An innovative fractional-order evolutionary game theoretical study of personal protection, quarantine, and isolation policies for combating epidemic diseases

Abstract

This study uses imposed control techniques and vaccination game theory to study disease dynamics with transitory or diminishing immunity. Our model uses the ABC fractional-order derivative mechanism to show the effect of non-pharmaceutical interventions such as personal protection or awareness, quarantine, and isolation to simulate the essential control strategies against an infectious disease spread in an infinite and uniformly distributed population. A comprehensive evolutionary game theory study quantified the significant influence of people's vaccination choices, with government forces participating in vaccination programs to improve obligatory control measures to reduce epidemic spread. This model uses the intervention options described above as a control strategy to reduce disease prevalence in human societies. Again, our simulated results show that a combined control strategy works exquisitely when the disease spreads even faster. A sluggish dissemination rate slows an epidemic outbreak, but modest control techniques can reestablish a disease-free equilibrium. Preventive vaccination regulates the border between the three phases, while personal protection, quarantine, and isolation methods reduce disease transmission in existing places. Thus, successfully combining these three intervention measures reduces epidemic or pandemic size, as represented by line graphs and 3D surface diagrams. For the first time, we use a fractional-order derivate to display the phase-portrayed trajectory graph to show the model's dynamics if immunity wanes at a specific pace, considering various vaccination cost and effectiveness settings.

Keywords: Evolutionary game; Fractional-order; Non-pharmaceutical intervention; Protection.

Figure 1

Flow diagram of the epidemiology model (1). The state variables and parameters of the model are described in Table 1.

Figure 2

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i), (*-ii), and (*-iii) show the results of awareness rate $a=0.1,0.5$ and $0.9,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.5,q=0.0,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,$ and $\theta =1/14$.

Figure 3

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the results of quarantine rate $q=0.1$ and $0.2,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.5,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0$ and $\theta =1/14$.

Figure 4

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the results of isolation rate $\mathrm{\zeta }=0.1$ and $0.2,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.5,q=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0$ and $\theta =1/14$.

Figure 5

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) shows the outcome of vaccine efficacy $\eta =0.5$ and $0.9$, respectively, whereas, remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,q=\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0$ and $\theta =1/14$.

Figure 6

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) shows the results of vaccine efficacy $\eta =0.5$ and $0.8$, respectively, whereas, remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\mathrm{q}=\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,a=0.5$ and $\theta =1/14$.

Figure 7

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i), (*-ii), and (*-iii) shows the results of government force to participate in vaccination program $A=0.1,0.5$ and $0.9,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.1,C_V=0.9,q=\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,m=0.1$ and $\theta =1/14$.

Figure 8

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i), (*-ii), and (*-iii) shows the results of government force to participate in vaccination program $A=0.1,0.5$ and $0.9,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.5,C_V=0.5,q=\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,m=0.1$ and $\theta =1/14$.

Figure 9

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i), (*-ii), and (*-iii) shows the results of government force to participate in vaccination program $A=0.1,0.5$ and $0.9,$ respectively, whereas the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =1/4,\eta =0.9,C_V=0.1,q=\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,m=0.1$ and $\theta =1/14$.

Figure 10

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the results of vaccination cost and efficacy $(C_V,\eta )=(0,9,0.1)$ and $(0,1,0.9)$ respectively, whereas, the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =0.25,q=\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.5,A=0.0,m=0.1$ and $\theta =1/14$.

Figure 11

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the outcomes of vaccination cost and efficacy $(C_V,\eta )=(0,9,0.1)$ and $(0,1,0.9)$ respectively, whereas, the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =0.25,q=0.1,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,A=0.0,m=0.1$ and $\theta =1/14$.

Figure 12

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the results of vaccination cost and efficacy $(C_V,\eta )=(0,9,0.1)$ and $(0,1,0.9)$ respectively, whereas, remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =0.25,,q=0.0,\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,A=0.0,m=0.1$ and $\theta =1/14$.

Figure 13

The effect of fractional-order $\alpha =0.8,0.85,0.9,0.95,1.0$ on the infected (Panel a-*), vaccinated (Panel b-*), and recovered (Panel c-*) individuals. Subpanels (*-i) and (*-ii) show the results under the vaccination cost and efficacy $(C_V,\eta )=(0,9,0.1)$ and $(0,1,0.9)$ respectively, whereas, the remaining parameter settings are $\beta =0.8333,\gamma =0.1,x=0.01,\sigma =0.25,q=0.1,\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,a=0.0,m=0.1,A=0.0$ and $\theta =1/14$.

Figure 14

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, and (C) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.0,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.5,a=0.0$ and $\theta =1/14$.

Figure 15

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, and (C) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.0,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.5$ and $\theta =1/14$.

Figure 16

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, (C) quarantined, and (D) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.1,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0$ and $\theta =1/14$.

Figure 17

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, (C) quarantined, and (D) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.1,\mathrm{\zeta }=0.0,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.3,$ and $\theta =1/14$.

Figure 18

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, (C) quarantined, and (D) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.0,\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 19

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, (C) quarantined, and (D) recovered populations, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.0,\mathrm{\zeta }=0.1,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.3,$ and $\theta =1/14$.

Figure 20

3D surface showing the effect of fractional order $(0\le \alpha \le 1)$ and time elapsed on the (A) infected, (B) vaccinated, (C) quarantined, and (D) recovered population, where parameters used are $\beta =0.8333,\gamma =0.1,\sigma =1/4,\eta =0.5,q=0.05,\mathrm{\zeta }=0.05,{\omega }_V=0.0,{\omega }_N=0.0,m=0.1,C_V=0.5,\eta =0.5,A=0.3,a=0.3,$ and $\theta =1/14$.

Figure 21

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of infected $(I(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.9,\eta =0.1,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 22

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of vaccinated $(V(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.9,\eta =0.1,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 23

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of quarantined and isolated $(J(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.9,\eta =0.1,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 24

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of recovered $(R(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.9,\eta =0.1,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 25

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of infected $(I(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 26

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of vaccinated $(V(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 27

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of quarantined and Isolated $(J(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 28

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of recovered $(R(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.5,\eta =0.5,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 29

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of infected $(I(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.1,\eta =0.9,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 30

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of vaaccinated $(V(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.1,\eta =0.9,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 31

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of quarantined and isolated $(J(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.1,\eta =0.9,A=0.0,a=0.0,$ and $\theta =1/14$.

Figure 32

The phase-portrayed fractional order $(\alpha =0.8,0.85,0.9,0.95,1.0)$ trajectories of recovered $(R(t))$ individuals concerning the waning rate of natural $\left({\omega }_N\right)$ and artificial $({\omega }_V)$ immunity. Subpanels (a*), (b*), and (c*) show the naturally achieved immunity waning rate $\left({\omega }_N,=,0.0,,,0.05,,,0.1\right),$ whereas (*i), (*ii), (*iii) artificial immunity waning rate $({\omega }_V=0.0,0.05,0.1)$, respectively. The remaining parameters settings are $\beta =0.8333,\gamma =0.1,\sigma =1/4,q=0.1,\mathrm{\zeta }=0.1,m=0.1,C_V=0.1,\eta =0.9,A=0.0,a=0.0,$ and $\theta =1/14$.