Local Immunodeficiency: Role of Neutral Viruses

Abstract

This paper analyzes the role of neutral viruses in the phenomenon of local immunodeficiency. We show that, even in the absence of altruistic viruses, neutral viruses can support the existence of persistent viruses and thus local immunodeficiency. However, in all such cases neutral viruses can maintain only bounded (relatively small) concentration of persistent viruses. Moreover, in all such cases the state of local immunodeficiency could only be marginally stable, while it is known that altruistic viruses can maintain stable local immunodeficiency. We also present an absolutely minimal cross-immunoreactivity network where a stable and robust state of local immunodeficiency can be maintained. It is now a challenge to synthetic biology to build such small networks with stable local immunodeficiency. Another important challenge for biology is to understand which types of viruses can play a role of persistent, altruistic and neutral ones and whether a role which a given virus plays depends on the structure (topology) of a given cross-immunoreactivity network.

Appendices

Appendix A. Computation of $P(\lambda )$ from the Characteristic Polynomial of the Jacobian for Fig. 1

$$\begin{aligned} P(\lambda )= & {} -\lambda \begin{vmatrix}-\lambda&\quad -p\beta x_3&\quad -px_3 \\ \frac{c\alpha r_2}{\alpha r_2+r_3}&\quad \frac{b\alpha r_3}{\alpha r_2+r_3}-b-\lambda&\quad -\frac{b\alpha r_2}{\alpha r_2+r_3} \\ \frac{cr_3}{\alpha r_2+r_3}&\quad -\frac{b\alpha r_3}{\alpha r_2+r_3}&\quad -\frac{br_3}{\alpha r_2+r_3}-\lambda \end{vmatrix}-p\beta x_1\begin{vmatrix} 0&\quad -\lambda&\quad -px_3 \\ c&\quad \frac{c\alpha r_2}{\alpha r_2+r_3}&\quad -\frac{b\alpha r_2}{\alpha r_2+r_3} \\ 0&\quad \frac{cr_3}{\alpha r_2+r_3}&\quad -\frac{br_3}{\alpha r_2+r_3}-\lambda \end{vmatrix}\\= & {} -\lambda \begin{vmatrix} -\lambda&\quad -p\beta x_3&\quad -px_3 \\ \frac{c\alpha r_2}{\alpha r_2+r_3}&\quad \frac{b\alpha r_3}{\alpha r_2+r_3}-b-\lambda&\quad -\frac{b\alpha r_2}{\alpha r_2+r_3} \\ c&\quad -b-\lambda&\quad -b-\lambda \end{vmatrix}+cp\beta x_1\begin{vmatrix} -\lambda&\quad -px_3 \\ \frac{cr_3}{\alpha r_2+r_3}&\quad -\frac{br_3}{\alpha r_2+r_3}-\lambda \end{vmatrix}\\= & {} cp\beta x_1\left( \lambda ^2+\frac{br_3}{\alpha r_2+r_3}\lambda +\frac{pcx_3r_3}{\alpha r_2+r_3}\right) +\lambda ^2\begin{vmatrix} \frac{b\alpha r_3}{\alpha r_2+r_3}-b-\lambda&\quad -\frac{b\alpha r_2}{\alpha r_2+r_3} \\ -b-\lambda&-b-\lambda \end{vmatrix}\\&-\lambda p\beta x_3\begin{vmatrix} \frac{c\alpha r_2}{\alpha r_2+r_3}&\quad -\frac{b\alpha r_2}{\alpha r_2+r_3} \\ c&\quad -b-\lambda \end{vmatrix}+\lambda px_3\begin{vmatrix} \frac{c\alpha r_2}{\alpha r_2+r_3}&\quad \frac{b\alpha r_3}{\alpha r_2+r_3}-b-\lambda \\ c&\quad -b-\lambda \end{vmatrix}\\= & {} bf_1(1-\alpha )\left( \lambda ^2+\frac{br_3}{\alpha r_2+r_3}\lambda +bpr_3\right) \\&+\lambda ^2(-b-\lambda )\left( \frac{b\alpha r_3}{\alpha r_3+r_3}-b-\lambda + \frac{b\alpha r_2}{\alpha r_2+r_3}\right) \\&-\lambda p\beta x_3\left( -\frac{c\alpha r_2}{\alpha r_2+r_3}\lambda - \frac{bc\alpha r_2}{\alpha r_2+r_3}+\frac{bc\alpha r_2}{\alpha r_2+r_3}\right) \\&+\lambda px_3\left( -\frac{c\alpha r_2}{\alpha r_2+r_3}\lambda -\frac{bc\alpha r_2}{\alpha r_2+r_3}-\frac{bc\alpha r_3}{\alpha r_2+r_3}+bc+c\lambda \right) \\= & {} bf_1(1-\alpha )\left( \lambda ^2+\frac{bf_3}{\alpha r_2+r_3}\lambda +bpr_3\right) +\lambda ^2(\lambda +b)\left( \lambda +b-\frac{b\alpha (r_2+r_3)}{\alpha r_2+r_3}\right) \\&+\lambda p\beta x_3\frac{c\alpha r_2}{\alpha r_2+r_3}\lambda \\&+px_3\lambda [\frac{cr_3}{\alpha r_2+r_3}\lambda +\frac{bcr_3(1-\alpha )}{\alpha r_2+r_3}]\\= & {} bf_1(1-\alpha )\left( \lambda ^2+\frac{br_3}{\alpha r_2+r_3}\lambda +bpr_3\right) + \lambda ^2(\lambda +b)\left( \lambda +\frac{br_3(1-\alpha )}{\alpha r_2+r_3}\right) \\&+\lambda ^2p\beta \alpha r_2b+px_3\lambda \left[ \frac{cr_3}{\alpha r_2+r_3}\lambda +\frac{bcr_3 (1-\alpha )}{\alpha r_2+r_3}\right] \\= & {} \alpha bf_1\lambda ^2+bf_1(1-\alpha )\left( \lambda ^2+\frac{br_3}{\alpha r_2+r_3}\lambda +bpr_3\right) \\&+\lambda ^2(\lambda +b)(\lambda +\frac{br_3(1-\alpha )}{\alpha r_2+r_3})+px_3\lambda \left[ \frac{cr_3}{\alpha r_2+r_3}\lambda +\frac{bcr_3(1-\alpha )}{\alpha r_2+r_3}\right] . \end{aligned}$$

Appendix B. Computation of the Characteristic Polynomial of the Jacobian for Fig. 3

Let $\lambda _1=\frac{cx_1}{\alpha r_2}-b$,

$$\begin{aligned} \det (J-\lambda I)= & {} \det \begin{pmatrix}-\lambda &{}\quad 0 &{}\quad -px_1 &{}\quad -p\beta x_1\\ 0 &{}\quad -\lambda &{}\quad 0 &{}\quad -px_2\\ 0 &{}\quad 0 &{}\quad \lambda _1-\lambda &{}\quad 0\\ c &{}\quad c &{}\quad -\frac{cx_1}{\alpha r_2} &{}\quad -b-\lambda \end{pmatrix}=(\lambda _1-\lambda )\det \begin{pmatrix}-\lambda &{}\quad 0 &{}\quad -p\beta x_1 \\ 0 &{}\quad -\lambda &{}\quad -px_2 \\ c &{}\quad c &{}\quad -b-\lambda \end{pmatrix}\\= & {} (\lambda -\lambda _1)\left[ \lambda (\lambda ^2+b\lambda +cpx_2)+p\beta x_1c\lambda \right] =\lambda (\lambda -\lambda _1)\left[ \lambda ^2+b\lambda +cp(x_2+\beta x_1)\right] \\= & {} \lambda (\lambda -\lambda _1)P(\lambda ).\end{aligned}$$

Appendix C. Computation of the Characteristic Polynomial of the Jacobian for Fig. 4

Let $\lambda _1=\frac{cx_1}{\alpha (r_2+r_3)}-b<\frac{b}{\alpha }-b$, then

By expanding along the fourth row, we get

$$\begin{aligned}|J-\lambda I|=(\lambda _1-\lambda )\begin{vmatrix} -\lambda&\quad 0&\quad 0&\quad -p\beta x_1&\quad -p\beta x_1 \\ 0&\quad -\lambda&\quad 0&\quad -px_2&\quad 0 \\ 0&\quad 0&\quad -\lambda&\quad 0&\quad -px_3 \\ \frac{cr_2}{r_2+r_3}&\quad c&\quad 0&\quad \frac{cx_1r_3}{(r_2+r_3)^2}-b-\lambda&\quad -\frac{cx_1r_2}{(r_2+r_3)^2} \\ c&\quad c&\quad c&\quad -b-\lambda&\quad -b-\lambda \end{vmatrix}. \end{aligned}$$

Expanding now along the second row, we obtain

$$\begin{aligned}|J-\lambda I|=(\lambda -\lambda _1)[\lambda D_1(\lambda )+px_2 D_2(\lambda )],\end{aligned}$$

where

$$\begin{aligned}&D_1(\lambda )=\begin{vmatrix} -\lambda&\quad 0&\quad -p\beta x_1&\quad -p\beta x_1 \\ 0&\quad -\lambda&\quad 0&\quad -px_3 \\ \frac{cr_2}{r_2+r_3}&\quad 0&\quad \frac{cx_1r_3}{(r_2+r_3)^2}-b-\lambda&\quad -\frac{cx_1r_2}{(r_2+r_3)^2} \\ c&\quad c&\quad -b-\lambda&\quad -b-\lambda \end{vmatrix}\\&\quad =-\lambda \begin{vmatrix}-\lambda&\quad -p\beta x_1&\quad -p\beta x_1\\ \frac{cr_2}{r_2+r_3}&\quad \frac{cx_1r_3}{(r_2+r_3)^2}-b-\lambda&\quad -\frac{cx_1r_2}{(r_2+r_3)^2}\\ c&\quad -b-\lambda&\quad -b-\lambda \end{vmatrix}-px_3\begin{vmatrix}-\lambda&\quad 0&\quad -p\beta x_1\\ \frac{cr_2}{r_2+r_3}&\quad 0&\quad \frac{cx_1r_3}{(r_2+r_3)^2}-b-\lambda \\ c&\quad c&\quad -b-\lambda \end{vmatrix}\\&\quad =-\lambda \begin{vmatrix}-\lambda&\quad 0&\quad -p\beta x_1\\ \frac{cr_2}{r_2+r_3}&\quad \frac{cx_1}{r_2+r_3}-b-\lambda&\quad -\frac{cx_1r_2}{(r_2+r_3)^2}\\ c&\quad 0&\quad -b-\lambda \end{vmatrix}+cpx_3\left[ \lambda ^2+\left( b-\frac{cx_1r_3}{(r_2+r_3)^2}\right) \lambda +cp\beta x_1\frac{r_2}{r_2+r_3}\right] \\&\quad =\lambda (\lambda +b-\frac{cx_1}{r_2+r_3}) \left[ \lambda ^2+b\lambda +cp\beta x_1\right] \\&\qquad +cpx_3\left[ \lambda ^2+(b-\frac{cx_1r_3}{(r_2+r_3)^2})\lambda +cp \beta x_1\frac{r_2}{r_2+r_3}\right] ,\\&\quad D_2(\lambda )=\begin{vmatrix}-\lambda&\quad 0&\quad 0&\quad -p\beta x_1 \\ 0&\quad 0&\quad -\lambda&\quad -px_3 \\ \frac{cr_2}{r_2+r_3}&\quad c&\quad 0&\quad -\frac{cx_1r_2}{(r_2+r_3)^2} \\ c&\quad c&\quad c&\quad -b-\lambda \end{vmatrix}=c\lambda \left[ \lambda ^2+(b-\frac{cx_1r_2}{(r_2+r_3)^2})\lambda +cpx_3\right] \\&\qquad +cp\beta x_1\lambda \frac{cr_3}{r_2+r_3}. \end{aligned}$$