Optimal control in a simple mathematical model of infectious disease
Tim Е. Seleznev 1,a.
1
Far Eastern Federal University, Sukhanova st. 8, 690950 Vladivostok, Russia
a
[email protected].
Keywords: optimal control, simple mathematical model of infections disease, The
Pontryagin Maximum Principle, numerical solution of optimal control problem.
Abstract.
This article discusses the possibility of setting an optimal control problem for a simple
mathematical model of infectious disease. And consider attempts to solve the control
problem analytically (for the linearized system) and numerically.
Intorduction
Nowadays clinicians and immunologists accumulate many materials about different
infectious diseases. Basing on analysis of this materials, many fundamental results
concerning the mechanisms interaction of antigens and antibodies at different levels
of detail are founded. These results allowed to approach the construction of
mathematical models immune processes [1], [2].
Using a mathematical model described in reference [1], [2] Chapter 1, we describe the
model and deliver it to the problem of optimal control.
In Chapter 2, optimality system is derived for a nonlinear system and conducted her
research. On the example the algorithm of solutions derived optimality systems, and
analyzed the influence of some parameters on the initial treatment strategy.
1. Formalization of the simple mathematical model of infections disease and
formulation of the optimal control problem.
We assume that the main variable factors of infection are the following values.
1) The concentration of pathogenic antigens V (t).
2) Concentration of antibody F (t). In this case, antibodies
is substrates of immune system, which neutralizing antigens (immunoglobulins, cell
receptors).
3) Concentration of plasma cells C (t). This is a population of carriers and producers
of antibodies (immune-competent cells and immunoglobulin producers).
4) Relating to the characteristics of the affected organ m (t), we define it as follows.
Let M - characteristic of a healthy body (mass or area), and M * - corresponding to
the characteristic of the part of affected body. Then, let m (t) = 1 - (M * / M).
So we can obtain (according to the reference [1],[2]) the following equations,
which determine the interaction of described variables.
The equation of change in the number of antigens in the body:
𝑑𝑉 = βVdt-γFVdt (i)
Where the first term on the left side of this equation describes the growth
dV antigens during the interval dt, at the expense of reproduction.
It is proportional to V and a number of , which we call the multiplication factor
antigens. FVdt describes the number of antigens neutralized by antibodies F over the
time interval dt. Where  - probability of meeting neutralizing antibody and antigen.
The second equation of change the number of plasma cells in the body:
𝑑𝐶 = ξ(m)αV(t-τ)F(t-τ)dt-𝜇𝐶 (C-𝐶 ∗ )dt (ii)
Here - the first term on the right-hand side describes the generation
plasma cells,  - time nedeed to form a cascade of plasma cells,  - the probability of
collision antigen with antibody. Sometimes performance antibody production falls, so
we added some function (m), which describes the speed reduction pruducing of
antibodies, depending of m. The second term in this formula describes the reduction
of number plasma cells due to aging, C - a coefficient equal to the reciprocal of their
lifetimes. C* - a constant level of plasma cells in a healthy body.
The third – is the balance equation for the number of antibodies that react with
the antigen:
𝑑𝐹 = ρCdt–ηγFVdt-𝜇𝐹 Fdt (iii)
The first term on the right describes the generation of antibodies by plasma
cells during the time interval dt,  - speed of production one antibody cell by plasma
cells. VFdt describes reducing the number of antibodies in the range of dt time due
to connection with antigens. Take into account that the neutralization of one antigen
cell requires  of antibodies. The third term describes the decrease in population of
antibodies due to aging, where 𝜇𝐹 - factor inversely proportional to the time decay of
antibodies.
The fourth equation:
𝑑𝑚 = σV-𝜇𝑚 m (iv)
The first term on the right-hand side describes the degree of damage to the
body. We expect, during the time interval dt, size of the affected organ change in
proportion to the number of antigens, which is described by V, where  - a constant,
different for each disease. Reduction of this characteristic is due to the replacement of
the organism. This term will depend on the aspect ratio of m. 𝜇𝑚 - describe inverse
value of the period of restoration of the body in the e times.
Combining equations (i) - (iv), considering, for simplicity, all the coefficients
are equal to 1, except for , C* = 0 and  - as a variable parameter. As well as we
adding the following initial conditions - V (0) = V0, C (0) = C0, F (0) = F0, m (0) =
m0. Also we add into the (ii) equation control u (t), which means the amount of drug u
(t), enters to the body per unit time, stimulating the growth of plasma cells in the
body. Control u(t) is a limited quantity 0 ≤ 𝑢(𝑡) ≤ 𝑢1 .
So, we obtain the following system of equations:
𝑉̇ = (1 − 𝐹)𝑉
𝐶̇ = 𝐹𝑉 − 𝐶
𝐹̇ = 𝐶 − (1 + 𝑉)𝐹
𝑚̇ = 𝜎𝑉 − 𝑚
V (0) = 𝑉0 ,
C (0) = 𝐶0 ,
F (0) = 𝐹0 ,
{ m (0) = 𝑚0
(1)
Extremal problem is to find the specified control from the minimization
condition of the affected body mass at time T:
𝐽(𝑇) = 𝑚(𝑇) → 𝑚𝑖𝑛, 0 ≤ 𝑢(𝑡) ≤ 𝑢1 .
(2)
2. Nonlinear optimal system solution
Appling the Pontryagin maximum principle [3], we obtain the following
conjugated system, where y1= V, y2 = C, y3= F, y4= m, and 1, 2, 3, 4 are
conjugate functions:
𝜓1̇ = −𝜓1 (1 − 𝑦3 ) − 𝜓2 𝑦3 − 𝜓3 𝑦3 − 𝜎𝜓4
𝜓2̇ = 𝜓2 − 𝜓3
𝜓3̇ = 𝜓1 𝑦1 − 𝜓2 𝑦2 + 𝜓3 (1 + 𝑦1 )
𝜓4̇ = 𝜓4
(3)
ψ1 (T) = 0,
ψ2 (T) = 0,
ψ3 (T) = 0,
{
ψ4 (T) = −1
The Hamiltonian in this case will look like H = ψ1(y1 − y1y3) + ψ2(y1y3 − y2 + u(t)) +
ψ3(y2 − y1y3 − y3) + ψ4(σy1 − y4) → max, to achieve optimal control. Then our control
depends of ψ2 like:
𝑢1 , 𝜓2 > 0
𝑢(𝑡) = {∀𝑣 ∈ [0 ], 𝜓 = 0
, 𝑢1 2
(4)
0, 𝜓2 < 0
This type of control allows to set that principle Bang-bang performed for considered
optimality system.
For solutions using an iterative method. Choose an initial guess for our control.
Solving at each step the original system, we substitute the values of the functions in
the conjugated system. At each step, we find a new approximate control. The
2
iterations are terminated when the integral norm ∫𝑇 (𝑟( )
𝑑𝑡 , where r(t) 𝑡 − 𝑟𝑛𝑒𝑤 (𝑡))
𝑡0
solution of optimality system at previous step, and rnew(t) - the next step solution,
becomes less than the specified . Optimality system is solved numerically using the
application package Mathematica.
For example, using V0=1, C0=1, F0=1, m0=1, u1=1, T = 5, = 1/3 as initial values,
optimal control value will be m(T) = 0,192805. This results observed in picture 1.
m
0.20
0.15
0.10
0.05
1
2
3
4
5
t
3. Conclusion
The research allowed to solve the control problem for the simplest model of an
infectious disease numerically. We have built and with the help of the application
package decided optimality system.
The calculations helped to establish the following facts:
1) Increase the amount of antigen enters the body, leads to a deterioration in the
clinical picture after treatment. Also extends the life of taking the medication.
2) Reducing the number of antibodies and plasma cells in the body at the beginning of
treatment also reduces the performance in T. Time of treatment increases.
3) More Serious illness requires an increase in the concentration of the drug.
4) In the case of increasing the concentration of the drug while it is receiving reduced.
The versatility of this model allows for a more detailed study of all the members of
the original model parameters to obtain more detailed study of various forms of
infectious diseases, using the method described.
References
[1] G.I. Marchuk, Simple mathematical model of infectious disease, RCSS USSRAS,
Novosibirsk, 1975
[2] G.I. Marchuk, Matematical models in immunology, Science, Moscow, 1991
[3] L.T. Ashepkov, B.V. Velichko, Optimal control, FENU, Vladivostok, 1989