Dynamical Analysis Of Viral Dynamic Models With Immune Response And Holling Ⅱ Incidence Rate

The study of the host body’s viral dynamic mainly is focused on persistent viral infection. After the virus invade the host, the host will instinctively evoke immune response. CTL immune and antibody immune play a key role in immune response. The virus will be erased or destroyed in the host body through the immune response. At the same time, it which is multiform functional reaction function in population dynamics, enlightens us to structure multiform contact rate in different diseases and environments. Thereby, when we study the viral dynamic model with immune response, in order to describe the dynamic characteristic of relevant viral infection and immune response as well as possible we will instead of bilinear incidence with functional reaction function. Meanwhile, it can also describe biological background, explain practical disease prevention and therapeutic strategy. In this paper, we built four kinds of viral dynamic models with immune response and Holling Ⅱ incidence rate, and discussed their dynamical behaviors and biological significances.In section1, we introduce viral infection, immune response, functional reaction function and research progress of viral dynamic models with immune response and Holling Ⅱ incidence rate.In section2, we discussed dynamical analysis of viral dynamic model with latent stage and Holling Ⅱ incidence rate. By means of the Lyapunov function and LaSalle invariance principle, we prove that, if R01≤1, then the disease-free equilibrium is globally asymptotically stable and the virus will be cleared in host; if R01>1, the endemic equilibrium is globally asymptotically stable, the viral can successfully invade the host.In section3, we discussed dynamical analysis of viral dynamic model with latent stage, antibody immune response and Holling Ⅱ incidence rate. By means of the Lyapunov function and LaSalle invariance principle, we prove that, if R01≤1, then the disease-free equilibrium is globally asymptotically stable and the virus will be cleared in host; if R01>1, R02≤1, the immune-free equilibrium is globally asymptotically stable and the viral can successfully invade the host, but antibody immune response can’t be built in host; if R02>1, t he en d em i c eq u i l i b r i u m i s gl ob a l ly a sym p tot i ca ll y st a ble, the vi r al can su cc es s fu l l y i n v a d et he ho st, an d an t i b o dy i m m un e r esp ons e can be bui lt i n host.I n se ct i o n4, w e disc u ss e d dyn am i ca l an a ly si s of vi r a l d y na m i c mo d el wi t h l at en tst ag e, C T L i mm u ne r esp on se a nd Ho l l in g Ⅱi nc id en ce r at e. B y mea n s o f t h e Lya p u no vf u n ct i o n a n d L a S al le inv ar ia nce pr i nc ip le, we pr ov e tha t, if R01≤1, t he n t h e di se as e-fr ee eq u i l i b r i u m i s gl o b al ly as ymp to ti ca ll y st a ble an d the vir u s wi l l be cl ea r e d i n hos t; i f R01>1,R03≤1, t h e i m m u ne-f r e e eq ui li br iu m is gl ob al ly as y m pt oti c a l l y st ab l e an d t h e vi r al can su c ce ssf u l l y i n v a d e the hos t, but CT L i m m un e r esp o nse ca n’t be b u i l t i n ho st; i f R03>1, t h een d em i c eq u i l i b ri u m i s gl ob a l ly a sym p tot i ca ll y st a ble, the vir al can su cc es s f u l l y i n v a d e t heh o s t, an d C T L i m mu ne re sp ons e can be bui lt i n host.I n se ct i o n5, w e disc u ss e d dyn am i ca l an a ly si s of vi r a l d y na m i c mo d el wi t h l at en tst ag e, an t i b o dy i mm u ne r esp o nse, C T L i m m un e re sp o n se a nd H o l l i n g Ⅱ i nc i d en ce ra t e. B yme an s o f t h e Lya p u no v fun ct io n a nd La S al l e i nva r i an ce pr i nc i p l e, w e pr o v e t h at, i f R01≤1,t he n t he d i s ea se-fr e e eq ui li br iu m i s gl ob al ly as y m pt ot ic al ly st a b l e a n d t h e v i r u s wi l l becl ear ed i n h o s t; if R01>1, R02≤1, R03≤1, the imm u ne-fr e e eq u i l i b r i u m i s gl o ba l l yas y m p t o t i c al l y st ab l e an d the vi ra l can suc ces s f ul ly i nva de t he ho st, bu t a n t i b o d y an d C T Li mm u n e r es p o n se can ’t be bu il t i n host; i f R02>1, R04≤1, a nti bo dy-d o mi n an t eq u i l i b ri u m i sg l o b a l l y as y m p t o t i c al ly st a ble, but a nti bo dy imm u ne r esp o nse can on l y be bu i l t i n h o st; i fR03>1, R’04≤1, CT L-dom i na nt equ il ib r i um i s gl ob a ll y a sym p t o t i c al l y st ab l e, b u t C T Li mm u n e r es p o n se ca n on l y be bu il t in ho st; if R04>1, R’04>1, po si t i ve eq u i l i b r i u m i s gl o ba l l yas y m p t o t i c al l y st ab le, a nti bo dy an d CT L imm u ne r esp on se ca n co nc u r re n t l y b e b u i l t i nh o s t.I n se ct i o n6, we co n cl ud e wi th disc u ss i o n for t he pap er.