig. 4

Dynamic belongingss of autoimmune disease theoretical accounts

The roots of are shown in fig 4. ( two ) by the intersection of the line and the curve. Here, it is shown that two interior equilbiria exist denoted as and on the diagram.

The larger root ( as shown in the diagram ) corresponds to the equation:


Consequently, an interior equilibria, , exists near to.

Following, leting to be the maximal value that satisfies the derivative, there exists two equilibria where.

Using the parametric quantities given at the start of subdivision 5.2, the boundary equilibrium is ever stable for all four instances. Hence, there is a tolerance to autoimmune disease if merely a few antigens exist. However, in the instances for A. , the interior equilibrium and are both unstable. Alternatively, is stable for the instances of B.. The stableness of represents whether the autoimmune disease shows repeated outbursts, or chronic symptoms. From detecting the forms of A. ( I ) and B. ( I ) , the important difference the parametric quantity makes is evident. Repeated outbursts and chronic symptoms are reliant on the density-dependant growing of mark cells. The consequence of the density-dependant growing is strong when is comparatively little ensuing in the perennial outbursts of disease symptoms, which remains unchanged6 when the parametric quantity is changed within a proper biological scope. Otherwise when is comparatively big, the consequence of density-dependant growing is weak, which therefore consequences in the autoimmune disease showing chronic symptoms.


Throughout this paper, there have been multiple distinctions of a basic theoretical account for autoimmune disease. Get downing off with the additive readings and stoping with non-linear maps.

The personal immune response map has been shown to promote tolerance or quiescence. For the additive map, the size of the variable determines whether there is tolerance of autoimmune disease, or whether it will be active. There is ever tolerance if is little, otherwise Autoimmune disease becomes active for big. Hence, with a additive personal immune response, merely a few antigens cause APC ‘s to show and convey about the activation of immune cells.

The non-linear map determines quiescence and tolerance of auto-immune disease. It is dependent on the initial population of antigens ( damaged cells, D ) . If the degree of damaged cells is little in the beginning, so there is a tolerance of autoimmune disease, in dependant of the size of. On the other manus, if the initial status of damaged cells is non little, so it induces a quiescence of autoimmune disease, which finally consequences in autoimmune disease activity. Hence, with a non-linear personal immune response, APC ‘s do non show for few antigens.

It turned out that the personal immune response map is closely related to autoimmune disease. Every patient is different, and has a different immune response, and therefore shows different symptoms. Therefore, in order to derive an accurate theoretical account for a peculiar patient for therapy of autoimmune disease, that individual ‘s immune response would necessitate to be investigated.

The mark cell growing map has been shown to do perennial outbursts of autoimmune disease. particularly shows repeated outbursts of the disease. If, so presents no perennial outbursts, merely chronic symptoms.

If the degree of mark cells is little, the generation of the logistic term of, [ ] , can increase their figure. Additionally, an addition of damaged and immune cells can be related to an addition in mark cells. This so has the consequence of the immune cells ‘attacking ‘ the mark cells, which decreases their figure. Hence, the form of repeated outbursts.

Using the map, maintaining little so the mark cell growing map can increase the figure of mark cells and therefore repeated outbursts are observed. However if is big, is unable to increase the population of mark cells, and no outburst of symptoms occur.

With the map, alterations to hold no consequence on whether flare-ups nowadays or non. However, if the parametric quantity is set to a little value, the density-dependant has a strong consequence, and therefore outbursts of symptoms occur.


The differences in the treatment suggest that it is the mark cell growing map that determine if flare-ups occur. Harmonizing to Iwami et Al. ( 2007 ) , it is thought that “ the difference of mark cell growing maps corresponds to the differences between internal variety meats which initiate autoimmune disease. ” . So, similar to the personal immune response map, each single internal organ ‘s mark cell growing needs to be investigated in order to derive an accurate theoretical account for therapy of autoimmune disease.

The Personal Immune Response, is merely that: personal. One specific map will non stand for the population. However, empirical surveies help to supply typical responses in order to give an thought of an immune response. Data is collected from a battalion of campaigners by medical professionals and passed to mathematicians. The mathematicians so construe it and make feasible theoretical accounts in order to stand for the phenomena of Autoimmune disease. ( Jo Stevenson, 2010 )

These theoretical accounts can besides be used to assist understand the consequence of drug therapy used for autoimmune disease patients. For illustration, immunosuppressor ‘s. These drugs, as suggested in their name, work in order to cut down the immune response from a patient. In the theoretical account investigated by this paper, these drugs consequence the personal response maps, , by cut downing the parametric quantities.

As already suggested, for tolerance with the immune response, needs to be a really little value. Hence, unless the drugs can win in this, there would be small consequence from them. However, for, drug therapy may be more effectual regardless of the size of.


Appendix A


map Dy = nlddtcgfA ( T, Y )

Dy = nothing ( 3,1 ) ; % a column vector

Dy ( 1 ) = 0.1-0.1*y ( 1 ) +3*y ( 1 ) * ( 1-y ( 1 ) /100 ) -0.5*y ( 1 ) *y ( 3 ) ;

Dy ( 2 ) = 0.5*y ( 1 ) *y ( 3 ) -1.1*y ( 2 ) ;

Dy ( 3 ) = ( ( 10*y ( 2 ) ^2 ) / ( ( 60^2 ) +y ( 2 ) ^2 ) ) -0.1*y ( 3 ) ;



map Dy = nlddtcgfB ( T, Y )

Dy = nothing ( 3,1 ) ; % a column vector

Dy ( 1 ) = 5-0.1*y ( 1 ) +3*y ( 1 ) * ( 1-y ( 1 ) /100 ) -0.5*y ( 1 ) *y ( 3 ) ;

Dy ( 2 ) = 0.5*y ( 1 ) *y ( 3 ) -1.1*y ( 2 ) ;

Dy ( 3 ) = ( ( 10*y ( 2 ) ^2 ) / ( ( 60^2 ) +y ( 2 ) ^2 ) ) -0.1*y ( 3 ) ;


Matlab Codec & gt ; Command window

A. ( I )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfA, [ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 ] , [ 100 4 0 ] )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfA, [ 0 100 ] , [ 100 4 0 ] ) ;

& gt ; & gt ; secret plan ( T, Y ( : ,1 ) , ‘- ‘ , T, Y ( : ,2 ) , ‘- ‘ , T, Y ( : ,3 ) , ‘- ‘ )

A. ( two )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfA, [ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 ] , [ 100 3 0 ] )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfA, [ 0 100 ] , [ 100 3 0 ] ) ;

& gt ; & gt ; secret plan ( T, Y ( : ,1 ) , ‘- ‘ , T, Y ( : ,2 ) , ‘- ‘ , T, Y ( : ,3 ) , ‘- ‘ )

B. ( I )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfB, [ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 ] , [ 100 4 0 ] )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfB, [ 0 100 ] , [ 100 4 0 ] ) ;

& gt ; & gt ; secret plan ( T, Y ( : ,1 ) , ‘- ‘ , T, Y ( : ,2 ) , ‘- ‘ , T, Y ( : ,3 ) , ‘- ‘ )

B. ( two )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfB, [ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 ] , [ 100 3 0 ] )

& gt ; & gt ; [ T, Y ] =ode45 ( @ nlddtcgfB, [ 0 100 ] , [ 100 3 0 ] ) ;

& gt ; & gt ; secret plan ( T, Y ( : ,1 ) , ‘- ‘ , T, Y ( : ,2 ) , ‘- ‘ , T, Y ( : ,3 ) , ‘- ‘ )

Appendix B