Stochastic dynamics of human papillomavirus delineates cervical cancer progression

Abstract

Starting from a deterministic model, we propose and study a stochastic model for human papillomavirus infection and cervical cancer progression. Our analysis shows that the chronic infection state as random variables which have the ergodic invariant probability measure is necessary for progression from infected cell population to cervical cancer cells. It is shown that small progression rate from infected cells to precancerous cells and small microenvironmental noises associated with the progression rate and viral infection help to establish such chronic infection states. It implicates that large environmental noises associated with viral infection and the progression rate in vivo can reduce chronic infection. We further show that there will be a cervical cancer if the noise associated with precancerous cell growth is large enough. In addition, comparable numerical studies for the deterministic model and stochastic model, together with Hopf bifurcations in both deterministic and stochastic systems, highlight our analytical results.

Appendix

To comprehend how environmental noises and randomness affect the dynamical behaviors of the deterministic system (4), we need to study how the threshold $\lambda$ is related to the reproduction number $R_0$. Since $R_0=\frac{\alpha n}{e(a+\delta )}$ is a decreasing function of the parameter $\delta$, which is the progression rate from infected cells to precancerous cells, we can consider the threshold $\lambda$ as a function of $\delta$ as well as a function of noise intensities. Propositions 2.1 and 2.2 give the behavior of the threshold $\lambda$ with respect to system parameters and noise intensities. Because Proposition 2.2 is a direct consequence of Proposition 2.1, so we only present the proof of Proposition 2.1

Proof of Proposition 2.1

Since $\dfrac{\Theta }{w}=\dfrac{a+\delta -e}{2\sqrt{\alpha n}}+\dfrac{\tau _1^2-\tau _2^2}{4\sqrt{\alpha n}}$, the threshold $\lambda$ can be rewritten as

$$\begin{aligned} \lambda = \sqrt{\alpha n}\left[ R_{\Theta }(w)-2\frac{\Theta }{w}\right] -e-\frac{\tau _2^2}{2}. \end{aligned}$$

Let $D_{\Theta }(w):=\dfrac{K_{\Theta +1}(w)K_{\Theta -1}(w)}{K^2_{\Theta }(w)}$ where $K_{\Theta }(\cdot )$ is the modified Bessel function of third kind with index $\Theta$. From Jorgensen (1982) in p. 172 and p. 175, we have $D_{\Theta }(w)=R_{\Theta }(w)R_{-\Theta }(w)\quad \text {and}\quad D_{\Theta }(w)=R_{\Theta }(w)\left[ R_{\Theta }(w)-2\dfrac{\Theta }{w}\right]$. Hence

$$\begin{aligned} \lambda = \sqrt{\alpha n}R_{-\Theta }(w)-e-\frac{\tau _2^2}{2}. \end{aligned}$$

Since $w>0$, a result in p. 173 in Jorgensen (1982) implies that $R_{\Theta }(w)$ is an increasing function of $\Theta$ and thus $\lambda$ is a decreasing function of $\Theta$. But, as $\Theta$ is an increasing function of $\delta$, so $\lambda$ is a decreasing function of $\delta$. Since $R_0$ is a decreasing function of $\delta$, $\lambda$ is an increasing function of $R_0$.

Next, since ${\overline{\lambda }}=\lim \limits _{(\tau _1,\tau _2)\rightarrow (0,0)}\lambda$ and $R_{\Theta }(w)=\dfrac{\Theta }{w}+\sqrt{\left( \dfrac{\Theta }{w}\right) ^2+D_{\Theta }(w)}$,

$$\begin{aligned} {\overline{\lambda }}=\lim \limits _{(\tau _1,\tau _2)\rightarrow (0,0)}\left\{ \left[ \dfrac{\Theta }{w}+\sqrt{\left( \dfrac{\Theta }{w}\right) ^2+D_{\Theta }(w)}\right] \sqrt{\alpha n}-a-\delta -\frac{\tau _1^2}{2}\right\} . \end{aligned}$$

As $\dfrac{\Theta }{w}\rightarrow \dfrac{a+\delta -e}{2\sqrt{\alpha n}}$ and $D_{\Theta }(w)\rightarrow 1$ as $(\tau _1,\tau _2)\rightarrow (0,0)$, so

$$\begin{aligned} {\overline{\lambda }}=\frac{a+\delta -e}{2}+\sqrt{\left( \frac{a+\delta -e}{2}\right) ^2+\alpha n}-a-\delta . \end{aligned}$$

It implies that ${\overline{\lambda }}=0$ iff $R_0=1$, ${\overline{\lambda }}<0$ iff $R_0<1$, and ${\overline{\lambda }}>0$ iff $R_0>1$. This completes the proof. $\square$