Dynamics of low and high pathogenic avian influenza in wild and domestic bird populations

Figures & data

Figure 1. Flow chart of model (1(1) $\begin{array}{rl}\frac{\mathrm{d}{S}_{\mathrm{w}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{w}}-{\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\mu }_{\mathrm{w}}{S}_{\mathrm{w}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{L}}_{\mathrm{w}}})I_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{w}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{r}_{{\mathrm{L}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{r}_{{\mathrm{L}}_{\mathrm{w}}},\\ r_{{\mathrm{L}}_{\mathrm{w}}}(0,t)& ={\alpha }_{{\mathrm{L}}_{\mathrm{w}}}{I}_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}+{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{w}}\left(\tau \right)r_{{\mathrm{L}}_{\mathrm{w}}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{H}}_{\mathrm{w}}}+{\nu }_{{\mathrm{H}}_{\mathrm{w}}})I_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{R}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\alpha }_{{\mathrm{H}}_{\mathrm{w}}}{I}_{{\mathrm{H}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{R}_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{S}_{\mathrm{d}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{d}}-{\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{S}_{\mathrm{d}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-({\mu }_{\mathrm{d}}+{\alpha }_{\mathrm{d}})I_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{d}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{r}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{r}_{\mathrm{d}},\\ r_{\mathrm{d}}(0,t)& ={\alpha }_{\mathrm{d}}{I}_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}+{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{d}}\left(\tau \right)r_{\mathrm{d}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{d}}+{\nu }_{{\mathrm{H}}_{\mathrm{d}}})I_{{\mathrm{H}}_{\mathrm{d}}}.\end{array}$(1) ).

Table 1. Definition of the variables of model (1(1) $\begin{array}{rl}\frac{\mathrm{d}{S}_{\mathrm{w}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{w}}-{\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\mu }_{\mathrm{w}}{S}_{\mathrm{w}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{L}}_{\mathrm{w}}})I_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{w}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{r}_{{\mathrm{L}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{r}_{{\mathrm{L}}_{\mathrm{w}}},\\ r_{{\mathrm{L}}_{\mathrm{w}}}(0,t)& ={\alpha }_{{\mathrm{L}}_{\mathrm{w}}}{I}_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}+{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{w}}\left(\tau \right)r_{{\mathrm{L}}_{\mathrm{w}}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{H}}_{\mathrm{w}}}+{\nu }_{{\mathrm{H}}_{\mathrm{w}}})I_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{R}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\alpha }_{{\mathrm{H}}_{\mathrm{w}}}{I}_{{\mathrm{H}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{R}_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{S}_{\mathrm{d}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{d}}-{\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{S}_{\mathrm{d}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-({\mu }_{\mathrm{d}}+{\alpha }_{\mathrm{d}})I_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{d}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{r}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{r}_{\mathrm{d}},\\ r_{\mathrm{d}}(0,t)& ={\alpha }_{\mathrm{d}}{I}_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}+{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{d}}\left(\tau \right)r_{\mathrm{d}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{d}}+{\nu }_{{\mathrm{H}}_{\mathrm{d}}})I_{{\mathrm{H}}_{\mathrm{d}}}.\end{array}$(1) ).

Variable	Meaning	Variable	Meaning
$S_{\mathrm{w}}$	Population of susceptible wild birds	$S_{\mathrm{d}}$	Population of susceptible domestic birds
$I_{{\mathrm{L}}_{\mathrm{w}}}$	Population of LPAI-infected wild birds	$I_{{\mathrm{L}}_{\mathrm{d}}}$	Population of LPAI-infected domestic birds
$I_{{\mathrm{H}}_{\mathrm{w}}}$	Population of HPAI-infected wild birds	$I_{{\mathrm{H}}_{\mathrm{d}}}$	Population of HPAI-infected domestic birds
$r_{{\mathrm{L}}_{\mathrm{w}}}$	Density of wild birds that have recovered from LPAI	$r_{\mathrm{d}}$	Density of domestic birds that have recovered from LPAI
$R_{{\mathrm{H}}_{\mathrm{w}}}$	Population of wild birds that have recovered from HPAI

Table 2. Definition of the parameters of model (1(1) $\begin{array}{rl}\frac{\mathrm{d}{S}_{\mathrm{w}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{w}}-{\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}-{\mu }_{\mathrm{w}}{S}_{\mathrm{w}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{w}}}{S}_{\mathrm{w}}-({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{L}}_{\mathrm{w}}})I_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{{\mathrm{L}}_{\mathrm{w}}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{w}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{r}_{{\mathrm{L}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{r}_{{\mathrm{L}}_{\mathrm{w}}},\\ r_{{\mathrm{L}}_{\mathrm{w}}}(0,t)& ={\alpha }_{{\mathrm{L}}_{\mathrm{w}}}{I}_{{\mathrm{L}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{S}_{\mathrm{w}}+{\lambda }_{{\mathrm{H}}_{\mathrm{w}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{w}}\left(\tau \right)r_{{\mathrm{L}}_{\mathrm{w}}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{w}}+{\alpha }_{{\mathrm{H}}_{\mathrm{w}}}+{\nu }_{{\mathrm{H}}_{\mathrm{w}}})I_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{R}_{{\mathrm{H}}_{\mathrm{w}}}}{\mathrm{d}t}& ={\alpha }_{{\mathrm{H}}_{\mathrm{w}}}{I}_{{\mathrm{H}}_{\mathrm{w}}}-{\mu }_{\mathrm{w}}{R}_{{\mathrm{H}}_{\mathrm{w}}},\\ \frac{\mathrm{d}{S}_{\mathrm{d}}}{\mathrm{d}t}& ={\mathrm{\Lambda }}_{\mathrm{d}}-{\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{S}_{\mathrm{d}},\\ \frac{\mathrm{d}{I}_{{\mathrm{L}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{L}}_{\mathrm{d}}}{S}_{\mathrm{d}}-({\mu }_{\mathrm{d}}+{\alpha }_{\mathrm{d}})I_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{r}_{\mathrm{d}}}{\mathrm{\partial }\tau }& =-q_{\mathrm{d}}\left(\tau \right){\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{r}_{\mathrm{d}}-{\mu }_{\mathrm{d}}{r}_{\mathrm{d}},\\ r_{\mathrm{d}}(0,t)& ={\alpha }_{\mathrm{d}}{I}_{{\mathrm{L}}_{\mathrm{d}}},\\ \frac{\mathrm{d}{I}_{{\mathrm{H}}_{\mathrm{d}}}}{\mathrm{d}t}& ={\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{S}_{\mathrm{d}}+{\lambda }_{{\mathrm{H}}_{\mathrm{d}}}{\int }_0^{\mathrm{\infty }}{q}_{\mathrm{d}}\left(\tau \right)r_{\mathrm{d}}(\tau ,t)\mathrm{d}\tau -({\mu }_{\mathrm{d}}+{\nu }_{{\mathrm{H}}_{\mathrm{d}}})I_{{\mathrm{H}}_{\mathrm{d}}}.\end{array}$(1) ).

Parameter	Meaning
${\mathrm{\Lambda }}_{\mathrm{d}}$	Birth/recruitment rate of domestic birds
${\mathrm{\Lambda }}_{\mathrm{w}}$	Birth/recruitment rate of wild birds
${\mu }_{\mathrm{d}}$	Natural death rate of domestic birds
${\mu }_{\mathrm{w}}$	Natural death rate of wild birds
${\nu }_{{\mathrm{H}}_{\mathrm{d}}}$	HPAI-induced mortality rate for domestic birds
${\nu }_{{\mathrm{H}}_{\mathrm{w}}}$	HPAI-induced mortality rate for wild birds
${\alpha }_{\mathrm{d}}$	Recovery rate of domestic birds from LPAI
${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}$	Recovery rate of wild birds from LPAI
${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}$	Recovery rate of wild birds from HPAI
${\beta }_{11}^{\mathrm{L}}/{\beta }_{11}^{\mathrm{H}}$	LPAI/HPAI transmission rate to susceptible wild birds from infected wild birds
${\beta }_{12}^{\mathrm{L}}/{\beta }_{12}^{\mathrm{H}}$	LPAI/HPAI transmission rate to susceptible wild birds from infected domestic birds
${\beta }_{22}^{\mathrm{L}}/{\beta }_{22}^{\mathrm{H}}$	LPAI/HPAI transmission rate to susceptible domestic birds from infected domestic birds
${\beta }_{21}^{\mathrm{L}}/{\beta }_{21}^{\mathrm{H}}$	LPAI/HPAI transmission rate to susceptible domestic birds from infected wild birds
$q_{\mathrm{w}}\left(\tau \right)$	Relative susceptibility of LPAI-recovered wild birds towards HPAI
$q_{\mathrm{d}}\left(\tau \right)$	Relative susceptibility of LPAI-recovered domestic birds towards HPAI

Table 3. Parameter ranges.

Parameter	Range	Average
${\mu }_{\mathrm{w}}$	0.027–1 year${}^{-1}$	0.1
${\mathrm{\Lambda }}_{\mathrm{w}}$	1000–3000 birds/year	1500
${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}$	365/21–365/2 year${}^{-1}$	36.5
${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}$	365/21–365/2 year${}^{-1}$	36.5
${\nu }_{{\mathrm{H}}_{\mathrm{w}}}$	365/21–365/2 year${}^{-1}$	36.5
$q_{\mathrm{w}}$	$0-1$	0.5
${\mu }_{\mathrm{d}}$	0.2–2 year${}^{-1}$	0.5
${\mathrm{\Lambda }}_{\mathrm{d}}$	1000–3000 birds/year	1500
${\alpha }_{{\mathrm{L}}_{\mathrm{d}}}$	365/21–365/2 year${}^{-1}$	36.5
${\nu }_{{\mathrm{H}}_{\mathrm{d}}}$	365/5–365/2 year${}^{-1}$	80.0
$q_{\mathrm{d}}$	0–1	0.5

Table 4. Source-sink status of birds to AI viruses.

	LPAI	HPAI
Wild birds	Source	?
Domestic birds	Sink	Source

Figure 2. Coexistence with realistic parameter values. The parameter values used in the figure are as follows: ${\mathrm{\Lambda }}_{\mathrm{w}}=2000$, ${\mu }_{\mathrm{w}}=0.25$, ${\nu }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}=73$, $q_{\mathrm{w}}=0.5$, ${\beta }_{11}^{\mathrm{L}}=.018776$, ${\beta }_{11}^{\mathrm{H}}=0.005$, ${\mathrm{\Lambda }}_{\mathrm{d}}=1020$, ${\mu }_{\mathrm{d}}=0.5$, ${\nu }_{{\mathrm{H}}_{\mathrm{d}}}=36.5$, ${\alpha }_{\mathrm{d}}=52.14$, $q_{\mathrm{d}}=0.5$, ${\beta }_{22}^{\mathrm{L}}=.02539$, ${\beta }_{22}^{\mathrm{H}}=0.04897$, ${\beta }_{12}^{\mathrm{L}}=0.006$, ${\beta }_{21}^{\mathrm{L}}=0.03$, ${\beta }_{12}^{\mathrm{H}}=0.002$, ${\beta }_{21}^{\mathrm{H}}=0.031$. The reproduction numbers are ${\mathcal{R}}_{\mathrm{L}}=2.54$ and ${\mathcal{R}}_{\mathrm{H}}=2.71$. The invasion coefficients are as follows: ${\hat{\mathcal{R}}}_{\mathrm{L}}^{\mathrm{H}}=1.75$ and ${\hat{\mathcal{R}}}_{\mathrm{H}}^{\mathrm{L}}=1.98$. The red line shows HPAI in wild birds, the orange dashed line shows HPAI in domestic birds, the blue line shows LPAI in wild birds, the green dashed line shows LPAI in domestic birds.

Figure 3. Oscillations with realistic parameter values. The parameter values used in the figure are as follows: ${\mathrm{\Lambda }}_{\mathrm{w}}=2000$, ${\mu }_{\mathrm{w}}=0.25$, ${\nu }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}=73$, $q_{\mathrm{w}}=0.45$, ${\beta }_{11}^{\mathrm{L}}=.018776$, ${\beta }_{11}^{\mathrm{H}}=0.015$, ${\mathrm{\Lambda }}_{\mathrm{d}}=1020$, ${\mu }_{\mathrm{d}}=0.5$, ${\nu }_{{\mathrm{H}}_{\mathrm{d}}}=36.5$, ${\alpha }_{\mathrm{d}}=52.14$, $q_{\mathrm{d}}=0.5$, ${\beta }_{22}^{\mathrm{L}}=.025$, ${\beta }_{22}^{\mathrm{H}}=0.04897$, ${\beta }_{12}^{\mathrm{L}}=0.006$, ${\beta }_{21}^{\mathrm{L}}=0.03$, ${\beta }_{12}^{\mathrm{H}}=0.0$, ${\beta }_{21}^{\mathrm{H}}=0.031$. The reproduction numbers are ${\mathcal{R}}_{\mathrm{L}}=2.54$ and ${\mathcal{R}}_{\mathrm{H}}=2.7$. The invasion coefficients are as follows: ${\hat{\mathcal{R}}}_{\mathrm{L}}=1.37$ and ${\hat{\mathcal{R}}}_{\mathrm{H}}=1.86$. The red line shows HPAI in wild birds, the orange dashed line shows HPAI in domestic birds, the blue line shows LPAI in wild birds, the green dashed line shows LPAI in domestic birds.

Figure 4. Coexistence with realistic parameter values. The parameter values used in the figure are as follows: ${\mathrm{\Lambda }}_{\mathrm{w}}=2000$, ${\mu }_{\mathrm{w}}=0.25$, ${\nu }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}=36.5$, ${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}=73$, $q_{\mathrm{w}}=0.426$, ${\beta }_{11}^{\mathrm{L}}=.0086$, ${\beta }_{11}^{\mathrm{H}}=0.005$, ${\mathrm{\Lambda }}_{\mathrm{d}}=1020$, ${\mu }_{\mathrm{d}}=0.5$, ${\nu }_{{\mathrm{H}}_{\mathrm{d}}}=36.5$, ${\alpha }_{\mathrm{d}}=52.14$, $q_{\mathrm{d}}=1$, ${\beta }_{22}^{\mathrm{L}}=.02539$, ${\beta }_{22}^{\mathrm{H}}=0.0166$, ${\beta }_{12}^{\mathrm{L}}=0.0043$, ${\beta }_{21}^{\mathrm{L}}=0.0131$, ${\beta }_{12}^{\mathrm{H}}=0.0014$, ${\beta }_{21}^{\mathrm{H}}=0.0332$. The reproduction numbers are ${\mathcal{R}}_{\mathrm{L}}=1.45$ and ${\mathcal{R}}_{\mathrm{H}}=1.29$. The invasion coefficients are as follows: $\widehat{{\mathcal{R}}_{\mathrm{L}}}=1.17$ and $\widehat{{\mathcal{R}}_{\mathrm{H}}}=1.22$. The red line shows HPAI in wild birds, the orange dashed line shows HPAI in domestic birds, the blue line shows LPAI in wild birds, the green dashed line shows LPAI in domestic birds.

Table

Reproduction numbers	Values	Invasion numbers	Values
${\mathcal{R}}_{11}^{\mathrm{L}}$	0.94	${\hat{\mathcal{R}}}_{11,\mathrm{L}}^{\mathrm{H}}$	0.46
${\mathcal{R}}_{12}^{\mathrm{L}}$	0.65	${\hat{\mathcal{R}}}_{12,\mathrm{L}}^{\mathrm{H}}$	0.26
${\mathcal{R}}_{21}^{\mathrm{L}}$	0.36	${\hat{\mathcal{R}}}_{21,\mathrm{L}}^{\mathrm{H}}$	0.92
${\mathcal{R}}_{22}^{\mathrm{L}}$	0.98	${\hat{\mathcal{R}}}_{22,\mathrm{L}}^{\mathrm{H}}$	0.91
${\mathcal{R}}_{11}^{\mathrm{H}}$	0.55	${\hat{\mathcal{R}}}_{11,\mathrm{H}}^{\mathrm{L}}$	0.86
${\mathcal{R}}_{12}^{\mathrm{H}}$	0.3	${\hat{\mathcal{R}}}_{12,\mathrm{H}}^{\mathrm{L}}$	0.6
${\mathcal{R}}_{21}^{\mathrm{H}}$	0.92	${\hat{\mathcal{R}}}_{21,\mathrm{H}}^{\mathrm{L}}$	0.26
${\mathcal{R}}_{22}^{\mathrm{H}}$	0.91	${\hat{\mathcal{R}}}_{22,\mathrm{H}}^{\mathrm{L}}$	0.69

Figure 5. Sustained oscillations in the wild birds only system. Parameter values are ${\mathrm{\Lambda }}_{\mathrm{w}}=2000$, ${\mu }_{\mathrm{w}}=0.14$, ${\nu }_{{\mathrm{H}}_{\mathrm{w}}}=49.5$, ${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}=51.6$, ${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}=73$, $q_{\mathrm{w}}=0.98$, ${\beta }_{11}^{\mathrm{L}}=.018776$, ${\beta }_{11}^{\mathrm{H}}=0.015$, $S_{\mathrm{w}}\left(0\right)=3449.72$, $I_{{\mathrm{L}}_{\mathrm{w}}}\left(0\right)=14.684$, $R_{{\mathrm{L}}_{\mathrm{w}}}\left(0\right)=3366.78$, $I_{{\mathrm{H}}_{\mathrm{w}}}\left(0\right)=7$, $R_{{\mathrm{H}}_{\mathrm{w}}}\left(0\right)=769.5$. The reproduction numbers are ${\mathcal{R}}_{\mathrm{L}}=3.66$ and ${\mathcal{R}}_{\mathrm{H}}=2.116$. The invasion coefficients are ${\hat{\mathcal{R}}}_{\mathrm{L}}=1.73$ and ${\hat{\mathcal{R}}}_{\mathrm{H}}=2.08$. The red line shows HPAI and the blue line shows LPAI.

Figure 6. Sustained oscillations in the wild birds only system. Parameter values are ${\mathrm{\Lambda }}_{\mathrm{w}}=3810$, ${\mu }_{\mathrm{w}}=0.054$, ${\nu }_{{\mathrm{H}}_{\mathrm{w}}}=87.5$, ${\alpha }_{{\mathrm{H}}_{\mathrm{w}}}=87.4$, ${\alpha }_{{\mathrm{L}}_{\mathrm{w}}}=69.4$, $q_{\mathrm{w}}=0.99$, ${\beta }_{11}^{\mathrm{L}}=.0131$, ${\beta }_{11}^{\mathrm{H}}=0.01$, $S_{\mathrm{w}}\left(0\right)=5000$, $I_{{\mathrm{L}}_{\mathrm{w}}}\left(0\right)=40$, $R_{{\mathrm{L}}_{\mathrm{w}}}\left(0\right)=12000$, $I_{{\mathrm{H}}_{\mathrm{w}}}\left(0\right)=15$, $R_{{\mathrm{H}}_{\mathrm{w}}}\left(0\right)=25,000$. The reproduction numbers are ${\mathcal{R}}_{\mathrm{L}}=13.3$ and ${\mathcal{R}}_{\mathrm{H}}=4.0$. The invasion coefficients are ${\hat{\mathcal{R}}}_{\mathrm{L}}=3.3$ and ${\hat{\mathcal{R}}}_{\mathrm{H}}=4$. The red line shows HPAI and the blue line shows LPAI.