The antiviral immune response model has been generalized to take into account the effect of adsorption therapy on the development of disease process under conditions of small diffuse scattering of acting factors based on the synthesis of perturbation theory and modeling ideas of the adsorption mass transfer process. A computational technology of step-by-step asymptotic approximation of the solution to the corresponding model problem with time delay as a perturbation of solutions to degenerate problems without delay has been developed. The numerical experiment results demonstrate a predictive decrease in the concentration of viral elements in the target organ through their neutralization into the organism by adsorbents introduced. It is noted that the efficiency of adsorption drugs depends, in particular, on the moment of time for their introduction, which must be considered when making decisions on the formation of complex treatment programs using appropriate therapy.
The practice of using adsorption agents for detoxification of the organism has a very long history
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The basic model of infectious disease and models of antiviral and antibacterial immune responses
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Let us note that the wide use of various micro- and nanoporous catalytic environments in many industries turned to the active study of adsorption processes and the development of corresponding mathematical models. In particular, in [10, 11] the single-component adsorption process in a certain catalytic medium of l length of microporous structure particles of the radius R is described by the following model problem: ct¢=Dintercx¢¢x -qintra (qr¢) r=R , qt¢=Dintra (qr¢¢r + 2qr¢ r) under conditions c(0,x)=0, c(t ,l) =c¥ , cx¢ (t ,0)=0 , q(0,x,r)=0 , q(t,x,R)=k ×c(t,x), qr¢ (t ,x,0)=0 . This approach is based on the idea of taking into account the interaction of mass transfer in the interparticle space (first equation) and mass transfer into the interior of particles with concentration q(t,x,r) (second equation), which is related to concentration c(t ,x) . This approach (in [12]) has been used for modifying the basic model of viral infection to consider adsorption therapy and diffuse scattering effects, which makes it possible to predict an additional decrease in the antigens’ concentration due to their absorption by adsorbents.
This work aims to generalize the antiviral immune response model to consider the effect of adsorbing substances on the disease dynamics in the conditions of diffusion perturbations and temperature of the organism. 2. Generalizing the antiviral immune response model to consider adsorption, diffusion perturbations and temperature
To predict the viral disease process development considering adsorption therapy we will generalize the modification of the immune response model to viral pathogens (in [9]) by introducing additional components that describe the diffusion mass transfer of antigens inside the adsorbent particles and its relationship with the mass transfer of antigens in the environment of the target organ. We will describe the corresponding immune response dynamics taking into account the effect of adsorbents under conditions of diffusion scattering and temperature of the organism, in the set G={(t,x): tÎR+ , xÎR} by such a singularly perturbed system of nonlinear differential equations with small parameter µ : ¶Vf =wV + h1(Q)×CV çae1-CCV** ÷ö+nh2 (Q)×CV E - h3 (Q)×FVf - h4 (Q)×MVf ¶t è V ø
2 -h5Vf (C* -m -CV )+µDV ¶¶xV2f -µDW* çèae¶¶Wrf øö r=R
÷ , ae¶2W ¶Wf =µ2DW çè ¶r2f + 2¶Wf ö÷,
¶t r ¶r ø ¶MV =h4 (Q)M ×Vf - h6MV +µ2DMV ¶¶2Mx2V ,
¶t ¶HE =h7 (Q)x(m)HE (t -t1o ,x)MV (t -t1o ,x)-h8 (Q)MV HE ¶t
¶2HE -h9 (Q)MV HE E + h10 (HE* - HE )+µ2DHE ¶x2 , ¶HB =h11(Q)x(m)HB (t -t2o ,x)MV (t -t2o,x) -h12 (Q)MV HB ¶t (1) ¶E ¶t
¶2HB -h13 (Q)MV HB B + h14 (HB* - HB )+µ2DHB ¶x2 , = h15 (Q)x(m)E(t -t3o ,x)HE (t -t3o ,x)MV (t -t3o ,x) -h16 (Q)MV HE E
-h2 (Q)CV E + h17 (E* - E)+µ2D ¶B ¶t = h18 (Q)x(m)B(t -t4o ,x)HB (t -t4o ,x)MV (t -t4o ,x)
¶2B -h19 (Q)MV HB B + h20 (B* - B) +µ2DB ¶x2
, ¶2E E ¶x2 , ¶P ¶t ¶2P = h21(Q)x(m)B(t -t5o ,x)HB (t -t5o ,x)MV (t -t5o ,x) + h22 (P* - P) +µ2DP ¶x2 , = h28F (t -t6o ,x)Vf (t -t6o ,x)(1-h29F (t -t6o ,x)Vf (t -t6o ,x)) -h30 (Q-Q*)+µDQ ¶2Q ¶x2 under ¶Q ¶t
Vf (0,x)=Vf0 (x), MV (0,x) = MV0 (x), HB (0,x) = HB0 (x), HE (0,x) = HE0 (x), E(0,x) = E0 (x),
P(0,x)= P0 (x), B(0,x)= B0 (x), F (0,x)= F0 (x), СV (0,x) =CV0 (x), m(0,x) = m0 (x), Q(0,x) =Q0 (x),
HE (t,x)MV (t,x) =j1(t,x), -t1o £t <0, HB (t,x)MV (t,x) =j2 (t,x), -t2o £t <0, E(t,x)HE (t,x)´
(2)
´MV (t,x)=j3 (t,x), -t3o £t <0, B(t,x)HB (t,x)MV (t,x) =j4 (t,x), -to £t <0, to =max{t4o,t5o},
¶Wf (t,x,0)
F (t,x)Vf (t,x) =j5 (t,x), -t6o £t <0, Wf (0,x,r)=Wf0 (x,r), Wf (t,x,R)=k×Vf (t,x), =0
¶r
where Vf =Vf (t,x) , Wf =Wf (t,x), MV = MV (t,x), HE = HE (t,x) , HB = HB (t,x) , E =E(t,x) , B= B(t,x)
, P= P(t,x) , F = F (t,x) , СV =СV (t,x) , m=m(t,x) , Q=Q(t,x) are accordingly the number of antigens
that are present in the intercellular space of the target organ, the number of antigens in the middle of
particles of the adsorbent, the number of stimulated macrophages, the number of T-helper-lymphocytes
of cellular immunity, the number of T-helper-lymphocytes of humoral immunity, the number of
Tcell-effectors, the number of B-lymphocytes, the number of plasma cells, the number of antibodies, the
number of viruses infected cells, non-functional part of the damaged target organ and temperature at
the moment of time t at point x; M is the number of all macrophages in the organism, determined by
homeostasis; h1(Q)=h10 (1+h1*(Q-Q*)) is temperature dependent parameter ( h1* =const >0 ), which is
related to antigen reproduction rate; hl (Q)=hl (1+hl*(Q-Q*)) , l ={2,3,4,7,8,9,11,12,13,15,16,18,19,21}
are temperature dependent parameters ( hl* =const >0), which are related to the immune system. The
parameter h28 characterizes the mechanism for increasing the organism’s temperature, which is related
to the number of Vf F -complexes: if their number does not exceed the certain threshold value (Vf F )* ,
then the temperature does not increase and h28 =0 . If the value of Vf F exceeds this threshold, then the
temperature increases h28 =h2*8 =const >0 . The other parameters hi of the model are determined
according to [
V produce antibodies; Vf0 (x) , MV0 (x) , HE0 (x) , HB0 (x) , E0 (x) , B0 (x) , P0 (x) , F0 (x) , CV0 (x) , m0 (x) , Q0 (x) , j1(t,x) , j2 (t,x), j3 (t,x), j4 (t,x), j5 (t,x), Wf0 (x,r) is smooth limited functions of necessary order. The function x(m) considers the effect of productivity decrease on the immune system due to damage to the target organ, 0£x(m)£1. And eDV , e2DW , e2DMV , e2DHE , e2DHB , e2DE , e2DB , e2D , P eDF , e2DCV , e2Dm are diffusion coefficients of the corresponding components of the process; µDQ is the thermal conductivity coefficient of the target organ medium; eDW* is the coefficient that characterizes the interaction of the diffusion redistribution of antigens in micropores of the adsorbent particles on their diffusional scattering in the intercellular space; µ is a parameter intended to describe the small influence of diffusion components compared to the influence of other process components. Functions wV (t,x) , wF (t,x) are intended to describe changes concentrated in space and time according to the number of antigens and antibodies [9]. 3. Computing technology of numerical-asymptotic approximation of the problem solution
Let us come to building a procedure for a step-by-step numerical asymptotic approximation of the
solution to the problem (1)-(2). We represent the delay values specified in the model as tio =si ×t, where
si ÎN , i =1,6, t>0 . It should be noted that according to the results of the parameter identification of
the immune response model to viral infection in [
We will also assume that equations in the system (1) is dimensionless [
E(s,i) =E(s,i) (t,x) , m(s,i) =m(s,i) (t,x) ,
B(s,i) =B(s,i) (t,x) , Q(s,i) =Q(s,i) (t,x)
P(s,i) =P(s,i) (t,x) , F(s,i) =F(s,i) (t,x) ,
( i =0,1,...,n ) required functions,
RnV(fs) = RnV(fs) (t,x,µ) , RnW(fs) = RnW(fs) (t,x,µ) , RnM(Vs) = RnM(Vs) (t,x,µ) , RnH(Es) = RnH(Es) (t,x,µ) , RnH(Bs) = RnH(Bs) (t,x,µ) ,
RnE(s) = RnE(s) (t,x,µ) , RnB(s) = RnB(s) (t,x,µ) , RnP(s) = RnP(s) (t,x,µ) , RnF(s) = RnF(s) (t,x,µ) , RnC(Vs) = RnC(Vs) (t,x,µ) ,
Rn(s) = Rn(s) (t,x,µ) , Rn(s) = Rn(s) (t,x,µ) are relevant residual members. We will perform a regularizing
m m Q Q
transformation r =r µ (0£r£R =R µ) [12] since the sizes of adsorption particles are small. We will
obtain problems for finding unknown functions (asymptotics) as a result of the "procedure of
equalization" [
As noted above, the introduction of adsorption agents can reduce the number of antigens in the organism as a result the severity of the viral infection will also decrease. Therefore, the assessment of the effect of certain drugs on the disease dynamics is an important task in developing a comprehensive treatment program using additional adsorption therapy. In general, we have focused the computer experiments on studying the features of the influence of adsorption agents on the predictive dynamics of the antiviral immune response.
Figure 1 illustrates the model dynamics of viral infection antigens in the infection locus in various situational conditions: without taking into account the diffusion scattering and the absence of adsorption therapy (curve 1); taking into account the effect of diffusion scattering and the absence of adsorption antigens (curve 2); taking into account diffusion scattering and the effect of adsorption therapy (curve 3). Here, the number of antigens at the initial moment was equal to zero ( Vf0 (x)=0 ), but at the moment tV =1 at the point xV =0 there is a concentrated increase in the model number of antigens: wV = AV e-aV (t-tV )2e-bV (x-xV )2 . As expected, the lowest level of predicted growth of antigens is observed in a situation when we consider the effects of diffusion scattering and absorption of antigens by adsorbent. Thus, the presented results confirm the practicability of the adsorption drugs for additional neutralization of antigens, which makes it possible to reduce, in particular, the critical level of disease exacerbation. In addition, it should be noted that the timely introduction of adsorption drugs into the body can more effectively control the growth of the antigen population. Figure 2 shows the predicted Fig. 1. Dynamics of antigens at: µDV =0.00 and µDW* =0.00 (curve 1); µDV =0.05 and µDW* =0.00 (curve 2); µDV =0.05 and µDW* =0.05 (curve 3).
Fig. 2. Dynamics of antigens at: tV =0.0 (curve 1); tV = 2.5 (curve 2); tV =5.0 (curve 3). dynamics of antigen concentration in cases if there is an increase in their number at different times: tV =0.0 (curve 1); tV = 2.5 (curve 2); tV =5.0 (curve 3). According to the numerical experiments, the largest increase in the predicted number of antigens was obtained in the case when the interval between the introduction of the adsorption drug and the concentrated increase in the number of antigens was the largest. It means a decrease in the effectiveness of the corresponding therapy. Let us also note that the results presented concern cases of one-time introduction of adsorbents into the organism at the initial point in time. However, similar results will be expected even with repeated introduction of adsorption drugs.
Based on the synthesis of approaches to modeling adsorption in the medium of porous particles and the ideas of perturbation theory, the basic model of the immune response to viral pathogens is generalized, that considers the effect of adsorption therapy in conditions of diffusion perturbations. We have formed a step-by-step computational technology for asymptotically approximating the solution as a perturbation of corresponding solutions to the degenerate problems by reducing the original model singularly perturbed problem with delay to a sequence of problems without delay.
The results of computer modeling illustrate the effect of an additional reduction in the predicted number of antigens as a result of their removal and neutralization by adsorbents. It has been shown that the use of adsorption drugs, in particular during periods of disease exacerbation, provides an additional reduction in the supercritical amount of antigens and accordingly alleviates the disease course. At the same time, it is emphasized that an important condition for increasing the effectiveness of adsorption substances is the timeliness of their introduction into the organism. Taking into account features of adsorption influence is important for decision-making systems for the formation of complex treatment programs using adsorption drugs to increase the effectiveness of traditional therapeutic agents.
In our opinion, it is promising to adapt and expand the limits of application of the presented our approach to consider the conditions of competitive adsorption and diffusion, convection mass transfer, mixed infections and biostimulation. It is also promising to take into account random factors [13,14] and more complex function dependencies, which take into account the decrease in the immune system efficiency with a significant number of target organ cells affected by the virus, and conditions of quasiperiodic introduction of adsorption drugs.
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