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Article Contents

# Tempered fractional order compartment models and applications in biology

• * Corresponding author

This work was supported by the National Natural Science Foundation of China under grant 41875084

• Compartment models with classical derivatives have diverse applications and attracted a lot of interest among scientists. To model the dynamical behavior of the particles that existed in the system for a long period of time with little chance to be removed, a power-law waiting time technique was introduced in the most recent work of Angstmann et al. [2]. The divergent first moment makes the power-law waiting time distribution less physical because of the finite lifespan of the particles. In this work, we take the tempered power-law function as the waiting time distribution, which has finite first moment while keeping the power-law properties. From the underlying physical stochastic process with the exponentially truncated power-law waiting time distribution, we build the tempered fractional compartment model. As an application, the tempered fractional SEIR epidemic model is proposed to simulate the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. (Colombia). Some analysis and numerical simulations are carried out around the equilibrium behavior.

Mathematics Subject Classification: Primary: 34A08, 60G22, 60K40, 92D30.

 Citation:

• Figure 1.  Flux flow of tempered fractional SEIR model

Figure 2.  The endemic steady state plotted as the functions of $\lambda_2$ $($Left$)$ and $\lambda_3$ $($Right$)$ for the tfrSEIR model

Figure 3.  The endemic steady state plotted as the functions of $\alpha_2$ $($Left$)$ and $\alpha_3$ $($Right$)$ for the tfrSEIR model

Figure 4.  Plots of Infected $I(t)$ with respect to time in the tfrSEIR model with $\alpha_2 = 0.5, 0.7, 0.9, \alpha_3 = 0.7, \lambda_2 = \lambda_3 = 0.0001$ $($Left$)$ and $\alpha_3 = 0.5, 0.7, 0.9, \alpha_2 = 0.7, \lambda_2 = \lambda_3 = 0.0001$ $($Right$)$

Figure 5.  Plots of Infected $I(t)$ with respect to time in the tfrSEIR model with $\lambda_2 = 0.0001, 0.001, 0.01, \lambda_3 = 0.0001, \alpha_2 = \alpha_3 = 0.7$ $($Left$)$ and $\lambda_3 = 0.0001, 0.001, 0.01, \lambda_2 = 0.0001, \alpha_2 = \alpha_3 = 0.7$ $($Right$)$

Figure 6.  The variation of Infected $I(t)$ in the tfrSEIR model for $\alpha_2 = 0.6,0.8,1$ $($Left$)$ and $\alpha_3 = 0.8,0.9,1$ $($Right$)$

Figure 7.  The variation of Infected $I(t)$ in the tfrSEIR model for $\beta = 2.69e-3,2.79e-3,2.89e-3.$

Figure 8.  The variation of Infected $I(t)$ in the tfrSEIR model for $\lambda_2 = 0.2,0.05,0.001$ $($Left$)$ and $\lambda_3 = 0.09,0.07,0.05$ $($Right$)$

Figure 9.  Comparing with the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. $($Colombia$)$, the numerical solutions of the classical SEIR model $($Left$)$ and the frSEIR model $($Right$)$ with $\alpha_2 = 0.6$, $\alpha_3 = 0.88$

Figure 10.  Numerical solution of the tfrSEIR model adjusted to real data of people infected with influenza AH1N1/09 in Bogotá D.C. $($Colombia$)$ with $\alpha_2 = 0.6$, $\alpha_3 = 0.88$, $\lambda_2 = 0.015$, $\lambda_3 = 0.09$

Table 1.  Data provided by the Secretaria be Salud Distrital de Bogotá D.C. First row corresponds to the week number of year 2009. Second row presents the number of infectious individuals by AH1N1/09 detected in each week

 Week 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Cases 4 1 0 1 2 5 12 17 22 16 15 53 55 45 Week 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Cases 84 102 127 261 210 155 109 116 125 76 52 19 15 1

Table 2.  Values of various parameters and initial conditions

 Model $\mu$ $\mu_1$ $\beta$ $\Omega_2$ $\Omega_3$ $\alpha_2$ $\alpha_3$ cSEIR 5.038e+01 1.0e-08 2.89e-03 2.25e-01 1.373e+00 1 1 frSEIR 5.038e+01 1.0e-08 2.89e-03 2.25e-01 1.373e+00 0.6 0.88 tfrSEIR 5.038e+01 1.0e-08 2.89e-03 2.25e-01 1.373e+00 0.6 0.88 Model $\lambda_2$ $\lambda_3$ $S_0$ $E_0$ $I_0$ $R_0$ cSEIR 0 0 100 0 1 0 frSEIR 0 0 100 0 1 0 tfrSEIR 0.015 0.09 100 0 1 0
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