How heterogeneity in density dependence affects disease spread: when lifestyle matters

Figures & data

Figure 1. Flow diagram reflecting infection dynamics for groups 1 and 2, interconnected by the infection rates via the weighted average proportion of contacts made with infectives, $q=\frac{{\text{β}}_1{I}_1+{\text{β}}_2{I}_{2}N}{{\text{β}}_1{N}_1+{\text{β}}_2{N}_{2}N}$.

This diagram shows new arrivals entering either the class S1 or the class S2. There are flows from S1 to I1, from I1 to R1, from R1 back to S1, and from all three of these classes leaving the study population altogether. Similar flows exist among S2, I2, and R2.

Table 1. State variable and parameter definitions and their units.

	Notation	Definition
State variables	$S_i\left(t\right)$	Number of susceptible individuals in group i at time t
	$I_i\left(t\right)$	Number of infected individuals in group i at time t
	$R_i\left(t\right)$	Number of recovered individuals in group i at time t
Demographic parameters	p	Fraction of new arrivals assigned to group 1 (dimensionless)
	$(1-p)$	Fraction of new arrivals assigned to group 2 (dimensionless)
	r	Maximum growth rate ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
	K	Carrying capacity ($\mathrm{\#}\mathrm{p}\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{l}\mathrm{e}/{\mathrm{k}\mathrm{m}}^2$)
	d	Natural death rate ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
Epidemiological parameters	${\text{β}}_1$	per capita infectious contact rate for group 1 ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
	${\text{β}}_{2}N\left(t\right)$	per capita infectious contact rate for group 2 ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
	δ	Infection induced death rate ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
	γ	Recovery rate ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)
	θ	Immunity waning rate for recovered individuals ($1/\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$)

Table 2. Summary of estimated model parameters.

Parameter (unit)	Value
d (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	7.283$\times {10}^{-6}$
r (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	3.128$\times {10}^{-4}$
K (people)	12,600; 794,600
δ (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	0.0833
γ (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	0.1548
θ (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	0.000913
${\text{β}}_1$ (${\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	0.05
${\text{β}}_2$ (${\mathrm{p}\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{l}\mathrm{e}}^{-1}{\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$)	$5\times {10}^{-7}$
p (dimensionless)	0–1

Figure 2. The basic reproduction number ${\mathcal{R}}_0$ as a function of p for high (Buraydah, solid curve) and low (Ash Shimasiyah, dashed curve) values of $N^{\ast }$.

Figure 3. Variation in ${\mathcal{R}}_0$ as a function of p and $N^{\ast }$, using ${\text{β}}_1=0.15$/day, ${\text{β}}_2=3\times {10}^{-6}$/day.

Figure 4. Effects of population density $N^{\ast }$ on ${\mathcal{R}}_0$ for different values of p. ${\text{β}}_1,{\text{β}}_2$ as in Figure 3.

Figure 5. The population size which minimizes ${\mathcal{R}}_0$, as a function of p. ${\text{β}}_1,{\text{β}}_2$ as in Figure 3.

Figure 6. Variation in ${\mathcal{R}}_0$ as a function of p and $N^{\ast }$, using ${\text{β}}_1=0.25$/day, ${\text{β}}_2=5\times {10}^{-6}$/day.

Figure 7. The dark shaded region shows combinations of p and $N^{\ast }$ for which ${\mathcal{R}}_0<1$, using ${\text{β}}_1=0.25$/day, ${\text{β}}_2=5\times {10}^{-6}$/day.