Marchuk’s Model: a simple mathematical model of infectious disease By Abdelaziz Gdoura & Katharina Riegler

The immune system, which is an example of a complex system in biology, defends organisms against viral and bacterial infections. Mathematical models in immunology are systems of nonlinear differential equations with retarded argument. Some of most illustrious studies in theoretical models in immunology have been elaborated by the famous mathematician, GI Marchuk. Examples of his constructed models are a basic model of an infectious disease and a mathematical model of the antiviral immune response. These models were used to investigate mechanisms of the development of unfavorable forms of infectious diseases such as hepatitis, influenza, pneumonia, tuberculosis, etc. Here we consider only the simplest mathematical model of an infectious disease published as early as 1980. This is a very important work which plays a similar role for mathematics applied to immunology as the Lotka-Volterra model for mathematics applied to ecology. Indeed, the immune response to a replicating antigen may be viewed as a problem of interacting populations of antigens and antibodies where antigen plays the role of prey and antibody the role of predator, with plasmacytes sensing the presence of the predator and producing the ad hoc antibodies. EQUATIONS

In the simplest form of Marchuk’s model we consider interactions between a pathogen (bacteria, virus,…) with the concentration (�[�]) and an immune system represented by 2 intricated elements :

- antibodies (�[�]) which are the neutralizing factor that impede the virus replication - antibody-secreting cells (�[�]) also called plasmacytes (or plasma cells) that produce the latter. -

Thus, the first equation describes changes of �[�] and has the same form as the equation for a predator in well-know predator-prey Lotka-Volterra: ��[�]

��= (� − ��[�])�[�]

� : the multiplication rate of viruses. γ : the coefficient describing the probability of the virus-antibody encounter and the efficiency of their interaction. In the absence of antigens, plasma cells stay in the so-called physiological state �∗. If some virus appears in the organism, then the process of recognition starts and there is a trigger of plasma cell division. However there is some time delay between the stimuli and new plasma cells appearance. It last approximately 1 day. It is assumed that this delay is constant. ��[�]

��= ��[� − �]�[� − �]− ��(�[�]− �∗)

�: the coefficient of the immune system stimulation or the immune reactivity ��: the value inverse or reciprocal to the the plasmacyte lifetime, it is a measure of the death rate for the plasmacytes) �: the time necessary for the plasmacyte cascade formation, formation of plasma cells and antibodies. C∗ : the constant level of plasma (immuno-competent) cells in a healthy body. Plasma cells produce antibodies with some constant rate �. Antibodies are used up in immune reactions and degrade with the constant rate ��. Therefore the change of antibodies’ concentration is governed by the following equation:

��[�]

��= ��[�]− ��� + ���[�]��[�]

� : is the antibody production rate per plasma cell due to the presence of virus. ��: the value inverse to the plasmacyte and antibody lifetime (degradation rate).

�: the number of antibodies needed for neutralization of one virus. Considering the evident influence of the damage on the dynamics of the immune system described by the Marchuk’s model, a variable reflecting the damage of the organ-target has been introduced, �. ��[�]

��= ��[�]− ���[�]

�: the coefficient of the body damage (per disease). ��: the coefficient of the recovery of the damaged body mass. When � = 0 there is no damage, when � = 1 the damage is maximum. To take into account the effect of the damage on the immune system himself the equation describing the production of plasmacytes has been modifying by integrating the fonction �(�). ��[�]

��= � ξ(m) �[� − �]�[� − �]− ��(�[�]− �∗)

When there is no damage ξ(m)= 1.

�(�)- the nonnegative continuous non-increasing function describing the failure of the normal functioning of the immune system. From the figure above we have thus :

�(�)=

1

���

�∗��

for � �[0,�∗]

for � �[�∗,1]

For simplicity, we will not take into account the ensuing damage inflicted to the immune system’s organism by the replicating virus (ξ(m)≡ 1). Therefore we will consider the following nonlinear (delay) differential equation (DDE) system. ��[�]

��= (� − ��[�])�[�]

��[�]

��= ��[� − �]�[� − �]− ��(�[�]− �∗) 1.1

��[�]

��= ��[�]− ��� + ���[�]��[�]

This system of equations with initial conditions describes the dynamics of pathologic infection development during immune response. We assume all the constants in the system of equations to be nonnegative. It follows directly from their biological meaning. It can describe 4 states of infectious disease as depicted in the following diagram.

STATIONARY SOLUTIONS

The system of equations 1.1 has stationary solutions. We have the following 2 solutions: �� and ��.

�� = ��[�]→ 0,�[�]→ �∗,�[�]→��∗

��= �∗�

The solution �� corresponds to the healthy state of the organism and is called healthy state (where the body is virus-free). Here the concentration of antigens (i.e virus or pathogen) is equal to zero but the quantities of plasma cells � and antibodies � correspond to the values of immunological status of a healthy man, {�∗, �∗}.

�� = ��[�]→��(�������∗)

�(�������),�[�]→

�����������∗

�(�������),�[�]→

�

��

The solution �� corresponds to the chronic from of the disease and is called chronic state, (where by definition the body contains a constant minimal amount of virus)

If we replace �∗by ��∗

�� (physiological status) then we have �[�]→

����(����∗)

�(�������)

Let’s now study the stability of these fixed points or equilibrium.

STABILITY

At the trivial equilibrium (healthy state)

�.

= �(� − ��[�])�[�]� = ��

�.

= ��[� − �]�[� − �]− ��(�[�]− �∗)= �� 1.2

�.

= ��[�]− ��� + ���[�]��[�]= ��

Because multivariate stability analysis makes use of the Jacobian matrix let’s calculate it for the simplified system :

��=

⎝

⎜⎛

���

���

���

���

���

���

���

���

���

���

���

���

���

���

���

���

���

���⎠

⎟⎞

= �

� − ��[�] 0 −��[�]

��[�] −�� ��[�]

−���[�] � −�� − ���[�]�

Let’s now evaluate the Jacobian matrix �� calculated for 1.2 at the first equilibrium �°

��(�) = �

� − ��∗ 0 −0��∗ −�� 0

−���∗ � −��

�

Let’s determine the characteristic polynomial of the matrix ��(�) ∶ Det���(�) − ����

Det ��

� − � − γF∗ 0 0��∗ −� − �� 0

−���∗ � −� − ��

�� = (� − � − γF∗)(�� + ��� + ��� + ����)

We are looking for the 3 roots (polynomial of degree 3) of: (� − γF∗ − �)(� + ��)�� + ��� obviously they are

{{� → −��},{� → −��},{� → � − γF∗}}

Here we can see that all the solutions are real and negative if � − γF∗ < 0. We have thus demonstrated that if � − γF∗ < 0 the stationary solution for the healthy state is locally asymptotically stable (stable node). Let’s now consider the same system with delay �:

�.

= �(� − � ∗ �[�])∗ �[�]�

�.

= � ∗ �[� − �]∗ �[� − �]− �� ∗ (�[�]− �∗)

�.

= � ∗ �[�]− ��� + � ∗ � ∗ �[�]��[�]

After linearization and �∼

= � − �∗, �∼

= � − �∗, �∼

= � − �∗ we have : ��

∼

��− �

∼� + v

∼��∗ + f

∼

��∗ = 0

��∼

��+ �

∼�� − � �v

∼�∗ + f

∼

�∗� − ��∗�∗ = 0

� �∼

��+ f(

∼

�� + ���∗)− c∼

� + v∼

���∗ + ���∗�∗ = 0

According to the theory of linear differential equations, the solution can be written as a superposition of terms of the form

���� where {��} is the set of eigenvalues of the Jacobian.

Let all the solution �∼

be of the form �∼

= ���� We have then the following system : �����

��− ����� + ������∗ + ������∗ = 0

�����

��+ ������ − �����(���)�∗ + ���(���)�∗� = 0

�����

��+ ����(�� + ���∗)− ������ + �������∗ = 0

since ���(���) = ���(�) ∗ ���(��) We have finally the following system (of linear differential equation) : �� − �� + ���∗ + ���∗ = 0 �����

��+ ������ − ���∗���� + �∗����� = 0

�� + �(�� + ���∗)− �� + ����∗ = 0

�.

= �∗δx where �∗ is the Jacobian evaluated at the equilibrium point. (Note that �∗ is a constant, so this is just a linear differential equation). From above, we obtain a characteristic matrix of the form:

�

� − ��∗ − � −��∗ 0

��∗���� ��∗���� −�� − �−���∗ −���∗ − �� − � �

�

Let’s evaluate it at �° ��∗ = 0

�∗

�∗� and calculate its determinant:

Det ��

� − � − ��∗ 0 0

������∗ 0 −� − ��

−���∗ −� − �� ��� =

����(−����� − ������ − ������ − �������)(� − � − ��∗) =(� + ��)(� + ��)(−� + � + ��∗)

Again, all roots are real and negatives provided that � − γF∗ < 0. We see here that the stability for that solution is independent of the delay. Because we are only keeping a locally linear approximation to the vector field, this analysis is called a linear stability analysis.

At the non-trivial equilibrium

�� = ��_

=����(����∗)

�(�� �����) ,�

_

=�����������∗

�(�������),�

_

=�

��; ����� �

If we want �_

=����(����∗)

�(�������)> 0

We should have have either : �� > ���� or �� < ���� � − γF∗ > 0 � − γF∗ < 0 (2)

Let’s recall the Jacobian matrix of the simplified system: �

� − γF[�] 0 −��[�]

��[�] −�� ��[�]

−���[�] � −�� − ���[�]�

Let’s evaluate the jacobian matrix at the second fixed point and determine its characteristic polynomial this time directly by using Mathematica. We must notice that calculating directly the eigenvalues of this evaluated Jacobian matrix provide too complicated formula to conclude on the sign of their real parts, the same is true for the roots of the characteristic polynomial, instead we should use some well-established stability criteria namely the Routh-Hurtwitz and the Mikhaïlov criterion based on the coefficients of the characteristic polynomial easily obtained.

We have: CharacteristicPolynomial

⎣⎢⎢⎢⎡

⎝

⎜⎛

� − � ∗�

�0 −� ∗ �

_

� ∗�

�−�� � ∗ �

_

−� ∗ � ∗�

�� −�� − ���

_

⎠

⎟⎞

,�

⎦⎥⎥⎥⎤

=−������

_

− ����_

� − �(� + ��)��� − ����_

+ ����_

+ ����

�

Let’s simplify the previous formula and we have :

�( � )= −�� + �����_

− �����_

− ����_

+ ����_

− ���� + ����_

�� − ����_

�� − ���� − �����

�( � )= �� + � �� + (� − �)� + ℎ − � = 0 with :

� = ���_

+ �� + ��

� − � = ��(�� + ���_

) +����_

− ���_

> 0

ℎ − � = ����_

− ����_

�� with ℎ = �� As stated previously one can say a lot about a stability of a polynomial �(�) without finding its roots. Routh-Hurtwitz criterion. If we use the usual notation for the coefficients of the polynomial �(�)= ���� + ������ + ������+. . . +����� + ��

the Routh-Hurwitz criteria for the polynomial �(�): Please see Gantmacher “the theory of matrices” (1964) for references. �(�)= ���� + ������ + ������+. . . +����� + �� � = 2: �� > 0 and �� > 0. � = �: �� > �,�� > � and ���� > �� � = 4: �� > 0,�� > 0�� > 0 and ������ > ��

� + �����

It follows from above that the following inequalities must be satisfied to meet stability: ℎ − � > 0 � − � > 0 �(� − �)> ℎ − � To fullfill inequality ℎ − � > 0 one need :

�����

> �������

�� > ���� Hence inequalities (2) implies � − γF∗ > 0 for the stability of the chronic state. 0 < ℎ − � < �(� − �)

0 < ����(� − γF∗)< (�� + �� + ηγ�_

)(���� + ηγμ��_

− αρ�_

− ηβγ�_

)

By algebraic manipulation we have :

(� − γF∗)< ����_

+ �� + ���

thus we have :

� < γF∗ + ���_

+ �� + ��

Hence the state B is locally stable only if the organism is strong (�� > ����) and the antigen reproduction rate is proper but not too strong (β < γF∗ + x

_ )

To obtain the stable chronic steady state one needs strong antigens with sufficient high reproduction rate � > γF∗ but this reproduction rate cannot be very large because it leads to instability. On the other hand immune system must be also strong, which is described by �� > ����.

(� − γF∗)< (���_

+ �� + ��)(��∗

�−

����_

����)

All these inequalities considered above imply that a chronic disease is possible only under very specific conditions. Mikhailov’s criterion of stability We’re going to make use of the Mikhailov stability criterion to define new conditions of stability for the positive steady state. Let’s now take into consideration the delay �: We have the characteristic polynomial (at the second equilibrium) :

�( � )= �� + � �� + � � − � − ��e��� + �e��� = 0 With the same coefficient as previously established. Indeed: The Jacobian matrix for the system of 3 nonlinear equations of MM is :

�

β − γF∗ − λ −γV∗ 0

αF∗e��� αV∗e��� −μ� − λ−ηγF∗ −ηγV∗ − μ� − λ ρ

�

Let’s evalute it at the non trivial equilibrium V,_

C_

,F_

�

β − γF_

− λ −γV_

0

αF_

e��� αV_

e��� −μ� − λ

−ηγF_

−ηγV_

− μ� − λ ρ

�

The characteristic equation is :

W( λ )= λ� + a λ� + b λ − d − gλe��� + fe��� = 0 With the corresponding coefficient:

a = (−β + γF_

+ γηV_

+ μ� + μ�)= (γηV_

+ μ� + μ�) with −β + γF_

= 0 since F_

=�

�

a = (γηV_

+ μ� + μ�)

b = (βγηV_

+ μ��−β + γF_

+ γηV_

+ μ�� + μ�(−β + γF_

))= (βγηV_

+ μ��γηV_

+ � b

d = βγηV_

μ� − μ�μ�(−β + γF_

) = βγηV_

μ�

g = αρV_

f= αβρV_

Z( λ )= λ� + a λ� + b λ − d − gλe��� + fe��� = 0

In 1938, Mikhailov proved a frequency response criterion, which is sufficient and necessary for the stability of processes described by known ��� order constant coefficient linear differential equations. It is equivalent to the Routh–Hurwitz criterion; however, it is geometric in character and does not require the verification of determinant inequalities. Consequently, stability depends on the shape of the so-called Mikhailov hodograph, that is the curve situated in a complex plane and connected with the locus of the characteristic equation. This criterion belongs to the class of methods applying the principle of argument to various problems in control and stability. All roots of a polynomial f(z)= a�z� + a�z��� + a�z���+. . . +a���z + a� with real coefficients have strictly negative real part if and only if the complex-valued function z = P(iω) of a real variable ω [0, ∞) describes a curve (the Mikhailov hodograph) in the complex z-plane which starts on the positive real semi-axis, does not hit the origin, crosses the imaginary axis at the positive part and then also decreases to −∞, successively generates an anti-clockwise motion through quadrants. Such a shape is presented in figure 1 and figure 2.

Figure 1 Figure 2 We had:

Z( λ )= λ� + a λ� + b λ − d − gλe��� + fe��� = 0

Z( λ )= −λ� − a λ� − b λ + d + gλe��� − fe��� = 0 By substiuting � by iw we have : −(i�)� + � − �(i�)� − �i� + �i�e���� − �e���� = � − i�� + ��� + i�� − �e���� + ���e���� With −e����� = −�Cos[��]+ i�Sin[��] ��� ���e���� = i��Cos[��]+ ��Sin[��] We have : � (i�)= Re� (i�)+ � Im� (i�) Where : Re �(i�)= � + ��� −�Cos[��]+ ��Sin[��] Im�(i�)= �� − �� −�Sin[��] +��Cos[��] Thus, for stability, considering the form of the curve one should find a sequence of frequencies satisfying the inequality: 0 = �� < �� < �� Where Re �(i��)= 0 and Im �(i��)= 0 and �� and �� can be estimated. It is demonstrated that :

��� ⩽ 1 and 0 <���

����< � − � − fτ are sufficient to satisfy 0 = �� < �� < ��

It follows that the chronic state is locally asymptotically stable if :

� − gτ >���

� ( implied by ��� ⩽ 1 )

0 <���

����< � − � − fτ

It can be demonstrated from below for � → ∞:

0 < � − γF∗ < �� +�

������

��

And according to a 2009 correction paper (cf references) from the same authors which posit that the necessary conditions of stability if � → ∞ are instead: ��� ⩽ 1

0 < � − γF∗ <��∗ �⁄

��(�������∗ �⁄ )��

DISEASE DYNAMICS and CLASSIFICATION

2 limit cases

We know that v[�]= v°��(����∗) At the time � = 0 a healthy organism is infected by an initial dose of a virus V° so that we have initial conditions �(0)=V°,�(0)= �∗,�(0)= �∗.

a) If F∗ = 0 implies v[�]= v°��� then �[�] increases exponentially. This will be a case of some immunodeficiencies or with some old patients whose immune systems fails to react to an antigen.

b) If � − γF∗ << 0 implies v[�]= v°�����∗ then �[�] decrease exponentially. This second limit case is realized

when the level and quality of specific antibodies in an organism to a given virus is sufficient for neutralization of all the antigens penetrating the body. (This is the most extreme subclinical case).

a) b)

Thus we have described two limiting solutions, which are boundaries of the solution V[t], corresponding to a lethal outcome and a high immunological barrier respectively. Interestingly, we could also have described limit cases based on the initial dose of virus V°, where we should have had, in the context of a normal immune system, for very small V° the most pronunced subclinical case and for a very high V° (in which case the immune system will be completely swamped) a lethal outcome, whathever the difference between � and ��∗. But this problem of initial conditions is beyond the scope of the present report. Let’s consider the case where: � − ��∗ < � The subclinical form arises when solution for the state A is stable and it is characterized by �[�]→ 0 at � → ∞. Whatever the nonzero initial number of viruses (under a certain threshold though) that have invaded the organism, the disease does not progress unless the immunological barrier is overcome (�. �. � > γF∗). This case corresponds to everyday contact of an organism with low doses of antigens getting into the organism. When the dose of infection increases to a considerable extent as compared to �* the strenght of immune response begins to play an important role depending on: αρ > ��ηγ (normal immune system) [lethal if V°>100�*]

αρ < ��ηγ (immunodeficiency) [lethal at least at V°=10�*] Thus the attraction domain of the state of healthy organism (“recovery zone”) with normal immune system is much larger than the same domain of an organism with immunodeficiency. Example of subclinical modeling:

Red : virus, Blue: plasmacytes, Green: Antibodies It is stated “that the antibody-producing’ mechanism is not switched on the case where � − γF∗ ≪ 0 but our modeling suggest that one has to set the level of intial virus dose at an extremely low level in order for any modelised immune reaction to vanish.

Lower level of virus (0.08) More lower level of initial dose of virus (0.008) However, one may interpret this very small immune response as so localized that it cannot trigger a more thorough immune response capable of generating memory cells for example. But when the triggering of the immune response will be sufficiently robust to generate an accumulation not only of effector cells but also of memory cells, it can fairly describes the situation of a healthy organism being vaccinated by weakened antigens. These latter cells persists over time and subsequently increase the level of C* which is equivalent to the increase of the immunological barrier since F* is linearily depend on C*. � − ��∗ > � Acute form of disease Let’s now assume that at t = 0 , � − γF∗ > 0 holds. We know that v[�]= v°��(����∗).

Thus, until the point F[t]= �

� is reached, the derivative

��

�� is positive, after this time �� , it becomes negative. F[t] will continue

to increase until it reaches its peak ���� after which point it begins to decrease until crossing again the line F[t]= �

� in the

reverse direction at time ��. As shown below, if the virus from �� to ��� has completely declined to zero before the time �� we are in the case of an acute infection (infection that is resolved).

Thus if we posit ΔT = �� − �� and Δt = ��� − �� it follows that the acute phase happens if ��

��> 1

The acute form has a pronounced dynamics of viruses (rapid growth and abrupt fall to “zero”) and is characterized by the effective immune reponse. If the number of viruses grows more rapidly than production of antibodies neutralizing the latter the curve of virus concentration begins to grow exponentially. However, after plasmacytes have formed and begun mass antibody production, the growth of the virus concentration decelerates and some time later it rapidly falls. At the same time there goes on a reproduction of new antibodies whose total number decelerates exponentially until the normal immunological level is reached. This means that the more the immune system is able to generate antibodies the more it will be able to combat an acute infection, but this has to happen in a fairly rapid pace (small delay, and small turn-over of antibodies and cells), to avoid the virus reaching such a level that the infection becomes incontrollable (which is also linked to the infection multiplicity rate).

Simulation with Python by Euler Method

Red : virus, Blue: plasmacytes, Green: Antibodies Simulation with Mathematica (and from now on) One should also note that the recovery case is not equivalent to a return to the initial case mathematically speaking and our modeling confirm that this behavior is not stable (cf figure below) but biologically speaking it is stable since Pasteur, who has shown that there is no “spontaneous generation” .

Hypertoxic form

if F[t]= �

� is not mathematically reached we have the lethal outcome case, indeed

��

��> 0 for an infinitely large interval of

time. Clearly if F* = 0 there exist no such restriction in the growth of the virus, as we’ve seen previously.

Red : virus, Blue: plasmacytes, Green: Antibodies Interestingly we can see that �[0]= �[0]= 0 from the beginning (which is not the case in reality, but follows from mathematical operation). Chronic form of the disease

if ��

��⩽ 1 that is if the virus has not been nullified before F[t]=

�

� then

��

�� can becomes positive again and we have then a

process of chronisation.

Here we can see that the 2 intervals are equals and we are in presence of the chronic state. It is clear that the higher is the maximal quantity of the antibodies ����the larger is ΔT and therefore, the smaller the probability for chronic form to develop. The chronic form has a and is characterized by the persistence of viruses in the body (i.e. �[�]→ const. at � → ∞ and solution �� is stable). To obtain the stable chronic steady state one needs strong antigens (�� > ����) with sufficient high reproduction rate � >

γF∗ but this reproduction rate should be proper, that is not too strong β < γF∗ + ���_

+ �� + ��.

This means that the physiological level of antibodies must not be too strong F∗ < �/γ. For a very strong immune immune system is very strong meaning that � → ∞ the chronic state is locally asymptotically stable only if : ��� ⩽ 1

0 < � − γF∗ <��∗ �⁄

��(�������∗ �⁄ )��

Thus the stable positive steady state which describe the chronic form of the disease con occur only for very specific paameter values, this explains why chronic diseases are not found very often in nature when an immune system is strong.

Red : virus, Blue: plasmacytes, Green: Antibodies The number of the virus number increases after infection, reached a maximum, and then decreases down to a minimum. The process repeats until the stationary level of antigen V in an organism is fixed. This results in an equilibrium between replicating stimulants and those eliminated by the immune system. Stability for positive delay and Hopf bifurcation A Hopf bifurcation (named after the mathematician Eberhard Hopf) occurs when an equilibrium’s stability properties change and periodic dynamics appear around the equilibrium as a parameter of the model is changed. The Hopf bifurcation theorem specifies conditions under which such periodic dynamics occur. Its analysis is too complicated to be described but we will present an example by modelisation. It was shown that stability in the case of positive delay is possible only under the condition of stability for the case with τ = 0, that is without delay. If the chronic form is stable for a delay = 0 then it exist a τ > τcrit where there is a hopf bifurcation with instability, we have either stable periodic orbits (limit cycle) or instable periodic orbits. The proof is based on the Mikhailov criterion. A Hopf bifurcation where a transition occurs between a stable equilibrium and periodic dynamic clearly appears when the stable positive steady state looses stability with increasing delay chosen as the destabilizing parameter. Delay: 0 delay : 0.1 hopf bifurcation delay: 0.2

Red : virus, Blue: plasmacytes, Green: Antibodies In conclusion: We thus observe either recovery, lethal outcome or a chronic dynamics (with periodic or not). Moreover, stable positive steady state, which describes the chronic form of the disease, can occur only for very specific parameter values, that is when

the difference between the antigen reproduction coefficient β and the parameters of the system γF∗ is sufficiently small.

This can be treated as a mathematical explanation of the fact that chronic diseases are not very often in nature, especially in natural conditions, when an immune system is strong. Then we expect one of the two types of the behaviour: for a small initial dose of infection there is recovery, while for larger ones the lethal outcome can appear. To obtain the chronic state in Marchuk’s model an antigen should have sufficiently large reproduction rate, � > γF∗ On the other hand, this rate cannot be too large in the stable case. Therefore, the range of parameters for which chronic dynamics can be observed is rather small. Typically, we observe either recovery or lethal outcome (the last one considers mainly the cases without treatment, of course).

Applications of the model

This model give rise to some proposition of treatment of virus infection. It turns out that with the increase of the initial dose of viruses the chronic process degenerates into a normal response (the acute from with recovery). This has allowed the authors to formulate, immunologically and mathematically, a method for treatment of chronic infections by aggravation. With higher doses of infection the chronic form turns into. Thus it was proposed that treatment of chronic form can be possibble through exacerbation of disease (considerable increase in the antigen quality compared to chronic from.) Second, the treatment of acute forms of disease by drugs that decrease the rate of virus multiplication promotes the disease becoming chronic. Also the use of antibiotics (that reduce the reproduction rate) can turn a lethal case into an acute one.

Limit and Improvement of the model

Marchuk’s model cannot be applied to all bacteria or viral infections. Immunologists agree with the fact that even if the immune system is weak (that is, its production of plasma cells and antibodies is small, which is expressed by the inequality) then the organism should recover for sufficiently small initial doses of the antigen. It seems that it is not possible in the case of Marchuk’s model. It may happen that the damage of the virus-affected organ is essential. In that case the normal functioning of the organs responsible for antibody formation is seriously upset. Then the number of viruses in the organism will continuously grow, which results in a lethal issue. That’s why Marchuk has added the fonction of damage in its model. Another extension to the model may be made by introducing the use of antibiotics, as disccused earlier. The antibiotic may be modeled as causing antigen death proportional to the concentration of the antibiotic in the blood stream. We can modify the model through the addition of others controls (extensively used in the field of optimal control) u1: pathogen killer u2: Plasma cell enhancer u3: Antibody enhancer u4: Organ health enhancer Also some other improvements are possible. For example one can study a role of interleukin in defense possibilities of the organism.

References

Papers:

- G. I. Marchuk, L. N. Belykh -- Mathematical model of an infectious disease (1979) - Urszula Fory´s, Hopf Bifurcation in Marchuk’s Model of Immune Reactions, Mathematical and Computer

Modelling 34 (2001) - Urszula Fory´s,Stability and bifurcations for the chronic state in Marchuk’s model of an immune system, Journal of

Mathematical Analysis and Applications (2009) Books:

- Guri I. Marchuk-Mathematical Modelling of Immune Response in Infectious Diseases-Springer Netherlands (1997) - Ernst Hairer, Syvert P. Norsett, Gerhard Wanner Solving Ordinary Differential Equations I_ Nonstiff Problems

(2009) - Felix R. Gantmacher-The Theory of Matrices, Vol. 1&2-American Mathematical Society (1990)

Annexes

Code Mathematica:

Manipulate[ Module[{eqns,soln,plot1,� = 1.2,� = 0.8,� = 10,� = 1,Ϛ = 1,� = 1,γη = 1,� = 1}, eqns = { ��[�]== (� − � ∗ �[�])∗ �[�], ��[�]== � ∗ �[� − �]∗ �[� − �]− Ϛ ∗ (�[�]− �), ��[�]== � ∗ �[�]− γη ∗ �[�]�[�]− � ∗ �[�],

�[�/; � ≤ 0]== 0.1,�[�/; � ≤ 0]== �,�[�/; � ≤ 0]==��

�};

soln = Quiet@NDSolve[eqns,{�,�,�},{�,0,time}]; plot1 = Plot[Evaluate[{�[�]∗ 20,�[�],�[�]}/. soln],{�,0,time}, PlotStyle → {Red,Blue,Darker@Green},ImageSize → 1.1{300,300},Frame→ True,PlotRange→ All,ImagePadding

→ {{40,20},{40,20}}]Grid[{{plot1}}]],{{time,200,Row[{"time",Style["t",Italic]}]},0,800,ImageSize→ Tiny,Appearance→ "Labeled"},{{�,0,"time delay τ"},0. ,100. ,0.1,ImageSize → Tiny,Appearance→ "Labeled"}

] Code Python: @author: Katharina import math import numpy as np import matplotlib.pyplot as plt beta=2; gamma=0.8; xsi=1; alpha=2; mc=1; Cet=2; rho=1; mf=0.8; sigma=1; mm=1; etha=1.5; tf=20;n=2000;h=tf/float(n); V=np.zeros([n,]);V[0]=9; C=np.zeros([n,]);C[0]=5; F=np.zeros([n,]);F[0]=8; M=np.zeros([n,]);M[0]=0; for i in range(n-1): V[i+1]=V[i]+h*V[i]*(beta-gamma*F[i]) C[i+1]=C[i]+h*(xsi*alpha*V[i]*F[i]-mc*(C[i]-Cet)) F[i+1]=F[i]+h*(rho*C[i]-(mf+etha*gamma*V[i])*F[i]) M[i+1]=M[i]+h*(sigma*V[i]-mm*M[i]) fig=plt.figure() plt.plot(C,label='C') plt.plot(F,label='F') plt.plot(V*3,label='V') plt.plot(M,label='M') plt.xlabel('Time') plt.legend()