Pulse HIV Vaccination: Feasibility for Virus Eradication and Optimal Vaccination Schedule

Abstract

We modify the classical virus dynamics model by incorporating an immune response with fixed or fluctuating vaccination frequencies and dosages to obtain a system of impulsive differential equations for the virus dynamics of both the wild-type and mutant strains. This model framework permits us to obtain precise conditions for the virus elimination, which are much more feasible compared with existing results, which require frequent vaccine administration with large dosage. We also consider the corresponding impulsive optimal control problem to describe when and how much of the vaccine should be administered in order to maximize levels of healthy CD4⁺ T cells and immune response cells. A gradient-based optimization method is applied to obtain the optimal schedule numerically. For a case study when the CTL vaccine is administered in a period of one year, our numerical studies support the optimal vaccination schedule consisting of vaccine administration three times, with the first dosage strong (to boost the immune system), followed by a second dosage shortly after (to strengthen the immune response) and then the third and final dosage long after (to ensure the immune system can handle viruses rebound).

Appendix

Proof of Lemma 3.1

By the continuity of solutions with respect to the initial values, ∀ϵ>0 there exists δ ₀>0 such that for all x ⁰∈X ₀ with ∥x ⁰−M ₁∥≤δ ₀, there holds ∥u(t,x ⁰)−u(t,M ₁))∥≤ϵ,∀t∈[0,τ]. We further claim that Eq. (5) holds. Assume, by contradiction, that (5) does not hold. Then we have

$$\limsup_{m\rightarrow \infty}d\bigl(P^{m}\bigl(x^{0} \bigr),M_{1}\bigr)< \delta_{0}, $$

for some x ⁰∈X ₀. Without loss of generality, we assume that d(P ^m(x ⁰),M ₁)<δ ₀, for all m≥0. It follows that

$$\bigl\|u\bigl(t,P^{m}\bigl(x^{0}\bigr)\bigr)-u(t,M_{1}) \bigr\| \leq \epsilon,\quad \forall t\in [0, \tau]. $$

For any t≥0, let t=mτ+t′, where t′∈[0,τ] and $m=[\frac{t}{\tau}]$ is the greatest integer less than or equal to $\frac{t}{\tau}$. Thus, we get

$$\bigl\|u\bigl(t,x^{0}\bigr)-u(t,M_{1}) \bigr\|= \bigl\|u \bigl(t',P^{m}\bigl(x^{0}\bigr)\bigr)-u \bigl(t',M_{1}\bigr) \bigr\|<\epsilon,\quad \forall t \geq 0. $$

Note that (T(t),T _w (t),V _w (t),T _r (t),V _r (t),C(t))=u(t,x ⁰). It then follows that T _w (t)<ϵ,V _w (t)<ϵ,T _r (t)<ϵ,V _r (t)<ϵ,∀t≥0. Then from the first and last equations of (1), we have

Consider an auxiliary system

$$ \everymath{\displaystyle} \begin{array}{l} \frac{dT}{dt}= \lambda-\delta_TT-\epsilon \beta_wT-\epsilon \beta_rT,\\[13pt] \frac{dC}{dt}= (2\epsilon\alpha-\delta_C)C,\quad t\neq i\tau,\\[13pt] C\bigl(i\tau^+\bigr)=C(i\tau)+\tilde{C},\quad t=i\tau. \end{array} $$

(20)

For any ϵ>0, system (20) admits a globally asymptotically stable solution $(\hat{T}(0,\epsilon), C^{*}(t,\epsilon))$, where $\hat{T}(0,\epsilon)=\lambda/(\delta_{T}+\epsilon \beta_{w}+\epsilon \beta_{r}), C^{*}(t,\epsilon)= \tilde{C}e^{(2\epsilon\alpha-\delta_{C})(t-n\tau)}/ (1-e^{(2\epsilon\alpha-\delta_{C})\tau}), t\in (i\tau, (i+1)\tau]$. Then for any ξ>0, there exists t ₃>0 such that $\hat{T}(t,\epsilon)\geq \hat{T}(0,\epsilon)-\xi, \hat{C}(t,\epsilon)\leq C^{*}(t,\epsilon)+\xi$ for t≥t ₃, $(\hat{T}(t,\epsilon), \hat{C}(t,\epsilon))$ is any solution of Eq. (20). Note that $\hat{T}(0,\epsilon) \rightarrow T_{0}, C^{*}(t,\epsilon)\rightarrow C^{*}(t)$ as ϵ→0. Then for any $\bar{\eta}>0$ there exists $\bar{\epsilon}>0$ such that $\hat{T}(0,\epsilon)\geq T_{0}-\bar{\eta}, C^{*}(t,\epsilon)\leq C^{*}(t)+\bar{\eta}$ for $\epsilon<\bar{\epsilon}$. It follows that for t≥t ₃ and ϵ small enough ($\epsilon<\bar{\epsilon}$)

It follows from Eq. (20) and comparison principles that for t≥t ₃ and ϵ small enough,

$$T(t)\geq \hat{T}(t,\epsilon)\geq T_0-\eta,\qquad C(t)\leq \hat{C}(t, \epsilon)\leq C^*(t)+\eta. $$

Consider the fourth and the fifth equations in system (1) with the nonnegativity of the solutions, there holds for t≥t ₃

$$ \everymath{\displaystyle} \begin{array} {l} T_r'(t) \geq \beta_r(T_0+\eta)V_r- \delta_IT_r-p_rT_r\bigl(C^*(t)+ \eta\bigr), \\[6pt] V_r'(t)= N_r\delta_IT_r- \delta_VV_r. \end{array} $$

(21)

Consider the corresponding comparison differential equations of system (21),

$$ \everymath{\displaystyle} \begin{array} {l} \hat{T}_r'(t)= \beta_r(T_0+\eta)\hat{V}_r- \delta_I\hat{T}_r-p_r\hat{T}_r \bigl(C^*(t)+\eta\bigr), \\[6pt] \hat{V}_r'(t)= N_r\delta_I \hat{T}_r-\delta_V\hat{V}_r, \end{array} $$

(22)

By Zhang and Zhao (2007) (Lemma 2.1), we know that there exists a positive, τ-periodic function p(t)=(p ₁(t),p ₂(t)), such that $e^{\mu_{2}t}p(t)$ is a solution of system (22), where $\mu_{2}=\frac{1}{\tau}\ln\rho(\varPhi_{F_{r}(\eta)-V_{r}(\eta)}(\tau))$. Since $\rho(\varPhi_{F_{r}(\eta)-V_{r}(\eta)}(\tau))$ is continuous for small η and $R_{0}^{r}>1$ indicates that $\rho(\varPhi_{F_{r}-V_{r}}(\tau))>1$, we can choose η small enough such that $\rho(\varPhi_{F_{r}(\eta)-V_{r}(\eta)}(\tau))>1$, that is μ ₂>0. Let t=nτ>t ₃, and n be nonnegative integer, we get

For any negative initial values (T _I (t ₃),V _I (t ₃))^T of system (21), there exits a sufficiently small z _∗>0, such that (T _r (t ₃),V _r (t ₃))^T≥z _∗(p ₁(0),p ₂(0))^T. By the comparison theorem, we have $(T_{r}(t),V_{r}(t))^{T}\geq z_{*}e^{\mu_{2}(t-t_{3})}(\hat{T}_{r}(t-t_{3}),\hat{V}_{r}(t-t_{3}))^{T}$, for all t≥t ₃. Thus, we obtain T _r (nτ)→∞,V _r (nτ)→∞, as n→∞, a contradiction. Hence, Eq. (5) holds. This completes the proof. □