Repulsion Effect on Superinfecting Virions by Infected Cells

Abstract

In this paper, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed show that the repulsion of superinfecting virion promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, the experimental result of Doceul et al. (Science 327:873–876, 2010).

Appendices

Appendix 1

Proofs of Theorems 2.1 and 2.3

Proof of Theorem 2.1

Note that the system (5) is normally elliptic and triangular (in fact diagonal). According to Theorem 1 Amann (1989) or Theorems 14.4 and 14.6 in Amann (1993), (5)–(4) has a unique classical solution $(T,I,V)$ defined on $[0,\tau _0)\times \Omega$ such that

$$\begin{aligned} (T,I,V)\in {\mathbf C}([0,\tau _0),{\mathbf X}_+)\cap {\mathbf C}^{2,1}((0,\tau _0)\times \bar{\Omega },{{\mathbb {R}}}^3), \end{aligned}$$

where $\tau _0>0$ is the maximal value for interval of existence of the solution. The nonnegativity of the solution follows from Theorem 15.1 by Amann (1993). In order to show that $\tau _0= \infty$, by Theorem 5.2 in Amann (1989) and the nonnegativeness of the solution confirmed above, it suffices to prove that the solution $(T,I,V)$ is bounded above by some positive values.

From the $T$ and $I$ equations in (2), we see that

$$\begin{aligned} \frac{\partial }{\partial t}(T+I)&= D_T\Delta (T+I)+h(x)-d_T T-d_{I}I\\&\le D_T\Delta (T+I)+\bar{h}-d_m (T+I), \end{aligned}$$

where $\bar{h}=\max _{x \in \Omega }h(x)$ and $d_m=\min \{d_T,d_{I}\}$. By Lemma 1 in Lou and Zhao (2011), $\bar{h}/d_m$ is the globally attractive steady state for the scalar parabolic equations

$$\begin{aligned} \frac{\partial w(t,x)}{\partial t}&= D_T\Delta w(t,x) +\bar{h}-d_m w(t,x),\quad x\in \Omega , \,t>0, \\ \frac{\partial w(t,x)}{\partial \nu }&= 0, \quad x\in \partial \Omega ,\, t>0. \end{aligned}$$

The parabolic comparison theorem (Smith 1995, Theorem 7.3.4) implies that $T+I$ is bounded. This together with the nonnegativity of $T$ and $I$ further implies that both $T(t,x)$ and $I(t,x)$ are bounded. We assume $0\le T(t,x)\le T_M$, $0 \le I(t,x) \le I_M$.

Let $\bar{\gamma }=\max _{x\in \Omega }\{\gamma (x)\}$, and $V_M= \bar{\gamma } I_M/d_V$. For any given $I$, define the operator ${\mathcal {P}}$ by

$$\begin{aligned} {\mathcal {P}} V=V_t- \nabla \cdot (D_V(I) \nabla V) - \gamma (x) I + d_V V. \end{aligned}$$

For any solution $(T,I,V)$ of the system (2)–(3)–(4), we have ${\mathcal {P}}V=0$. On the other hand,

$$\begin{aligned} {\mathcal {P}} V_M= d_V V_M- \gamma (x) I \ge d_V V_M-\bar{\gamma }I_M=0 =PV. \end{aligned}$$

On the boundary $\partial \Omega$, we have $\frac{\partial V_M}{\partial \nu }=0$. Thus, $V=V_M$ is an upper solution of the $V$ equation in the system (2)–(3). By the comparison principle, we obtain that $V(t,x)\le V_M$. Therefore, the solution $(T,I,V)$ is bounded, and hence, it exists globally.

Proof of Theorem 2.3

Linearizing (2) at $\bar{E}=(\bar{T}, \bar{I},\bar{V})$ gives

$$\begin{aligned} \frac{\partial }{\partial t}u(t,x)= (\bar{\mathrm{D}} \Delta + \bar{\mathrm{A}})u(t,x), \end{aligned}$$

where

$$\begin{aligned} \bar{\mathrm{D}} =\left( \begin{array}{ccc} D_T &{} 0 &{} 0 \\ 0 &{} D_T &{} 0 \\ 0 &{} 0 &{} D_V(\bar{I}) \end{array} \right) ,\, \bar{\mathrm{A}}= \left( \begin{array}{ccc} -d_T-\beta \bar{V} &{} 0 &{} -\beta \bar{T} \\ \beta \bar{V} &{} -d_{I} &{} \beta \bar{T} \\ 0 &{} \gamma &{} -d_V \end{array} \right) , \quad u=\left( \begin{array}{c} u_1 \\ u_2 \\ u_3 \end{array} \right) . \end{aligned}$$

The corresponding characteristic polynomial of this linearized system is

$$\begin{aligned} \mid \lambda I + \bar{\mathrm{D}} k^2-\bar{\mathrm{A}} \mid =0, \end{aligned}$$

(27)

where $k$ is the wavenumber, $\lambda$ is the eigenvalue which determines temporal growth (Murray 2000). The positive steady state $\bar{E}$ is linearly stable if all eigenvalues have negative real parts.

Substituting the two matrices $\bar{\mathrm{A}}$ and $\bar{\mathrm{D}}$ into (27), we obtain

$$\begin{aligned} \left| \begin{array}{ccc} \lambda + D_T k^2+d_T +\beta \bar{V}&{} 0 &{} \beta \bar{T} \\ -\beta \bar{V} &{} \lambda + D_T k^2+d_{I} &{} - \beta \bar{T} \\ 0 &{} -\gamma &{} \lambda + D_V(\bar{I}) k^2+d_V \end{array} \right| =0, \end{aligned}$$

that is,

$$\begin{aligned} \lambda ^3 +b_1(k^2)\lambda ^2+b_2(k^2)\lambda + b_3(k^2)=0, \end{aligned}$$

(28)

where

$$\begin{aligned} b_1(k^2)&= (2D_T+D_V(\bar{I}))k^2+(d_T+d_{I}+d_V)>0,\\ b_2(k^2)&= [D_T^2 +2D_T D_V(\bar{I})]k^4\\&+ \,[D_T d_{I} +D_T (d_T +\beta \bar{V})+D_T d_V+ D_V(\bar{I})(d_T +\beta \bar{V})+ D_T d_V\\&+ \,D_V(\bar{I})d_{I}]k^2 +(d_T +\beta \bar{V})d_{I}+(d_T +\beta \bar{V})d_V >0,\\ b_3(k^2)&= D_T^2D_V({\bar{I}})k^6+[ D_T^2d_V + D_T D_V(\bar{I}) d_{I}+D_T D_V(\bar{I}) (d_T +\beta \bar{V})]k^4\\&+\, [D_T(d_T +\beta \bar{V})d_V+D_V(\bar{I})(d_T +\beta \bar{V})d_{I}]k^2\\&+\, \beta \bar{V}d_{I}d_V >0. \end{aligned}$$

$$\begin{aligned}&b_1(k^2)b_2(k^2)-b_3(k^2)\\&\quad =(2D_T+D_V(\bar{I}))[D_T^2 +2D_T D_V(\bar{I})]k^6\\&\quad \quad \quad +\, \{[(2D_T+D_V(\bar{I}))][D_T d_{I} +(D_T+D_V(\bar{I})) (d_T +\beta \bar{V})\\&\quad \quad \quad +\, 2D_T d_V+D_V(\bar{I})d_{I}]\\&\quad \quad \quad + \,[D_T^2 +2D_T D_V(\bar{I})](d_T+d_{I}+d_V)\}k^4 \\&\quad \quad \quad +\, \{(2D_T+D_V(\bar{I}))[(d_T +\beta \bar{V})d_{I}+(d_T +\beta \bar{V})d_V]\\&\quad \quad \quad +[D_T d_{I} +(D_T+D_V(\bar{I})) (d_T +\beta \bar{V})+2D_T d_V\\&\quad \quad \quad +D_V(\bar{I})d_{I}](d_T+d_{I}+d_V)\}k^2\\&\quad \quad \quad +\, (d_T+d_{I}+d_V)[(d_T +\beta \bar{V})d_{I}+(d_T +\beta \bar{V})d_V]\\&\quad \quad \quad - \,D_T^2D_V({\bar{I}})k^6-[ D_T^2 d_V + D_T D_V(\bar{I}) d_{I}+D_T D_V(\bar{I}) (d_T +\beta \bar{V})]k^4\\&\quad \quad \quad -\, [D_T(d_T +\beta \bar{V})d_V+D_V(\bar{I})(d_T +\beta \bar{V})d_{I}]k^2\\&\quad \quad \quad -\, \beta \bar{V}d_{I}d_V \end{aligned}$$

$$\begin{aligned}&\ge 4D_T^2 D_V(\bar{T})k^6+\{2D_T[D_T d_{I}\\&+\, D_V(\bar{I}) (d_T +\beta \bar{V})+2D_T d_V+ D_V(\bar{I})d_{I}]+[D_T^2+2D_T D_V(\bar{I})]d_T\}k^4\\&+\, \{D_T[(d_T +\beta \bar{V})d_{I}+(d_T +\beta \bar{V})d_V]\\&+ \,[D_T d_{I} +(D_T+D_V(\bar{I})) (d_T +\beta \bar{V})+2D_T d_V+D_V(\bar{I})d_{I}](d_T+d_{I}+d_V)\}k^2\\&+ \,(d_T+d_{I})[(d_T +\beta \bar{V})d_{I}+(d_T +\beta \bar{V})d_V]\\&> 0. \end{aligned}$$

By the Routh–Hurwitz Criterion, we know that all eigenvalues of (28) have negative real parts, and therefore, the positive steady state $\bar{E}$ is linearly stable if it exists.

Appendix 2

Proof of nonexistence of traveling wavefront solutions for $c\in (0,c^*)$.

The Jacobian matrix of (16) at $E'_0$ is

$$\begin{aligned} J_0=\left( \begin{array}{cccc} -1/c &{} 0 &{} -1/c &{} 0 \\ 0 &{} - \rho _1/c &{} 1/c &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} -\rho _2/D_0&{} \rho _3/D_0 &{} c/D_0 \\ \end{array} \right) . \end{aligned}$$

It has an eigenvalue $\lambda =-1/c$ which is negative for all $c>0$. So, we only need to consider other eigenvalues which are determined by

$$\begin{aligned} P(\lambda ):=\lambda ^3+a_1\lambda ^2+a_2\lambda +a_3=0, \end{aligned}$$

where

$$\begin{aligned} a_1=\frac{D_0\rho _1-c^2}{c D_0},\, a_2=-\frac{\rho _1+\rho _3}{D_0}<0,\,a_3=\frac{\rho _2-\rho _1\rho _3}{c D_0}>0. \end{aligned}$$

Since $P(0)=a_3>0$ and $P(-\infty )=-\infty$, $P(\lambda )=0$ has a negative root. By the Descarte’s rule of signs and by the Routh–Hurwitz criterion, the other two roots of $P(\lambda )=0$ are either positive and real or a pair of conjugate complex numbers. In the latter case, the complex eigenvalues imply the oscillations of solutions of (16) near $E_0'$, implying the $w$ and $v$ will take negative values (making solutions biologically meaningless), and thus, (16)–(17) cannot have positive solutions, meaning that (14) cannot have traveling wavefronts connecting $E_0$ and $\bar{E}$. Therefore, in order for (14) to have traveling wavefronts connecting $E_0$ and $\bar{E}$, it is necessary $P(\lambda )=0$ to have a pair of positive real roots (counting multiplicity).

Note that $P'(\lambda )=3\left( \lambda ^2+\frac{2a_1}{3}\lambda +\frac{a_2}{3}\right)$ and $P'(\lambda )=0$ has a unique positive root

$$\begin{aligned} \lambda ^*=\frac{1}{3}\left( -a_1+\sqrt{a^2_1-3a_2}\right) . \end{aligned}$$

Since $P(0)=a_3>0$ and $P'(0)=a_2<0$, we conclude that $P(\lambda )=0$ has two positive real roots if and only if $P(\lambda ^*)<0$. From $P'(\lambda ^*)=0$, we obtain that

$$\begin{aligned} {\lambda ^*}^2+\frac{2a_1}{3}\lambda ^*+\frac{a_2}{3}=0,\quad {\lambda ^*}^3+\frac{2a_1}{3}{\lambda ^*}^2+\frac{a_2}{3}\lambda ^*=0. \end{aligned}$$

Using these equations to simplify the form of $P(\lambda ^*)$, we obtain

$$\begin{aligned} P(\lambda ^*)&= \frac{a_1}{3}{\lambda ^*}^2+\frac{2a_2}{3}\lambda ^*+a_3\\&= \frac{2}{3}\left( a_2-\frac{a_1^2}{3}\right) \lambda ^*+a_3-\frac{a_1a_2}{9}\\&= \frac{1}{27}\left[ -2 \left( a^2_1-3a_2 \right) ^{3/2}+27a_3+2a^3_1-9a_1a_2\right] . \end{aligned}$$

It then follows that

$$\begin{aligned} P(\lambda ^*)&< 0 \Leftrightarrow 27a_3+2a^3_1-9a_1a_2\le 0, \mathrm{{OR}} \,\,\, 27a_3+2a^3_1-9a_1a_2\\&> 0\, \,\mathrm{{AND}}\,\, 4\left( a^2_1-3a_2\right) ^3>\left( 27a_3+2a^3_1-9a_1a_2\right) ^2;\\ P(\lambda ^*)&> 0\Leftrightarrow 27a_3+2a^3_1-9a_1a_2>0\, \,\mathrm{{AND}}\,\, 4\left( a^2_1-3a_2\right) ^3\\&< \left( 27a_3+2a^3_1-9a_1a_2\right) ^2. \end{aligned}$$

Let $Q_1(c):=27a_3+2a^3_1-9a_1a_2$, then $Q_1(c)=\frac{1}{D_0^3c^3}(d_0c^6 +d_1c^4+d_2c^2+d_3)$, where $d_0=-2,\, d_1=-3D_0(\rho _1-3\rho _3)$, $d_2=3D_0^2(9\rho _2-6\rho _1\rho _3+\rho ^2_1)>0$ and $d_3=2\rho ^3_1D_0^3>0$. By the Descarte’s rule of signs, $\bar{Q}_1(c):=d_0c^6+d_1c^4+d_2c^2+d_3=0$ has a unique positive root $c_0^*>0$. Since $\bar{Q}_1(0)=d_3>0$, we see that $\bar{Q}_1(c)>0$ if $0<c<c^*_0$, and $\bar{Q}_1(c)<0$ if $c>c^*_0$. Furthermore, $Q_1(c^*_0)=0,\, Q_1(c)>0$ if $0<c<c^*_0$, and $Q_1(c)<0$ if $c>c^*_0$.

Let $Q_2(c):=4\left( a^2_1-3a_2\right) ^3-\left( 27a_3+2a^3_1-9a_1a_2\right) ^2$, then $Q_2(c)=\frac{27}{D_0^4c^4}(b_0c^6+b_1c^4+b_2c^2+b_3)$ where $b_i,\, i=0,1,2,3$, are given by (18).

Note that $b_0>0,\, b_1>0,\, b_3<0$. Again by the Descarte’s rule of signs, $Q(c)$ given by (18) has a unique positive root $c^*>0$. Since $Q(0)=b_3<0$, we see that $Q(c)<0$ if $0<c<c^*$, and $Q(c)>0$ if $c>c^*$. Therefore, $Q_2(c^*)=0,\, Q_2(c)<0$ if $0<c<c^*$, and $Q_2(c)>0$ if $c>c^*$.

Note that $a_1^2(c)-3a_2>0$ for all $c>0$. Thus, $Q_2(c_0^*)=a_1^2(c_0^*)-3a_2>0$, implying $c^*<c^*_0$. In summary, we have obtained:

$$\begin{aligned} P(\lambda ^*)<0&\Leftrightarrow Q_2(c)\le 0, \,\,\mathrm{{OR}},\,\,\, Q_2(c)>0 \,\, \mathrm{{AND}}\,\,Q_1(c)>0\\&\Leftrightarrow c\ge c^*_0 \,\,(\text{ hence }\,\, c>c^*), \,\,\, \mathrm{{OR}},\,\,\, 0<c<c^*_0 \,\,\mathrm{{AND}}\,\, c>c^*\\&\Leftrightarrow c>c^*; \\ P(\lambda ^*)>0&\Leftrightarrow Q_2(c)>0 \,\, \mathrm{{AND}}\,\,Q_1(c)<0 \\&\Leftrightarrow 0<c<c^*_0 \,\,\mathrm{{AND}} \,\, c<c^*\\&\Leftrightarrow 0<c<c^*. \end{aligned}$$

Thus, for any $c\in (0, c^*)$, system (14) has no traveling wavefront solutions with speed $c$ that connects $E_0$ and $\bar{E}$.

From the definition of $Q(c)$, we obtain

$$\begin{aligned} Q(\sqrt{D_{0}\rho _{1}})=D_{0}^{3}\rho _1\left[ 4\rho _1(\rho _1+\rho _3)^3-27 (\rho _2-\rho _1\rho _3)^2\right] . \end{aligned}$$

It is easy to see that $Q_{3}(\rho _{1}):=4\rho _{1}(\rho _{1}+\rho _{3})^3$ is strictly increasing function of $\rho _1$, and $Q_3(0)=0,\, Q_3(+\infty )=+\infty ;\, Q_4(\rho _1):=27(\rho _2-\rho _1\rho _3)^2$ is strictly decreasing function of $\rho _1$ when $\rho _2>\rho _1\rho _3$, and $Q_4(0)=27\rho ^2_2$. Therefore, $Q(\sqrt{D_0 \rho _1})$ has a unique positive root $\rho _1^*$, such that $Q(\sqrt{D_0\rho _1})<0$ for $0<\rho _1<\rho ^*_1$, and $Q(\sqrt{D_0\rho _1})>0$ for $\rho _1>\rho ^*_1$. By the property of $Q(c)$, we have $\sqrt{D_0\rho _1}<c^*$ for $0<\rho _1<\rho ^*_1$; $\sqrt{D_0\rho _1}>c^*$ for $\rho _1>\rho ^*_1$ and $c^*=\sqrt{D_0\rho _1^*}$, where $\rho _1$ is determined by (21). Therefore, we obtain the information (20) about $c^*$.