Demographic population cycles and ℛ₀ in discrete-time epidemic models

Figures & data

Figure 1. Period-doubling bifurcations in the disease-free Equation (15(15) $S_{t+1}=(r{e}^{-b{S}_t}+(1-d))S_t.$(15) ), where d=0.5, b=0.1, on the horizontal axis $20\le r\le 140$ and on the vertical axis $20\le S\le 120.$

Figure 2. Bifurcation diagram for SIR model: As ${\mathcal{R}}_0$ increases from values less than 1 to values greater than 1, the model dynamics changes from disease extinction to disease persistence on a locally asymptotically stable period 2 population cycle.

Figure 3. Model (7(7) $\begin{array}{rl}{S}_{t+1}& =g\left(N_t\right)+(1-d)S_t\varphi \left(I_t\right)\\ I_{t+1}& =(1-d)S_t(1-\varphi \left(I_t\right))+(1-\gamma )(1-d)I_t\\ R_{t+1}& =\gamma (1-d)I_t+(1-d)R_t,\end{array}$(7) ) with $g\left(N_t\right)=r{N}_t{e}^{-b{N}_t}$ has a locally asymptotically stable endemic period-2 population cycle when $b=0.1,\gamma =d=0.5,\beta =0.05\mathrm{and}r=e^4$.

Figure 4. Susceptible and recovered disease-free populations of Model (19(19) $\begin{array}{rl}{S}_{t+1}& =(1-p)r(S_t+R_t)e^{-b(S_t+R_t)}+(1-d)S_t\\ R_{t+1}& =pr(S_t+R_t)e^{-b(S_t+R_t)}+(1-d)R_t.\end{array}$(19) ) undergo period-doubling bifurcations as r is varied between 20 and 140, where p=0.78, d=0.5 and b=0.1.

Figure 5. Bifurcation diagram for ISAv Model (20(20) $\begin{array}{rl}{S}_{t+1}& =r{S}_t{e}^{-b{S}_t}+\hat{d}{S}_t(\theta {\varphi }_I\left(I_t\right)+\hat{\theta }{\varphi }_V\left(V_t\right))\\ I_{t+1}& =\hat{d}(S_t(\theta {\hat{\varphi }}_I\left(I_t\right)+\hat{\theta }{\hat{\varphi }}_V\left(V_t\right))+\hat{\mu }{I}_t)\\ V_{t+1}& ={\hat{d}}_V(V_t+\delta I_t),\end{array}$(20) ): As ${\mathcal{R}}_0$ increases from values less than 1 to values greater than 1, the model dynamics changes from disease extinction to disease persistence on a periodic or chaotic population cycle.

Figure 6. Model (20(20) $\begin{array}{rl}{S}_{t+1}& =r{S}_t{e}^{-b{S}_t}+\hat{d}{S}_t(\theta {\varphi }_I\left(I_t\right)+\hat{\theta }{\varphi }_V\left(V_t\right))\\ I_{t+1}& =\hat{d}(S_t(\theta {\hat{\varphi }}_I\left(I_t\right)+\hat{\theta }{\hat{\varphi }}_V\left(V_t\right))+\hat{\mu }{I}_t)\\ V_{t+1}& ={\hat{d}}_V(V_t+\delta I_t),\end{array}$(20) ) with $g\left(S_t\right)=r{S}_t{e}^{-b{S}_t}$ has a locally asymptotically stable period 4 population cycle when ${\beta }_I=0.056$ and all the other parameter values are kept fixed at their current values in Figure 5.