Spatiotemporal Dynamics of a Generalized HBV Infection Model with Capsids and Adaptive Immunity

A generalized distributed delay model for hepatitis B virus infection with two modes of transmission and adaptive immunity: A mathematical study

“…$$ Using the hypotheses ( $$$ {A}_1 $$$)–( $$$ {A}_4 $$$), we have $$$ {\Phi}_2^{\prime }(H)>0 $$$ which implies that $$$ {\Phi}_2 $$$ is a strictly increasing function of $$$ H $$$. Now, we define the reproduction number for antibody immune response by $$$$ {R}_1:= \frac{a}{\sigma }{V}_1, $$$$ which provides a measure of the average number of antibodies activated by virus when CTL immune response has not been activated 3,6 . Here, $$$ {V}_1 $$$ represents the density of virions at the immune‐free equilibrium level, while other parameters $$$ a $$$ and $$$ \frac{1}{\sigma } $$$ respectively indicate the activation rate of antibody immune response and the average life expectancy of antibody immune cells.…”

“…System () is considered to be a generalized model describing HBV infection in vivo with both modes of infection transmission and three distributed delay terms. Of course, the biologically relevant assumptions ( $$$ {A}_1 $$$)–( $$$ {A}_4 $$$) on incidence functions are valid for a wide variety of commonly used incidence functions 3,16,23 . Also, the probability distributions used to describe distributed delays can take different forms such as gamma distribution and Dirac delta functions 23–25 .…”

“…Hattaf et al 5,6 extended the model of Yousfi et al 2 in order to describe the dynamics of other viral infections such as human immunodeficiency virus (HIV) that often causes acquired immunodeficiency syndrome (AIDS). More recently, Manna and Hattaf 3 proposed an immunological model that incorporates the intracellular HBV DNA‐containing capsids, three delays, adaptive immunity, and general incidence rate that covers the bilinear incidence rate, the standard incidence, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, and the Hattaf–Yousfi functional response. Recently, Elaiw and Al Agha studied the global dynamics of a generalized HBV infection model with capsids and adaptive immunity 7 …”

Spatiotemporal dynamics of a fractional model for hepatitis B virus infection with cellular immunity

“…Their work was an extension to the work of Manna and Chakrabarty [14]. Manna and Hattaf [15] studied an HBV infection model which contains two arms of immunity, three time delays, capsids, general incidence rate, and allow the movement of capsids and viruses by diffusion. Manna [16] investigated the role of the CTL immune response in a reaction-diffusion model of HBV with capsids.…”

Positivity Preserving Technique for the Solution of HIV/AIDS Reaction Diffusion Model With Time Delay

“…Delay is a comprehensive term used in physical models of epidemiology. Various authors have proposed several epidemic models of delay differential equations for the study of the control of the dynamics of infectious diseases with the help of the delay factor [7,19,[30][31][32][33][34][35][36][37][38]. Vaccination may be used as a delay factor for some diseases.…”