Spatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activation

Abstract

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction number R0 and the CTL immune response reproduction number R1. The stability of the last equilibrium depends on R0 and R1 as well as time delay τ in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when τ passes through a certain critical value.

Mathematical Model Describing HIV Infection with Time-Delayed CD4 T-Cell Activation

“…The susceptible CD4 T cell population consists of resting T cells that have not been immunologically activated, and of activated T cells, also known as helper T-lymphocytes. The importance of this distinction is that HIV infects resting lymphocytes, but only 1% actively replicate, which indicates that, in 99% the proviral genome is housed and dormant [1,27,28,31,35].…”

“…A wide variety of mathematical models have been formulated to study HIV infection at the cellular level [2,3,18,20,21,23,24,26,28,[31][32][33][34][35]37,38,40,[46][47][48] as well as its spread in susceptible populations [7,10,22,39,44,45]. Such models approach the study of infection from different perspectives, and have made it possible to evaluate the effect that preventive measures or prophylaxis and diagnosis may have in reducing transmission [10,11,17,39,41,43], the effectiveness of antiretrovirals in controlling viral loads, in this case, optimal control models have been very important tools [20,23,31,[41][42][43][44]46].…”

“…One of the reasons for the slow development of the infection in people carrying the virus is the state of latency in which a proportion of the infected cells is found; in deed this cell population consists of resting T cells that have not been immunologically activated, different from the activated T cells, also known as helper T-lymphocytes. The importance of this distinction is that HIV infects resting lymphocytes, but only 1 % actively replicate viruses, which indicates that, in 99 % the proviral genome is housed and dormant [1][2][3][26][27][28]31,32,35,47,48]. Considering the above, and with the objective of studying CD4 T cell interaction with viral HIV particles, two mathematical models, based on ordinary differential equations, are proposed here, and will be studied from a qualitative perspective.…”

Mathematical Model Describing HIV Infection with Time-Delayed CD4 T-Cell Activation

“…A wide variety of mathematical models has been formulated to study HIV infection at cellular level [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], as well as its spread in uninfected populations [28][29][30][31][32][33]. Such models address the study of infection from different perspectives and have made it possible to evaluate the possible effect of preventive measures or prophylaxis and diagnosis on reducing transmission [29,31,[34][35][36][37].…”

“…One of the reasons for the slow development of the infection in people carrying the virus is the state of latency in which a proportion of the infected cells is found; indeed, this cell population consists of resting T cells that have not been immunologically activated, different from the activated T cells, also known as helper T-lymphocytes. The importance of this distinction is that HIV infects resting lymphocytes, but only 1% actively replicate viruses, which indicates that in 99% the pro-viral genome is housed and dormant [7][8][9][15][16][17][18]21,26,27,39].…”