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August  2020, 25(8): 2867-2893. doi: 10.3934/dcdsb.2020044

## Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection

 1 Colorado School of Mines, Department of Applied Mathematics and Statistics, 1500 Illinois St., Golden, CO 80401, USA 2 Los Alamos National Laboratory, Information Systems and Modeling, Los Alamos, NM 87544 USA

* Corresponding author: Stephen Pankavich

Received  August 2018 Revised  April 2019 Published  February 2020

Fund Project: The first author is supported by NSF grants DMS-1551229 and DMS-1614586

Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability analysis, which displays additional complexity in comparison to previous models, is conducted. The current study extends previous theoretical and computational work on the early stages of the disease and leads to interesting nonlinear dynamics, including a parameter region featuring bistability of infectious and viral clearance equilibria and the appearance of a Hopf bifurcation within biologically relevant parameter regimes.

Citation: Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044
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A representative simulation of (3CM) with parameter values given in Table 1, and a comparison with the full three-stage model in [11] - T-cell count (left) and viral load (right)
Regions of existence (i.e. real and positive values) for the uninfected state, $E_u$, and infected states, $E_i^{+}$ and $E_i^{-}$ in the $(R_m, R_0)$ plane
Regions of stability in the $(R_m, R_0)$ plane when $\alpha_1, \alpha_2, \beta$ are fixed to the values in Table 1
Changes to stability of equilibria given differing values of $\beta$ where $\beta^* = 0.73$ is the fitted value for $\beta$ in Table 1
Small changes in initial data, $T_0$ (top) and $V_0$ (bottom), yield changes in asymptotic behavior within the bistable parameter region - T-cell count (left) and viral load (right). Within both simulations, dimensionless parameters are fixed to $R_m = 0.6$ and $R_0 = 3$. Recall that $\overline{V}_u \equiv 0$ is the equilibrium viral load for $E_u$, and note that $V(t)$ is represented on a log scale
Simulation results for $R_m, R_0$ values within the bistable region. A simulation was conducted at each point in the displayed parameter space with initial conditions $T_0 = 1, I_0 = 0, V_0 = \overline{V}_+$, where $\overline{V}_+$ is the equilibrium viral load of the $E_i^+$ infective equilibrium evaluated at $(R_m, R_0)$. Green dots indicate solutions which tend toward the uninfected steady state $E_u$, while red crosses indicate solutions tending to $E_i^{+}$, the infected steady state
The dependence of equilibria on initial values of healthy T-cells and the initial viral load at arbitrary points in the $(R_m, R_0)$ plane. Corresponding results are shown in Figure 8
Basins of attraction within the bistable region. For each $(R_0, R_m)$ point, the initial T-cell count and viral load (both dimensionless) were set to the $E_i^+$ steady state evaluated at $(R_0, R_m)$. This forms the central point in each plot, and then both initial conditions are shifted $\pm 40\%$. Green dots denote simulation convergence to $E_u$, and red crosses to $E_i^+$. A visual summary of the points chosen in the bistable region is provided by Figure 7
Real and imaginary parts of eigenvalues $\eta_2$ and $\eta_3$ as functions of the bifurcation parameter $R_m$ with $R_0 = 1.25$ fixed. These curves cross the imaginary axis at $R_m = R_m^{*} \approx 9$ when using parameter values from Table 1
Simulation of the dimensionless system for $R_m = 6.5$ and $R_0 = 1.25$. The parameter location is in the (dark) red region of Figure 3. Hence, complex eigenvalues of the system linearized about $E_i^+$ are of the form $\alpha \pm i \beta$ where $\alpha < 0$, yielding a stable steady state. Green and red horizontal lines indicate corresponding population values of the $E_i^+$ and $E_u$ steady states, respectively
Simulation of the dimensionless system for $R_m = 9$ and $R_0 = 1.25$. The parameter location is at the border of the (dark) red and unshaded regions of Figure 3. Hence, complex eigenvalues of the system linearized about $E_i^+$ are of the form $\pm i \beta$, signaling a transition in the asymptotic behavior of solutions and the emergence of a stable orbit
Simulation of the dimensionless system for $R_m = 9.25$ and $R_0 = 1.25$. The parameters now lie within the unshaded region (Figure 3). Complex eigenvalues of the system linearized about $E_i^+$ are of the form $\alpha \pm i \beta$ with $\alpha > 0$ yielding instability of the equilibrium. With initial conditions taken near the bifurcation, a stable periodic orbit appears
Variables and Parameters
 Quantity Values / Initial Values References Original Populations $T$ Uninfected CD4$^+$ T-cells $1000\; \mathrm{mm}^{-3}$ [12] $I$ Infected CD4$^+$ T-cells $0\; \mathrm{mm}^{-3}$ [12] $V$ Wild-type HIV virions $10^{-2}\; \mathrm{mm}^{-3}$ [12] Dimensionless Populations ($^*$ omitted in exposition) $T^* = \frac{pk}{d_I d_V} T$ $T^*_0 = 1.81$ $I^* = \frac{pk}{d_T d_V} I$ $I^*_0 = 0$ $V^* = \frac{k}{d_T} V$ $V^*_0 = 4.57\times 10^{-5}$ $t^* = d_T t$ Original Parameters $\lambda$ Rate of supply of T-cells $10\; \mathrm{mm}^{-3}\; \mathrm{day}^{-1}$ [27] $\rho$ Maximum homeostatic growth rate $0.01\; \mathrm{day}^{-1}$ [12] $C$ Homeostatic half-velocity $300\; \mathrm{copies}\; \mathrm{mm}^{-3}$ [12] $k$ Infection rate $4.57\times10^{-5}\; \mathrm{mm}^{3}\; \mathrm{day}^{-1}$ [12,11] $d_T$ Death rate of uninfected T-cells $0.01\; \mathrm{day}^{-1}$ [11,12] $d_I$ Death rate of infected T-cells $0.40\; \mathrm{day}^{-1}$ [27,12] $p$ Rate of viral production $38\; \mathrm{virions}\; \mathrm{per\; cell}\; \mathrm{day}^{-1}$ [12,11] $d_V$ Clearance rate of free virus $2.4\; \mathrm{day}^{-1}$ [12,11] Dimensionless Parameters $R_0 = \frac{\lambda k p}{d_T d_I d_V}$ $1.81$ $R_m = \frac{\rho}{Ck}$ $0.73$ $\alpha_1 = \frac{d_I}{d_T}$ $40$ $\alpha_2 = \frac{d_V}{d_T}$ $240$ $\beta = \frac{d_T}{Ck}$ $0.73$
 Quantity Values / Initial Values References Original Populations $T$ Uninfected CD4$^+$ T-cells $1000\; \mathrm{mm}^{-3}$ [12] $I$ Infected CD4$^+$ T-cells $0\; \mathrm{mm}^{-3}$ [12] $V$ Wild-type HIV virions $10^{-2}\; \mathrm{mm}^{-3}$ [12] Dimensionless Populations ($^*$ omitted in exposition) $T^* = \frac{pk}{d_I d_V} T$ $T^*_0 = 1.81$ $I^* = \frac{pk}{d_T d_V} I$ $I^*_0 = 0$ $V^* = \frac{k}{d_T} V$ $V^*_0 = 4.57\times 10^{-5}$ $t^* = d_T t$ Original Parameters $\lambda$ Rate of supply of T-cells $10\; \mathrm{mm}^{-3}\; \mathrm{day}^{-1}$ [27] $\rho$ Maximum homeostatic growth rate $0.01\; \mathrm{day}^{-1}$ [12] $C$ Homeostatic half-velocity $300\; \mathrm{copies}\; \mathrm{mm}^{-3}$ [12] $k$ Infection rate $4.57\times10^{-5}\; \mathrm{mm}^{3}\; \mathrm{day}^{-1}$ [12,11] $d_T$ Death rate of uninfected T-cells $0.01\; \mathrm{day}^{-1}$ [11,12] $d_I$ Death rate of infected T-cells $0.40\; \mathrm{day}^{-1}$ [27,12] $p$ Rate of viral production $38\; \mathrm{virions}\; \mathrm{per\; cell}\; \mathrm{day}^{-1}$ [12,11] $d_V$ Clearance rate of free virus $2.4\; \mathrm{day}^{-1}$ [12,11] Dimensionless Parameters $R_0 = \frac{\lambda k p}{d_T d_I d_V}$ $1.81$ $R_m = \frac{\rho}{Ck}$ $0.73$ $\alpha_1 = \frac{d_I}{d_T}$ $40$ $\alpha_2 = \frac{d_V}{d_T}$ $240$ $\beta = \frac{d_T}{Ck}$ $0.73$
Calculated $ARE$ of each parameter with $N = 1000$ trials
 Noise level $\delta$ in $\%$ Calculated $ARE$ in $\%$ of fitted value $\alpha_1$ $\alpha_2$ $\beta$ $R_0$ $R_m$ 5 2.6066 5.1185 8.5814 3.2080 4.1637 10 3.5020 6.6600 14.9092 4.7260 6.0953 15 4.2652 7.4536 20.0601 6.5109 8.0292 20 4.6269 8.5138 24.2512 7.7800 8.6645 25 5.2935 9.6199 27.6901 5.5697 9.7956 30 5.6840 9.9405 30.8474 9.9440 10.9832
 Noise level $\delta$ in $\%$ Calculated $ARE$ in $\%$ of fitted value $\alpha_1$ $\alpha_2$ $\beta$ $R_0$ $R_m$ 5 2.6066 5.1185 8.5814 3.2080 4.1637 10 3.5020 6.6600 14.9092 4.7260 6.0953 15 4.2652 7.4536 20.0601 6.5109 8.0292 20 4.6269 8.5138 24.2512 7.7800 8.6645 25 5.2935 9.6199 27.6901 5.5697 9.7956 30 5.6840 9.9405 30.8474 9.9440 10.9832
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