Optimal control for an age-structured model for the transmission of hepatitis B

Abstract

One of the characteristics of HBV transmission is the age structure of the host population and the vertical transmission of the disease. That is the infection is transmitted directly from infected mother to an embryo, fetus, or baby during pregnancy or childbirth (the perinatal infection). We formulated an age-structured model for the transmission dynamics of HBV with differential infectivity: symptomatic and asymptomatic infections. The model without intervention strategies is completely analyzed. We compute the basic reproduction number which determines the outcome of the disease. We also compute equilibria and study their stability. The sensitivity analysis of the initial model parameters is performed (to determine the impact of control-related parameters on outbreak severity). Using optimal control theory, we determine the cost-effective balance of three interventions methods which minimizes HBV-related deaths as well as the costs associated with intervention.

Appendices

Proof of Claim 2.3.2

The positive operator T(0) has the Perron-Frobenius properties, roughly speaking, T(z) is irreducible and $r\left( T(z)\right)$ is decreasing for real $z\in (-v_{min},+\infty )$. Moreover, $\lim _{z\rightarrow -v_{min}}r\left( T(z)\right) =+\infty$ and $\lim _{z\rightarrow +\infty }r\left( T(z)\right) =0$; then the first half of the proposition is the direct consequence of this monotonicity of $r\left( T(z)\right)$. Next we show the dominant property of $z_0$. For any $z\in \varSigma ^0{\setminus }\{z_0\},$ there is an vector $\psi _z$, such that $T(z)\psi _z=\psi _z$. Then we have $|\psi _z|=|T(z)\psi _z| \le T(R_ez)|\psi _z|$. The eigenspace corresponding to the eigenvalue $r\left( T(R_ez)\right)$ is one-dimensional subspace of $\mathbb {R}^4$ spanned by a strictly positive functional $F_{R_ez}$. We obtain that

$$\begin{aligned} r\left( T(R_ez)\right) [F_{R_ez}, |\psi _z|]= [F_{R_ez}, T(R_ez)|\psi _z|]\ge [F_{R_ez},|\psi _z|], \end{aligned}$$

where we write the value of $F_{R_ez}$ at $\psi _z$ as $[F_{R_ez},\psi _z]$. Hence we have $r\left( T(R_ez)\right) \ge 1$ and $R_ez\le z_0$ because $r\left( T(z)\right)$ is strictly deceasing for $z\in (-\mu _1,+\infty )$ and $r\left( T(R_ez_0)\right) =1$. This end the proof of Claim 2.3.2.

Proof of Theorem 4

The coordinates of $\mathbf {E}^*$ satisfied

$$\begin{aligned} s(a)= & {} \theta b(1-\nu C) e^{-\int _0^a (\beta (\sigma )(I+\tau C)+p(\sigma ))d\sigma }\nonumber \\&+\psi \int _0^a v(\eta ) e^{-\int _\eta ^a (\beta (\sigma )(I+\tau C)+p(\sigma ))d\sigma }d\eta ,\end{aligned}$$

(45)

$$\begin{aligned} L_i= & {} \frac{I+\tau C}{\mu _1+\gamma } \int _0^\omega \beta (a)\alpha (a)l(a)h(I,C,a)da, \nonumber \\ L_c= & {} \frac{I+\tau C}{\mu _1+\delta } \int _0^\omega \beta (a)(1-\alpha (a))l(a)h(I,C,a)da+ \frac{b\theta \nu C}{\mu _1+\delta }, \nonumber \\ I= & {} \frac{\gamma (I+\tau C)}{(\mu _1+\gamma )(\mu _1+\mu _I+\gamma _1)} \int _0^\omega \beta (a)\alpha (a)l(a)h(I,C,a)da, \end{aligned}$$

(46)

$$\begin{aligned} C= & {} \frac{\delta (I+\tau C)}{(\mu _1+\delta )(\mu _1+\mu _c+\gamma _2)} \int _0^\omega \beta (a)(1-\alpha (a))l(a)h(I,C,a)da\nonumber \\&+\frac{\delta b\theta \nu C}{(\mu _1+\delta )(\mu _1+\mu _c+\gamma _2)}, \nonumber \\ v(a)= & {} b(1-\theta ) e^{-\psi a}+\int _0^a p(\eta ) s(\eta ) e^{-\psi (a-\eta )}d\eta ,\nonumber \\ R= & {} \frac{\gamma _1I+\gamma _2C}{\mu _1}. \end{aligned}$$

(47)

wherein h(I, C, a) is the right-hand side of (45).

Using Eqs. (46) and (47) we have the following fixed point equation $H(I,C)^T=(I,C)^T;$ where $H(I,C)^T= \left( \begin{array}{c@{\quad }c} H_1(I,C), H_2(I,C) \end{array} \right) ^T$ and $H_1(I,C)$; $H_2(I,C)$ are respectively the right-hand side of Eqs. (46) and (47).

Thus the equilibrium points are fixed points of H given by

$$\begin{aligned} H(I,C)^T=(I,C)^T. \end{aligned}$$

(48)

The Eq. (48) implies that at the endemic steady state the infected population simply reproduce itself. Therefore we can call H the next generation operator at the endemic steady state. This fact is used to show the stability of the endemic steady state in Sect. 2.3.3.

We use (48) to prove existence and uniqueness of an endemic equilibrium point. Then we use a theorem for the existence and uniqueness of a positive fixed point of a multi-variable function (see Hethcote and Thieme 1985, Theorem 2.1).

In fact H(I, C) is continuous, bounded function. Since $h(0,0,.)=s^0(.)$ (the disease-free steady state) and H infinitely differentiable, then the Jacobian at point (0, 0) is given by

$$\begin{aligned} H'(0,0)=\left( \begin{array}{c@{\quad }c} \frac{\gamma \mathscr {K}_i}{(\mu _1+\gamma )(\mu _1+\mu _I+\gamma _1)} &{} \frac{\gamma \mathscr {K}_i}{(\mu _1+\gamma )(\mu _1+\mu _I+\gamma _1)} \\ &{} \\ \frac{\delta \mathscr {K}_c}{(\mu _1+\delta )(\mu _1+\mu _c+\gamma _2)} &{} \frac{\delta (\mathscr {K}_c+b\theta \nu )}{(\mu _1+\delta )(\mu _1+\mu _c+\gamma _2)} \\ \end{array} \right) \end{aligned}$$

Thus the function H(I, C) is monotone non-decreasing and $H(0,0)=(0,0)$. Note that $\rho (H'(0,0))=\mathscr {R}_0>1$. Thanks the graph theory, we claim that $H'(0,0)$ is irreducible because the associated graph of the matrix is strongly connected.

Let us now prove that H is strictly sub linear, i.e., $H(rI,rC)>rH(I,C)$, for any $(I,C)>0$ and $r\in (0,1)$. For instance

$$\begin{aligned} \frac{rH_1(I,C)}{H_1(rI,rC)}= \frac{r\int _0^\omega \beta (a)(1-\alpha (a))l(a)h(I,C,a)da }{\int _0^\omega \beta (a)(1-\alpha (a))l(a)h(rI,rC,a)da } \le r <1; \end{aligned}$$

and the same argument gives that $\frac{rH_2(I,C)}{H_2(rI,rC)}<1$. In this way we end the proof of Theorem 4.

Proof of Lemma 6

• 1. Let us set $w^h:=(S^h,V^h,L_i^h,I^h,L_c^h,C^h)$ and the same for $w^v$. Using the Volterra integral formulation and system (31), we find that

  $$\begin{aligned}&||S^{h}-S^{v}||_{L^1(\mathscr {Q}_1)}+ ||V^{h}-V^{v}||_{L^1(\mathscr {Q}_1)}\le T_fC_4( ||w^{h}-w^{v}||_{L^1(\mathscr {X})}\\&\quad +||{h}-{v}||_{L^1(\mathscr {Q})}+ ||h_1(.,0)-v_1(.,0)||_{L^1(\mathscr {Q}_2)}). \end{aligned}$$

  We also find that

  $$\begin{aligned}&||(L_i^h,I^h,L_c^h,C^h)-(L_i^v,I^v,L_c^v,C^v)||_{L^1(\mathscr {Q}_2)} \le T_fC_5 (||w^{h}-w^{v}||_{L^1(\mathscr {X})}\\&\quad +||{h}-{v}||_{L^1(\mathscr {Q})}+||h_1(.,0) -v_1(.,0)||_{L^1(\mathscr {Q}_2)}). \end{aligned}$$

  Then, for $T_f$ sufficiently small,

  $$\begin{aligned} ||w^{h}-w^{v}||_{L^1(\mathscr {X})}\le T_f C_1 ||{h}-{v}||_{L^1(\mathscr {Q})}. \end{aligned}$$

  The same arguments can be apply for the second estimate of item 1. and for item

• 2. It remains to check item 3.

• 3. We suppose that $h_n:=(h_{1n},h_{2n},h_{3n})\rightarrow h:=(h_1,h_2,h_3)$ in $L^1(\mathscr {Q}$. Possibly along a subsequence (using the same notation), $h_n^2\rightarrow h^2$ a.e. on $\mathscr {Q}$ by (see Evans and Gariepy 1992, p. 21). By Lebesgue’s dominated convergence theorem, it comes $\lim _{n\rightarrow \infty } ||h_n^2||_{L^1(\mathscr {Q})}= ||h^2||_{L^1(\mathscr {Q})}$. We have the similar arguments for $||v^2||_{L^1(\mathscr {Q})}$. These handle the convergence of the squared terms in our functional.

Next, we illustrate the convergence of one term in the functional,

Therefore,

$$\begin{aligned} \left| \mathscr {J}(h_n)-\mathscr {J}(h)\right| \le C_7(T_f) ||{h_n}-h|| _{L^1(\mathscr {Q})}. \end{aligned}$$

Hence we have the lower semi-continuity, $\mathscr {J}(h)\le \liminf _{n\rightarrow \infty } \mathscr {J}(h_n)$.