Effects of superinfection and cost of immunity on host-parasite co-evolution

Figure 1.  Dependence of the value of evolutionary singular point on the cost of immunological up-regulation $k_1$ and the superinfection rate $\varphi$, where $\delta= 0.095$, $b= 0.6$, $c_2=0.3$, and $\bar{g}=0.15$. From two figures, both $c_1^*(k_1)$ and $c_1^*(\varphi)$ are decreasing functions in first quadrant. In (a) and (b), the four curves are obtained by varying the value of $\mu$, respectively. In (a), the curves are moved up when $\mu$ increases. However, the movement in (b) are in two direction and more complicated than it in (a)

Figure 2.  Singularity and Isoclinic stability: when $\delta= 0.95$, $b=10$, $\beta=0.4$, $\mu=0.2$, $k_1=0.5$, and $k_2=0.8$. We only observe the regions in first quadrant. In figure (a) and (b), we plot the solutions when n (26) and (27) are equal to zero. In figures (c) and (d), the red solid curves represents function (35) and the blue dash curves represent function (36). In shadows, both conditions (33) and (34) for isoclinic stability can be met. We adjust the value of superinfection rates $\varphi$ to observe its effects. When superinfection rate increase, the values of $\tilde{c}_1^*$ and $\tilde{c}_2^*$ also increase. The shadow area has significant change when superinfection rate changes

Figure 3.  Absolute stability: when $\delta= 0.3$, $\varphi=10$, $b=2$, $\beta=0.4$, $\mu=0.2$, $k_1=0.1$, and $k_2=0.8$. The red dot curve represents function k1 = 0.1, and k2 = 0.8. The red dot curve represents function (35) and the blue dash curve represents function (36), too. The golden solid line stands for the formula in inequality (37). In two shadows, the conditions for absolute stability can be satisfied