Mathematical modeling of viral infection dynamics in spherical organs

Abstract

A general mathematical model of viral infections inside a spherical organ is presented. Transported quantities are used to represent external cells or viral particles that penetrate the organ surface to either promote or combat the infection. A diffusion mechanism is considered for the migration of transported quantities to the organ inner tissue. Cases that include the effect of penetration, diffusion and proliferation of immune system cells, the generation of latently infected cells and the delivery of antiviral treatment are analyzed. Different antiviral mechanisms are modeled in the context of spatial variation. Equilibrium conditions are also calculated to determine the radial profile after the infection progresses and antiviral therapy is delivered for a long period of time. The dynamic and equilibrium solutions obtained in this paper provide insight into the temporal and spatial evolution of viral infections.

Appendices

Appendix A: Model parameters and variables

The uninfected host cells $X$ proliferate at a rate of $\lambda$, and are infected at a rate of $\beta$. The calculation of $\beta$ considers that 70 % of the virion successfully infect host cells. A productive infected cell lives five times less than a latently infected cell. For convenience, the expectancy life time of a latently infected cell is made equivalent to the uninfected cell, $a_L = d$. Therefore, an uninfected cell is expected to live 10 days, which makes 2 days the life of a productive infected cell. The lifetime of an immune cell is around 5 days. Immune cells are stimulated by infected cells and the production of immune cells is a function of the infected cells concentration. In the case immune cell production is assumed inside the infected organ, the generation term $c Y Z$ is included in $\mathbf{G}$. Nevertheless, infected cells at the organ surface may trigger the generation of immune cells at the surface and their penetration, measured by $p_{ZY}$, to the infected organ. Table 1 shows these parameters and literature sources.

Appendix B: Numerical solution approach

A numerical solution for the system defined by Eqs. (2, 3) is obtained by the discretization of the spherical organ into $n$ equal volume shells. This discretization originates $n+1$ nodes at radial locations defined by $\eta _i$, where $i = 0, 1, \ldots , n-1, n$. The center and surface nodes are given by $\eta _0 = 0$ and $\eta _n = 1$, respectively. The following recursive expression is used to calculate the nodes locations for an equal volume shell configuration,

$$\begin{aligned} \eta _{i+1}^3 = \eta _{i}^3 + \frac{1}{n} \end{aligned}$$

Two contiguous shell elements, which are equivalent to three consecutive nodes, are used to approximate the diffusion terms inside the spherical organ. This approximation consists of making $T(\eta ,t)$ a piecewise quadratic function. The expression for the quadratic transported function defined by three consecutive nodes $i-1$, $i$ and $i+1$ is given by,

$$\begin{aligned} T(\eta , t) = \frac{ \delta _{i} \delta _{i+1} }{\varDelta _i ( \varDelta _i + \varDelta _{i+1} )}T_{i-1} - \frac{ \delta _{i-1} \delta _{i+1} }{\varDelta _i \varDelta _{i+1} } T_{i} + \frac{ \delta _{i-1} \delta _{i} }{( \varDelta _i + \varDelta _{i+1} ) \varDelta _{i+1}} T_{i+1} \end{aligned}$$

(15)

where $\varDelta _j = \eta _j - \eta _{j-1}$, $\delta _j = \eta - \eta _j$, and the node $i$ is chosen such that $\eta \in [\eta _{i-1} \quad \eta _{i+1}]$. Notice that $T_{i} \equiv T(\eta _i, t)$. The partial derivative with respect to $\eta$ for the quadratic approximation in Eq. (15) is given by,

$$\begin{aligned} \frac{\partial T(\eta , t)}{\partial \eta } \!=\! \frac{ \delta _{i} \!+\! \delta _{i+1} }{\varDelta _i ( \varDelta _i + \varDelta _{i+1} )}T_{i-1} - \frac{ \delta _{i-1} + \delta _{i+1} }{\varDelta _i \varDelta _{i+1} } T_{i} + \frac{ \delta _{i-1} \!+\! \delta _{i} }{( \varDelta _i + \varDelta _{i+1} ) \varDelta _{i+1}} T_{i+1}\qquad \end{aligned}$$

(16)

Note that the two boundary conditions at $\eta _0$ and $\eta _n$ can be substituted in the previous equation to obtain the correspondent expressions for $T_{-1}$ and $T_{n+1}$ necessary for the calculation of $T(\eta , t)$ in Eq. (15) when $\eta < \eta _{1}$ and $\eta > \eta _{n-1}$, respectively. The second derivative of the quadratic approximation in Eq. (15) is used in the calculation of the diffusion term,

$$\begin{aligned} \frac{1}{2}\frac{\partial ^2 T(\eta , t)}{\partial \eta ^2} \!=\! \frac{ 1}{\varDelta _i ( \varDelta _i + \varDelta _{i+1} )}T_{i-1} \!-\! \frac{1}{\varDelta _i \varDelta _{i+1} } T_{i} + \frac{ 1 }{( \varDelta _i +\varDelta _{i+1} ) \varDelta _{i+1}} T_{i+1}\qquad \end{aligned}$$

(17)

Substitution of Eqs. (16, 17) in the diffusion term of any transported quantity gives:

$$\begin{aligned} \frac{1}{\eta ^2} \frac{\partial }{\partial \eta } \left( \eta ^2 \frac{\partial T }{d \eta } \right) \approx h_{i-1} T_{i-1} - h_i T_{i} + h_{i+1} T_{i+1} \end{aligned}$$

(18)

where the coefficients are a function of $\eta$,

$$\begin{aligned} h_{i-1}&= 2 \frac{ \delta _{i} + \delta _{i+1} + \eta }{\eta \varDelta _i ( \varDelta _i + \varDelta _{i+1} )}\\ h_i&= 2 \frac{ \delta _{i-1} + \delta _{i+1} + \eta }{ \eta \varDelta _i \varDelta _{i+1} }\\ h_{i+1}&= 2 \frac{ \delta _{i-1} + \delta _{i} + \eta }{\eta ( \varDelta _i + \varDelta _{i+1} ) \varDelta _{i+1}} \end{aligned}$$

It is important to determine a minimum number of nodes $n+1$ to reliably approximate the diffusion term. This condition can be estimated by comparing the numerical approximation solution for different number of nodes at equilibrium condition.

Figure 11 illustrates de effect of the number of shell elements for the equilibrium solution of the basic virus model. The continuous line represents the solution provided by the Runge–Kutta 23 solver with variable step size. The solution provided by 25 shell elements shows acceptable as it seems to lie very close to the Runge–Kutta 23 solution profile.

Fig. 11

Effect of the number of shell elements for the equilibrium solution. The continuous line represents the solution provided by Runge–Kutta 23 solver