A Host–Parasite System with Multiple Parasite Strains and Superinfection Revisited: The Global Dynamics

Abstract

In this paper, we revisit a host–parasite system with multiple parasite strains and superinfection proposed by Nowak and May (Proc R Soc Lond B 255(1342):81–89, 1994), and study its global dynamics when we relax the two strict conditions assumed therein. As for system with two parasite strains, we derive that the basic reproduction number $R_0$ is the threshold condition for parasite extinction and the invasion reproduction number $R_i^j\ (i,j=1,2,i\ne j)$ is the subthreshold condition for coexistence of two parasite strains. As for system with three parasite strains, we are surprised to discover the global stability of parasite-free and coexistence equilibrium, which is distinct from the previous result. Furthermore, for system with n strains, we obtain the global asymptotical stability of the parasite-free equilibrium, conjecture a general result on the global stability of coexistence equilibrium and provide two numerical examples to testify our conjecture.

Appendices

Appendix A: Equivalency of Inequality (7)

Proof

Firstly, we show that $R_1^2>1$ implies that $x^*>0$. Note that $R_1^2>1$ can be rewritten as $\left[\frac{R_1}{R_2}>1+\frac{su(R_2-1)}{u+v_1}\right]$, which yields that $\left[R_1-R_2>\frac{su(R_2-1)}{u+v_1}\cdot \frac{\beta_2 k}{u(u+v_2)}\right]$. Thus, we have $\left[\frac{u}{k}(u+v_1)(u+v_2)(R_1-R_2)+us{\beta_2}>su{\beta_2} R_2>0\right]$, which can guarantee that $x^*>0$.

Next, we show the conclusion holds under the condition $x^*>0$. We now show that $x^*>{(u+v_1)}/{\beta _1}$ is equivalent to the inequality $R_2^1>1$. It follows from $x^*>{(u+v_1)}/{\beta _1}$ that we have

$$\begin{aligned} \frac{ks\beta _2}{\frac{u}{k}(u+v_2)(u+v_1)(R_1-R_2)+us\beta _2} >\frac{u+v_1}{\beta _1}, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} R_1>1+\frac{(u+v_2)(u+v_1)}{ks\beta _2}(R_1-R_2). \end{aligned}$$

That is,

$$\begin{aligned} R_1>1+\frac{u+v_2}{s\beta _2}\cdot \frac{\beta _1}{u}\left( 1-\frac{R_2}{R_1}\right) . \end{aligned}$$

Thus, one has

$$\begin{aligned} (R_1-1)\cdot \frac{s\beta _2}{u+v_2}\cdot \frac{u}{\beta _1}>1-\frac{R_2}{R_1}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} R_2^1=\frac{R_2}{R_1} +\frac{s\beta _2}{u+v_2}\cdot \frac{u}{\beta _1}(R_1-1)>1. \end{aligned}$$

Similarly, $x^*<{(u+v_2)}/{\beta _2}$ is equivalent to $R_1^2>1$. $\square$

Appendix B: Proof of Theorem 2

Proof

The Jacobian matrix at $E_0$ for the right hand side of system (2) is given by

$$\begin{aligned} J(E_0)=\begin{bmatrix} -u&-\beta _1 x_0&-\beta _2 x_0\\ 0&(u+v_1)(R_1-1)&0\\ 0&0&(u+v_2)(R_2-1) \end{bmatrix}. \end{aligned}$$

It is easy to see that the eigenvalues of $J(E_0)$ are $\lambda _1=-u,\lambda _2=(u+v_1)(R_1-1),\lambda _3=(u+v_2)(R_2-1)$. Noting that $R_0<1$, we can conclude that all eigenvalues of $J(E_0)$ are negative. When $R_0>1$, $J(E_0)$ has at least one positive eigenvalue. Hence, the parasite-free equilibrium $E_0$ is locally asymptotically stable if $R_0<1$ and is unstable if $R_0>1$. This completes the proof. $\square$

Appendix C: Proof of Theorem 3

Proof

The Jacobian matrix at $\bar{E}_1$ for the right hand side of system (2) is given by

$$\begin{aligned} J(\bar{E}_1)=\begin{bmatrix} -uR_1&-\beta _1 \bar{x}_1&-\beta _2\bar{x}_1\\ \beta _1\bar{y}_1&0&-s\beta _2\bar{y}_1\\ 0&0&(u+v_2)(R_2^1-1) \end{bmatrix} \end{aligned}$$

and its second additive compound matrix is

$$\begin{aligned} J^{[2]}(\bar{E}_1)= \begin{bmatrix} -uR_1&-s\beta _2\bar{y}_1&\beta _2\bar{x}_1\\ 0&-uR_1+(u+v_2)(R_2^1-1)&-\beta _1 \bar{x}_1\\ 0&\beta _1\bar{y}_1&(u+v_2)(R_2^1-1) \end{bmatrix}. \end{aligned}$$

Then, one has

$$\begin{aligned} \mathrm{tr}(J(\bar{E}_1))&=-uR_1+(u+v_2)(R_2^1-1)<0,\\ \det (J(\bar{E}_1))&=(u+v_2)(R_2^1-1)\cdot \beta _1^2\bar{x}_1\bar{y}_1<0,\\ \det (J^{[2]}(\bar{E}_1))&=-uR_1[(u+v_2)^2(R_2^1-1)^2+\beta _1^2\bar{x}_1\bar{y}_1]<0 \end{aligned}$$

hold if $R_2^1<1$. Thus, it follows from Lemma 2.1 that $J(\bar{E}_1)$ is stable when $R_2^1<1$. When $R_2^1>1$, $J(\bar{E}_1)$ has at least one positive eigenvalue. Thus, $\bar{E}_1$ is locally asymptotically stable if $R_2^1<1$ and it is unstable if $R_2^1>1$. This finishes the proof of Theorem 3. $\square$

Appendix D: Proof of Existence of ${E}^*$

Proof

By using (22), one has

$$\begin{aligned} y_2^*&=\frac{1}{s}\left( x^*+sy^*-\frac{u+v_2}{\beta _2}\right) :=h_2 \end{aligned}$$

(23)

if $h_2>0$. Furthermore, we have

$$\begin{aligned} y_1^*&=\frac{1}{s}\left[ x^*+sy^*-\frac{u+v_1}{\beta _1}-s\left( 1+\frac{\beta _2}{\beta _1}\right) y^*_2\right] \nonumber \\ \quad&=\frac{1}{s}\left[ -\frac{\beta _2}{\beta _1}(x^*+sy^*)+\frac{u+v_2}{\beta _2}+\frac{v_2-v_1}{\beta _1}\right] :=h_1 \end{aligned}$$

(24)

if $h_1>0$. Substituting (23) and (24) into (20) yields the first relation of $x^*$ and $y^*$.

It follows from system (2) that we obtain

$$\begin{aligned} y^*=y_1^*+y_2^*=\frac{1}{s\beta _2}\left[ (v_2-v_1)-(\beta _2-\beta _1)x^*\right] , \end{aligned}$$

(25)

which is derived according to the subtracting between the second and third equations. Thus, we have the other relation of $x^*$ and $y^*$. Combining those two relations, we can solve the positive solution $E^*$ of system (2) in this specific way.

Appendix E: Proof of Existence of $\widetilde{E}^*$

Proof

It follows from (22) that we have

$$\begin{aligned} \widetilde{y}_3^*&=\frac{1}{s}\left( \widetilde{x}^*+s\widetilde{y}^*-\frac{u+v_3}{\beta _3}\right) :=H_3 \end{aligned}$$

(26)

if $H_3>0$. Furthermore, we have

$$\begin{aligned} \widetilde{y}_2^*&=\frac{1}{s}\left[ \widetilde{x}^*+s\widetilde{y}^*-\frac{u+v_2}{\beta _2}-s\left( 1+\frac{\beta _3}{\beta _2}\right) \widetilde{y}^*_3\right] \nonumber \\ \quad&=\frac{1}{s}\left[ -\frac{\beta _3}{\beta _2}(\widetilde{x}^*+s\widetilde{y}^*)+\frac{u+v_3}{\beta _3}+\frac{v_3-v_2}{\beta _2}\right] :=H_2 \end{aligned}$$

(27)

if $H_2>0$ and

$$\begin{aligned} \widetilde{y}_1^*&=\frac{1}{s}\left[ \widetilde{x}^*+s\widetilde{y}^*-\frac{u+v_1}{\beta _1}-s\left( 1+\frac{\beta _3}{\beta _2}\right) \widetilde{y}^*_3 -s\left( 1+\frac{\beta _2}{\beta _1}\right) \widetilde{y}^*_2\right] \\ \quad&=\frac{1}{s}\left[ \frac{\beta _3}{\beta _1}(\widetilde{x}^*+s\widetilde{y}^*)-\frac{u+v_1}{\beta _1}+\frac{u+v_2}{\beta _2} -\frac{v_3-v_2}{\beta _1}-\frac{\beta _2}{\beta _1}\cdot \frac{u+v_3}{\beta _3}\right] :=H_1\nonumber \end{aligned}$$

(28)

if $H_1>0$. Substituting (26)–(28) into (20) yields the first relation of $\widetilde{x}^*$ and $\widetilde{y}^*$.

It is easy to derive that

$$\begin{aligned} \widetilde{y}_1^*+\widetilde{y}_2^*&=\frac{1}{s\beta _3}\left[ (u+v_3)-\beta _3\widetilde{x}^*\right] ,\\ \widetilde{y}_1^*+\widetilde{y}_2^*&=\frac{1}{s\beta _2}\left[ (v_2-v_1)-(\beta _2-\beta _1)\widetilde{x}^*\right] . \end{aligned}$$

Here, the second equation is derived from the subtracting between the fourth and third equations. Combining the above two equations, we can solve

$$\begin{aligned} \widetilde{x}^*=\frac{\beta _2}{\beta _1}\left( \frac{u+v_3}{\beta _3}-\frac{v_2-v_1}{\beta _2}\right) :=H_0. \end{aligned}$$

(29)

if $H_0>0$. Substituting (29) into the first relation of $\widetilde{x}^*$ and $\widetilde{y}^*$ yields that

$$\begin{aligned} A\widetilde{y}^*=B, \end{aligned}$$

where

$$\begin{aligned} A&=s\left( 1+\frac{\beta _3}{\beta _1}v_1-\frac{\beta _3}{\beta _2}v_2+v_3\right) ,\nonumber \\ B&=\frac{ks}{u}-s\left( \frac{\beta _2}{\beta _1}\cdot \frac{u+v_3}{\beta _3}-\frac{v_2-v_1}{\beta _1}\right) +v_3\left( \frac{u+v_3}{\beta _3}+\frac{v_2-v_1}{\beta _1}-\frac{\beta _2}{\beta _1}\cdot \frac{u+v_3}{\beta _3}\right) \\&\quad +v_2\left( \frac{u+v_3}{\beta _1}-\frac{u+v_3}{\beta _3}-\frac{v_3-v_2}{\beta _2}-\frac{\beta _3}{\beta _2}\cdot \frac{v_2-v_1}{\beta _1}\right) \\&\quad +v_1\left( \frac{\beta _3}{\beta _1}\cdot \frac{v_2-v_1}{\beta _1} -\frac{u+v_2}{\beta _2}-\frac{v_2-v_1}{\beta _1}+\frac{\beta _2}{\beta _1}\cdot \frac{u+v_3}{\beta _3}\right) . \end{aligned}$$

Thus, there exists a unique positive $\widetilde{y}^*$ if and only if $A\ne 0$, $AB>0$ and $H_i>0, i=0,1,2,3,4$. $\square$