Interntionl Journl of Biomedicl Mterils Reserch 8 6(): -7 http://www.sciencepublishinggroup.com/j/ijbmr doi:.648/j.ijbmr.86. ISSN: 33-756 (Print) ISSN: 33-7579 (Online) Anlysis of Regultory of Interrelted Activity of Heptocyte nd Heptitis B Viruses Mohiniso Bxromovn Hidirov *, Abrorjon Mxmtsoliyevich Turgunov Scientific nd Innovtion Center of Informtion nd Communiction Technologies, Tshkent University of Informtion Technologies Nmed After Muhmmd Al-Khwrizmi, Tshkent, Uzbekistn Emil ddress: mhidirov@yndex.ru (M. B. Hidirov), brorjon-7@mil.ru (A. M. Turgunov) * Corresponding uthor To cite this rticle: Mohiniso Bxromovn Hidirov, Abrorjon Mxmtsoliyevich Turgunov. Anlysis of Regultory of Interrelted Activity of Heptocyte nd Heptitis B Viruses. Interntionl Journl of Biomedicl Mterils Reserch. Vol. 6, No., 8, pp. -7. doi:.648/j.ijbmr.86. Received: October 5, 7 Accepted: November 6, 7 Published: Jnury 6, 8 Abstrct: In this rticle, we will present the results of the stbility nlysis of the equilibrium point of the mthemticl model of the regultory of the interrelted ctivity of the heptocyte nd heptitis B viruses. The nlysis of this model used the conditions of the Hyes criterion. In this study, the generl condition of the Hyes criterion is obtined. If the generl condition of the Hyes criterion is stisfied, then the equilibrium point is stble. If the generl condition of the Hyes criterion is not fulfilled, then the equilibrium point is not stble, nd hence thiscn describe modes "limit cycle", "chos" nd "blck hole" mthemticl models of the interrelted ctivity of the liver cell nd heptitis B viruses. The results of the computtionl experiment on the quntittive nlysis of the regultory of liver cell nd HBV re presented. Keywords: Regultory, Mthemtics Model, Equilibrium Points, Stbility, Qulittive nd Quntittive Anlysis. Introduction Heptitis B is virl infection tht ffects the liver. According to WHO estimtes, 4 million people re chroniclly infected with the heptitis B virus (HBV) []. Currently, this virus remins the third most common fter diseses like crdiovsculr system nd oncologicl pthologies []. Approximtely 9.4% of deths in recent yers hve been cused by infections, nd this is grdully incresing. The most dedly virus on the erth ws vrious strins of the heptitis virus, which kills more thn.3 million people every yer. The mortlity from heptitis incresed by bout % compred with HIV, tuberculosis nd mlri. One of the resons for this is tht people do not even know tht they hve virus. According to reserchers, only 5% of developed nd developing countries re wre of their dignosis nd tretment in medicl institutions [3]. Heptitis B virus (HBV) is one of the smllest enveloped DNA viruses tht cuses cute nd chronic infections. HBV is ble to evde the immune system of the host nd persist lifelong within infected heptocytes. During ctive repliction, HBV produces enormous virl lods in the blood [4]. Therefore, it is necessry to study the functioning of heptitis B in liver cell using methods of mthemticl nd computer modeling.. Mterils nd Methods Mthemticl modeling nd model nlysis of the dynmics of the heptitis B virus re very importnt for the study of regultory mechnisms nd the dynmic behvior of the process of virl infection. Mny scientific ppers hve proposed mthemticl models describing the dynmics of virl heptitis B in the liver cell. In these studies, mthemticl modeling plys n importnt role in understnding nd quntifying the biologicl mechnisms tht control the dynmics of the heptitis B virus. Abu O. nd Onlo S. E. considered mthemticl model of the dynmics of trnsmission of the heptitis B virus, which includes vccintion nd tretment s control prmeters. With the use of vlues of model prmeters, the properties of the disese-free nd the endemic equilibrium were numericlly studied [5].
Mohiniso Bxromovn Hidirov nd Abrorjon Mxmtsoliyevich Turgunov: Anlysis of Regultory of Interrelted Activity of Heptocyte nd Heptitis B Viruses Moneim I. A. nd Khlil H. A. studied the globl behvior of HBV spred using the SEIR model with constnt vccintion rte. Infectivity during the incubtion period is considered s the second mode of trnsmission [6]. These mthemticl models describe the dynmics of the heptitis B virus in the liver cell, t the cellulr level. In ccordnce with the biologicl regulrity of these processes, the development of n infectious disese occurs in the reltionship between the genomes of the heptitis B virus nd liver cell. In the modeling of the regultory of liver cell nd heptitis B virus, functionl-differentil equtions were used. Equtions re built on the bsis of firmly estblished biologicl fcts nd regulrities. A mthemticl model for the functioning of the regultory of the heptocyte (liver cell) nd HBV t the moleculr-genetic level hs been developed by B. N. Hidirov [7, 8], nd cn be presented in the following form [9-]: dx ( t ) X ( t ) = X( t) dt + X ( t ) + c ( t ) d( t) bx( t ) ( t ) = ( t) dt + dx ( t ) + ( t ) X( t) = ϕ( t) ( t) = ϕ( t) t t t t + where X( t ), ( t ) the vlues chrcterizing the ctivity of the moleculr genetic systems of the liver cell nd heptitis B viruses, respectively,,, b, c, d prmeters regultory of the model of the heptocyte nd HBV ϕ( t), ϕ ( t) continuous functions on t t + t the beginning of reserch All prmeters re positive. This model describes the dynmics of the interrelted ctivity of liver cell nd heptitis B viruses t the moleculrgenetic level. The system of functionl-differentil equtions () is nonliner system nd is closed system. Therefore, we qulittively investigte the equilibrium position of the (). To nlyze the stbility of the equilibrium position of (), it is necessry to look t the equilibrium position of () round the equilibrium position [3]. By introducing smll vrible chnges nd we hve the following X( t) = X + x( t) ( t) = + y( t) X( t ) = X + x( t ) ( t ) = + y( t ), dx( t) ( X + x( t )) dt + ( X + x( t )) + c ( + y( t )) dt + d( X + x( t )) + ( + y( t )) = ( X + x( t)) dy( t) b( X + x( t ))( + y( t )) = ( + y( t)). () () With the implementtion of the simplifiction (), we get the following eqution: dx( t) c = X x( t ) y( t ) x( t) dt (3) dy( t) d = x( t ) y( t ) y( t). dt + X b This (3) is linerized eqution for smll round equilibrium positions (). Let us find the chrcteristic equtions for the linerized (3). To obtin the chrcteristic equtions, we introduce the bove expression on (4): λ c X e λ e X b λ d e λ e λ λ b X nd we hve the chrcteristic equtions (4). If then λ X e λ = λ e λ =, =, (4) =, X bc we obtin (5) nd (6). First, for the nlysis of (5), using the conditions of the Hyes criterion [4], it is necessry to rrive t the λ trnscendentl eqution ( λ + ) e + b = : (5) (6) λ λ + e + X =. (7) For the (7) we pply the condition of the Hyes criterion: >. Given the positive vlue of the generl prmeters (), then > is dischrged. + X >. In this cse we obtin X >. X ξsinξ cosξ <.
Interntionl Journl of Biomedicl Mterils Reserch 8 6(): -7 3 Where ξ - root of the eqution ξ = tgξ for, < ξ < π. According to condition ξ = tgξ, y = ξ, y = tgξ. the following generl conditions: < X <. (8) We continue the nlysis of (6) using the conditions of Hyes' criterion, s in (5), nd it is necessry to rrive t the λ trnscendentl eqution ( λ + ) e + b =. λ λ + e + =. (9) For the (9) we pply the condition of the Hyes criterion: >. Figure. Fulfillment of the third condition of the Hyes criterion. Due to is.3.), ξ π, π, ξ, π X. <. Herewith π >. X. < Given the positive vlue of the generl prmeters (), then > is stisfied. if y + >. Wherein X >. < ξsinξ cosξ. Where ξ - root of the eqution ξ = tgξ t < ξ < π,. According to condition ξ = tgξ, y = ξ, = tgξ. By simplifying X <, we get X <. Reducing the conditions of Hyes criteri > X > π > X <. To nlyze the condition of the Hyes criterion, we hve Figure. Fulfillment of the third condition of the Hyes criterion. π, ξ,
4 Mohiniso Bxromovn Hidirov nd Abrorjon Mxmtsoliyevich Turgunov: Anlysis of Regultory of Interrelted Activity of Heptocyte nd Heptitis B Viruses Due to is π <.. Herewith π >., ξ π, <. By simplifying <, we get b X <. Thus, we give the condition of the Hyes criterion:.) >.) X > π.3.) >.3.) b. X < To nlyze the condition of the Hyes criterion, we hve the following generl conditions: < < b. () X Thus, if the fulfillment of the generl conditions of the Hyes criterion (8) nd () is then stble to the equilibrium position (). Otherwise, it is not stble. This describes tht you cn observe the following regimes of functioning of the regultory of the interrelted ctivity of the moleculr genetic systems of heptocyte nd HBV: Poincre type limit cycles, chos nd "blck hole" effects. To crry out computtionl experiments, we creted computer model on the bsis of equtions () using the Runge-Kutt method [5, 6]. The following expressions re used to perform the clcultion [7]: nd X( t ) t ti < t ( σi X( t j ) + σi X( t j+ ))/ h, X( t i ) = σi = ( ti tj), σi = h σi t tj < ti < t j + j =,,..., i i =,,..., N, ( t ) t ti < t ( σi ( t j ) + σi ( t j+ ))/ h, ( t i ) = σi = ( ti tj), σi = h σi t tj < ti < t j + j =,,..., i i =,,..., N. () () The vlues of X( t ) nd ( t ) in the mthemticl model of the regultory of the heptocyte nd HBV t the moleculr-genetic level cn be clculted using expressions () nd (). 3. Results Computtionl experiments on the quntittive nlysis of the regultory of the heptocyte nd HBV show the presence following regime: Figure 3. The regime of dominnt functioning of the regultory of heptocyte nd heptitis B viruses.
Interntionl Journl of Biomedicl Mterils Reserch 8 6(): -7 5 Figure 3 shows the dominnt functioning of the genetic system of the heptocyte with the following vlues of the prmeters ( =., =.5, = 4, b =.9, c = 3.6, d =.8 nd X = 3.9, = 8), thus the ctivtion of only the moleculr genetic system of the heptocyte with loss of HBV ctivity is described. The new virus will stop growing in the liver cell nd the liver will be helthy. Figure 4. The regime of the limit cycle of regultory of heptocyte nd heptitis B viruses. Figure 4 shows the limit cycle of the functioning of the genetic system of the heptocyte with the following vlues of the prmeters ( =., =.5, = 7, b = 3.5, c =.6, d =. nd X =, = 6). This describes the ctivtion of the moleculr genetic system of heptocyte nd HBV, s well s the onset of n infectious disese in the liver. Figure 5. The regime of the chos regultory of heptocyte nd heptitis B viruses.
6 Mohiniso Bxromovn Hidirov nd Abrorjon Mxmtsoliyevich Turgunov: Anlysis of Regultory of Interrelted Activity of Heptocyte nd Heptitis B Viruses Figure 5 shows the regime of the chos regultory of moleculr genetic systems of liver cell nd heptitis B viruses with the following vlues of prmeters ( =., =.4, = 7.9, b = 4., c =.5, d =.5 nd X = 3.9, = 6). The results in the figure show tht the irregulr functioning of the moleculr genetic systems of the liver cell nd HBV, nd this describes the ctive infectious disese of virl heptitis B in the liver. Figure 6. The regime of the "blck hole" regultory of heptocyte nd heptitis B viruses. In figure 6 shows the regime of the "blck hole" regultory of moleculr genetic systems of liver cell nd heptitis B viruses with the following vlues of prmeters ( =., =.4, =., b =.6, c = 4.6, d = 5.758 nd X = 3.9, = 6). The obtined results in the figure show tht the functioning of the moleculr genetic systems of the liver cell nd HBV tends to zero nd destroys the body. 4. Conclusions Thus, the developed mthemticl nd computer models for the study of the functioning of the heptocyte nd HBV regultory mechnisms llow one to ssess the stte of the interrelted ctivity of the moleculr genetic systems of the liver cell nd heptitis B viruses Estblish the moleculr genetic bsis of pthogenesis To ssess nd predict the occurrence of chrcteristic stges of the course of the disese with virl heptitis B. Computtionl experiments cn be performed using the computer model developed. Bsed on the results of computtionl experiments, it is possible to develop prmetric portrit of the functioning of regultory mechnisms of the interrelted ctivity of heptocyte nd heptitis B viruses. The prmetric portrit includes ll the regimes of the regultory of liver cell nd heptitis B viruses. The prmetric portrit llows nlyzing the regimes of the interrelted ctivity of liver cell nd heptitis B viruses t the moleculr genetic level. References [] World Helth Orgniztion (6). Fct Sheet, July, Avilble t http://www.who.int/topics/heptitis/en/. [] Perov I. G., Khludeev O. I. (5). Mngement of the development of infectious diseses. System of informtion boxes. (6): 74-76. [3] Lncet 7. 39: 5. http://www.thelncet.com/journls/lncet/rticle/piis4-6736 (7)35-9/fulltext. [4] Sunbul M. (4). Heptitis B virus genotypes: globl distribution nd clinicl importnce. World J. Gstroenterology. (8): 547-5434. [5] Abu O., Onlo S. E. (7). Numericl Anlysis of Mthemticl Model of Heptitis B Virus Trnsmission Dynmics in the Presence of Vccintion nd Tretment. Journl of Scientific nd Engineering Reserch. 4 (9): 95-3. [6] Moneim I. A., Khlil H. A. (5). Modeling nd Simultion of the Spred of HBV Disese with Infectious Ltent. Applied Mthemtics. 6: 745-753. http://dx.doi.org/.436/m.5.657 [7] Hidirov B. N., Turgunov A. M. (). Modeling of moleculr genetics mechnisms of control of virl heptitis B. Uzbek journl of the Problems of Informtics nd Energetics. -3: 3-8. [8] Sidliev M., Hidirov M. B., Turgunov A. M. (4). Ares of homogeneous solutions of the equtions of the mthemticl model of the regultory of liver in heptitis B. Uzbek journl of the Problems of Informtics nd Energetics. 6: 3-8.
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