Source-Sink Dynamics in a Two-Patch SI Epidemic Model with Life Stages and No Recovery from Infection

Abstract

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies basic reproduction numbers $R_0$ for each of the patches, and conditions for the basic reproduction numbers to produce a receiving patch under which its population. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Appendices

Appendix A Basic Reproductive Number computations

We compute the basic reproductive number $R_{0,i}$ for $i=1,2$. Because calculations are analogous, we provide only the details for the first patch. We follow the next-generation matrix approach, discussed in Castillo-Garsow and Castillo-Chavez (2020).

For the first patch the transmission matrix in infected F and the transition matrix V are given by

$$\begin{aligned} F := \begin{bmatrix} \beta _1 J_{S,1} \frac{J_{I,1}+A_{I,1}}{N_1} \\ \beta _1 A_{S,1} \frac{J_{I,1}+A_{I,1}}{N_1} \end{bmatrix} \quad V := \begin{bmatrix} -\mu _I J_{I,1} \\ -\mu _I A_{I,1} \end{bmatrix}. \end{aligned}$$

(A1)

Their corresponding Jacobian matrices with respect to $I= (J_{I,1}, A_{I,1})$ are given by

$$\begin{aligned} \mathcal {F}&= \frac{\partial F}{\partial I} = \begin{bmatrix} \beta _1 J_{S,1} \frac{J_{S,1}+ A_{S,1}}{N_1^2} &{} \beta _1 J_{S,1} \frac{J_{S,1}+ A_{S,1}}{N_1^2} \\ \beta _1 A_{S,1} \frac{J_{S,1}+ A_{S,1}}{N_1^2} &{} \beta _1 A_{S,1} \frac{J_{S,1}+ A_{S,1}}{N_1^2} \end{bmatrix} \nonumber \\ \mathcal {V}&= \frac{\partial V}{\partial I} = \begin{bmatrix} -\mu _I &{} 0 \\ 0 &{} - \mu _I \end{bmatrix}. \end{aligned}$$

(A2)

At the disease free equilibrium point we have that $J_{S,1} + A_{S,1} = N_1$, therefore at this point we have that

$$\begin{aligned} \mathcal {F}\mathcal {V}^{-1} = \begin{bmatrix} \beta _1 \frac{J_{S,1}}{N_1} &{} \beta _1 \frac{J_{S,1}}{N_1} \\ \beta _1 \frac{A_{S,1}}{N_1} &{} \beta _1 \frac{A_{S,1}}{N_1} \end{bmatrix} \begin{bmatrix} -\mu _I &{} 0 \\ 0 &{} -\mu _I \end{bmatrix} = \begin{bmatrix} -\frac{\beta _1}{\mu _I}\frac{J_{S,1}}{N_1} &{} -\frac{\beta _1}{\mu _I}\frac{J_{S,1}}{N_1} \\ -\frac{\beta _1}{\mu _I}\frac{A{S,1}}{N_1} &{} -\frac{\beta _1}{\mu _I}\frac{A_{S,1}}{N_1} \end{bmatrix}. \end{aligned}$$

(A3)

A simple calculation using the fact that $N_1= A_{S,1}+J_{S,1}$ gives us that the characteristic polynomial of this matrix equals $p(\lambda ) = \lambda \left( \lambda + \frac{\beta _1}{\mu _I}\right)$. Therefore, the expectral radius $\rho (\mathcal {F}\mathcal {V}^{-1})$ equals $\beta _1/\mu _I$, thus providing the aforementioned formula for the basic reproductive number of the first patch. As mentioned before, the computation of $R_{0,2}$ is completely analogous.

Appendix B First Patch Disease Free Equilibrium Jacobian Calculation

Summing the fourth row to the third and the second row to the first equals

Transposing this matrix, we must compute

(B4)

To compute determinant (A), we employ the second row.

$$\begin{aligned} (A)&= - (\mu _S + \lambda ) \left( \left( \beta _1 \frac{J_{S,1}}{N_1} - \mu _I -\lambda \right) \cdot \left( \beta _1 \frac{A_{S,1}}{N_1} - \mu _I -\lambda \right) - \beta _1 \frac{J_{S,1}}{N_1}\cdot \beta _1 \frac{A_{S,1}}{N_1} \right) \nonumber \\&= - (\mu _S + \lambda )\left( - \frac{\beta _1}{N_1}(\mu _I+ \lambda )(A_{S,1}+J_{S,1}) + (\mu _I+ \lambda )^2\right) \end{aligned}$$

(B5)

Likewise, for computing the determinant (B), we use the second row.

(B6)

Therefore, the final determinant equals

(B7)

simplifying and using $N_1 = A_{S,1} + J_{S,1}$ at the disease-free equilibrium, this determinant is equal to

(B8)

After some algebraic manipulations, this expression is simplified to

(B9)

Note that $B_1>0$. A condition to ensure $C_1>0$ is $\mu _S(\alpha + \mu _S) - \alpha pr >0$.

Appendix C First Patch Endemic Equilibrium Calculation

Using $I_1:= J_{I,1}+A_{I,1}$ we have that

$$\begin{aligned} \dot{I_1}&= \dot{J_{I,1}} + \dot{A_{I,1}} = \beta _1J_{S,1}\frac{(J_{I,1}+A_{I,1})}{N_1}\nonumber \\&\quad -\mu _IJ_{I,1} + \beta _1A_{S,1}\frac{(J_{I,1}+A_{I,1})}{N_1}-\mu _IA_{I,1} \end{aligned}$$

(C10)

$$\begin{aligned}&=\beta _1(J_{S,1}+A_{S,1})\frac{I_1}{N_1}-\mu _I I_1 \end{aligned}$$

(C11)

We can then consider the sub-model for the first patch,

$$\begin{aligned} \begin{aligned} \dot{J_{S,1}}=&pr\left( 1- \frac{N_1}{k}\right) A_{S,1}-\beta _1J_{S,1}\frac{I_1}{N_1}-\alpha J_{S,1}-\mu _SJ_{S,1}\\ \dot{A_{S,1}}=&\alpha J_{S,1}-\beta _1A_{S,1}\frac{I_1}{N_1}-\mu _SA_{S,1}\\ \dot{I_1}=&\beta _1(J_{S,1}+A_{S,1})\frac{I_1}{N_1}-\mu _I I_1. \end{aligned} \end{aligned}$$

(C12)

We want to find an equilibrium point in this sub-model, that is, a point in which the following equations are satisfied,

$$\begin{aligned} 0= & {} pr\left( 1-\frac{N_1}{k}\right) A_{S,1}-\beta _1J_{S,1}\frac{I_1}{N_1}-\alpha J_{S,1}- \mu _SJ_{S,1},\end{aligned}$$

(C13)

$$\begin{aligned} 0= & {} \alpha J_{S,1}-\beta _1A_{S,1}\frac{I_1}{N_1}-\mu _SA_{S,1},\end{aligned}$$

(C14)

$$\begin{aligned} 0= & {} \beta _1(J_{S,1}+A_{S,1})\frac{I_1}{N_1}-\mu _I I. \end{aligned}$$

(C15)

In particular, we are interested in finding an endemic equilibrium, which is an equilibrium in which $I_1 \ne 0$. Using Eq. (C15) and canceling $I_1$, we obtain that

$$\begin{aligned} N_1 = \frac{\beta _1}{\mu _I}(J_{S,1} + A_{S,1}). \end{aligned}$$

(C16)

Furthermore, using (C16) we get

$$\begin{aligned} I_1 :&= N_1 - A_{S,1} - J_{S,1} \nonumber \\&= \frac{\beta _1}{\mu _I}(J_{S,1} + A_{S,1}) - (J_{S,1} + A_{S,1}) \nonumber \\&= \left[ \frac{\beta _1}{\mu _I} -1 \right] (J_{S,1} + A_{S,1}) \nonumber \\&= \frac{\beta _1 - \mu _I}{\mu _I}(J_{S,1} + A_{S,1}) \end{aligned}$$

(C17)

As $I \ne 0$ we must have $\beta _1 \ne \mu _I$. If we multiply Eq. (C14) by $N_1$ and use (C16) and (C17), we obtain

$$\begin{aligned} 0&=\alpha J_{S,1}N_1-\beta _1A_{S,1}I_1-\mu _SA_{S,1}N_1 \Rightarrow \nonumber \\ 0&=\alpha J_{S,1} \cdot \frac{\beta _1}{\mu _I}(J_{S,1} + A_{S,1}) - \beta _1 A_{S,1} \cdot \frac{\beta _1 - \mu _I}{\mu _I}(J_{S,1} + A_{S,1}) \nonumber \\ {}&\quad -\mu _SA_{S,1} \cdot \frac{\beta _1}{\mu _I}(J_{S,1} + A_{S,1}) \Rightarrow \nonumber \\ 0&=(J_{S,1} + A_{S,1}) \left[ \alpha J_{S,1} \cdot \frac{\beta _1}{\mu _I} - \beta _1 A_{S,1} \cdot \frac{\beta _1 - \mu _I}{\mu _I} -\mu _SA_{S,1} \cdot \frac{\beta _1}{\mu _I}\right] \nonumber \\ {}&\Rightarrow \left( \text {{factoring} } \frac{\beta _1}{\mu _I}\right) \nonumber \\ 0&=(J_{S,1} + A_{S,1}) \biggl ( \alpha J_{S,1} - A_{S,1} (\beta _1 - \mu _I +\mu _S) \biggr ). \end{aligned}$$

(C18)

Note that $J_{S,1}+A_{S,1} \ne 0$ because otherwise (C15) would imply $I_1=0$. Then it follows that $\alpha J_{S,1} = A_{S,1} (\beta _1 - \mu _I +\mu _S)$. Moreover, if we assume $\beta _1 - \mu _I +\mu _S \ne 0$ we arrive at the equality

$$\begin{aligned} \frac{\alpha }{\beta _1 - \mu _I +\mu _S}J_{S,1} = A_{S,1}. \end{aligned}$$

(C19)

Call $\lambda _1 := \frac{\alpha }{\beta _1 - \mu _I +\mu _S} = \frac{\alpha }{\mu _I \left( R_{0,1} + \frac{\mu _S}{\mu _I} - 1 \right) }$ and note that

$$\begin{aligned} N_1 = A_{S,1} + J_{S,1} + I_1 = \lambda _1 J_{S,I} + J_{S,1} + I_1 = (\lambda _1+1)J_{S,1} + I_1. \end{aligned}$$

(C20)

Modifying (C17) we get

$$\begin{aligned} I_1 = \frac{\beta _1 - \mu _I}{\mu _I}(J_{S,1} + A_{S,1}) = \frac{\beta _1 - \mu _I}{\mu _I}(\lambda _1+1)J_{S,1} \Rightarrow \frac{\mu _I}{\beta _1 - \mu _I} I_1 = (\lambda _1+1)J_{S,1} \end{aligned}$$

(C21)

meaning that

$$\begin{aligned} N_1 = (\lambda _1+1)J_{S,1} + I_1 = \frac{\mu _I}{\beta _1 - \mu _I} I_1 + I_1 = \frac{\beta _1}{\beta _1 - \mu _I}I_1. \end{aligned}$$

(C22)

We turn to the first equation of the model, Eq. (C13), and substitute Eq. (C19) to obtain

$$\begin{aligned} 0=pr\left( 1- \frac{N_1}{k}\right) \lambda _1 J_{S,1} -\beta _1J_{S,1}\frac{I_1}{N_1}-\alpha J_{S,1}-\mu _SJ_{S,1}. \end{aligned}$$

(C23)

Note that $J_{S,1}=0$ would imply $A_{S,1}$ because of (C19), which implies $J_{S,1}+A_{S,1}=0$, an impossibility when assuming $I_1 \ne 0$ as we commented above. Therefore we can cancel $J_{S,1}$ in this last equation to get

$$\begin{aligned} 0=pr\lambda _1\left( 1- \frac{N_1}{k}\right) -\beta _1\frac{I_1}{N_1}-(\alpha + \mu _S). \end{aligned}$$

(C24)

If we multiply by $N_1$, we get

$$\begin{aligned} 0=pr\lambda _1\left( 1- \frac{N_1}{k}\right) N_1 -\beta _1I-(\alpha + \mu _S)N_1. \end{aligned}$$

(C25)

Using (C22) we obtain

$$\begin{aligned} 0=pr\lambda _1\left( 1- \frac{\beta _1}{k(\beta _1 - \mu _I)}I_1\right) \frac{\beta _1}{\beta _1 - \mu _I}I_1 -\beta _1I_1-(\alpha + \mu _S)\frac{\beta _1}{\beta _1 - \mu _I}I_1. \end{aligned}$$

(C26)

As $I_1 \ne 0$ we can cancel all $I_1$ values and arrive to

$$\begin{aligned} 0=pr\lambda _1\left( 1- \frac{\beta _1}{k(\beta _1 - \mu _I)}I_1\right) \frac{\beta _1}{\beta _1 - \mu _I} -\beta _1-\frac{\beta _1(\alpha + \mu _S)}{\beta _1 - \mu _I}. \end{aligned}$$

(C27)

We note that $\frac{\beta _1}{\beta _1 - \mu _I} = \frac{R_{0,1}}{R_{0,1} - 1}$, so we have that

$$\begin{aligned} 0=pr\lambda _1\left( 1- \frac{R_{0,1}}{k(R_{0,1}-1)}I_1\right) \frac{R_{0,1}}{R_{0,1} - 1}-\mu _I R_{0,1} -\frac{R_{0,1}(\alpha + \mu _S)}{R_{0,1}-1}. \end{aligned}$$

(C28)

Factoring $I_1$ we get

$$\begin{aligned} I_1^* = \frac{k\left( R_{0,1}-1\right) }{pr\alpha R_{0,1}}\biggl (pr\alpha - \bigl (\mu _IR_{0,1} + \mu _S - \mu _I + \alpha \bigr )\bigl (\mu _I R_{0,1} + \mu _S - \mu _I\bigr )\biggr ).\nonumber \\ \end{aligned}$$

(C29)

Further normalizing with respect to $\mu _I$, we obtain that

$$\begin{aligned} I_1^* = \frac{k\left( R_{0,1}-1\right) }{r \mathcal {R}_p \mathcal {R}_{\alpha } R_{0,1}}\biggl (r\mathcal {R}_p\mathcal {R}_{\alpha } - (R_{0,1} + \mathcal {R}_s + \mathcal {R}_{\alpha } -1)(R_{0,1} + \mathcal {R}_s -1)\biggr ),\nonumber \\ \end{aligned}$$

(C30)

where

$$\begin{aligned} \mathcal {R}_p := \frac{p}{\mu _I}, \quad \mathcal {R}_{\alpha } := \frac{\alpha }{\mu _I}, \quad \mathcal {R}_s = \frac{\mu _S}{\mu _I}. \end{aligned}$$

(C31)

We can obtain the values of $J_{S,1}$ and $A_{S,1}$ at the equilibrium with the previous equations. These are given by

$$\begin{aligned} J_{S,1}^*&= \frac{\mu _I}{(\beta _1 - \mu _I)(\lambda _1+1)}I_1^* = \frac{1}{(R_{0,1}-1)(\lambda _1+1)}I_1^*,\nonumber \\ A_{S,1}^*&= \frac{\mu _I\lambda _1}{(\beta _1 - \mu _I)(\lambda _1+1)}I_1^* = \frac{\lambda _1}{(R_{0,1}-1)(\lambda _1+1)}I_1^*. \end{aligned}$$

(C32)

Finally, we can return to the original model and find the values of $J_{I,1}$ and $A_{I,1}$ at this equilibrium. Using $\dot{J_{S,1}}=0$ in the second Equation (1) and plugging the values of $I_1^*, J_{S,1}^*$ and $N_1^* = R_{\beta _1}I_1^*$ we get

$$\begin{aligned} 0&= \beta _1 \cdot \frac{\mu _IR_{\beta _1}}{\beta _1(\lambda _1+1)}I_1^* \cdot \frac{I_1^*}{R_{\beta _1} I_1^*} - \mu _I J_{I,1}^* \Rightarrow J_{I,1}^* = \frac{1}{\lambda _1+1}I_1^* \end{aligned}$$

(C33)

$$\begin{aligned}&\Rightarrow A_{I,1}^* = \frac{\lambda _1}{\lambda _1+1}I_1^*. \end{aligned}$$

(C34)

Appendix D Second Patch Disease Free Equilibrium Jacobian Calculation

By adding the term $-\frac{(1-p)r}{k}A_{S,1}$ to each entry of the first row and transposing this matrix, we must compute

(D35)

To compute determinant (A), we employ the second row.

$$\begin{aligned} (A)&= - (\mu _S + \lambda ) \left( \left( \beta _2 \frac{J_{S,2}}{N_2} - \mu _I -\lambda \right) \cdot \left( \beta _2 \frac{A_{S,2}}{N_2} - \mu _I -\lambda \right) - \beta _2 \frac{J_{S,2}}{N_2}\cdot \beta _2 \frac{A_{S,2}}{N_2} \right) \nonumber \\&= - (\mu _S + \lambda )\left( - \frac{\beta _2}{N_2}(\mu _I+ \lambda )(A_{S,2}+J_{S,2}) + (\mu _I+ \lambda )^2\right) \end{aligned}$$

(D36)

Likewise, for computing the determinant (B), we use the second row.

(D37)

We note that by grou** the two terms $\frac{(1-p)r}{k}A_{S,1}$ cancel out, and the final form of the determinant comes to be

(D38)

We then can follow the same logic as in the appendix B.