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Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment

  • Received: 01 August 2020 Revised: 01 October 2020 Published: 24 November 2020
  • Primary: 35C07, 35B40, 35K57; Secondary: 92D30

  • A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number R0 and wave speed c, is firstly proved as R0>1 and cc via the Schauder fixed point theorem, where c is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when R01 or R0>1 with c(0, c), we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.

    Citation: Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment[J]. Electronic Research Archive, 2021, 29(3): 2325-2358. doi: 10.3934/era.2020118

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  • A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number R0 and wave speed c, is firstly proved as R0>1 and cc via the Schauder fixed point theorem, where c is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when R01 or R0>1 with c(0, c), we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.



    In 2014, an outbreak of Ebola virus (Ebola) decimated many people in Western Africa. With more than 16,000 clinically confirmed cases and approximately 70% mortality cases, this was the more deadly outbreak compared to 20 Ebola threats that occurred since 1976 [1]. In Africa, and particularly in the regions that were affected by Ebola outbreaks, people live close to the rain-forests, hunt bats and monkeys and harvest forest fruits for food [2], [3].

    In [4] develop a SIR type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses with stability and numerical analysis is discussed. A number of mathematical models have been developed to understand the transmission dynamics of Ebola and other infectious diseases outbreak from various aspects [5], [6]. A commonly used model for characterising epidemics of diseases including Ebola is the susceptible-exposed-infectious-recovered (SEIR) model [7], and extensions to this basic model include explicit incorporation of transmission from Ebola deceased hosts [1], [8] or accounting for mismatches between symptoms and infectiousness [9], [10].

    Many researchers and mathematicians have shown that fractional extensions of mathematical integer-order models are a very systematic representation of natural reality [11], [12], [13]. Recently, a non-integer-order idea is given by Caputo and Fabrizio [14]. The primary goal of this article is to use a fresh non-integer order derivative to study the model of diabetes and to present information about the diabetes model solution's uniqueness and existence using a fixed point theorem [15]. Atangana and Baleanu [16] then proposed another non-singular derivative version using the Mittag Leffler kernel function. In many apps in the actual globe, these operators have been successful [17], [18], [19]. The few existing works [4], [8], [9], [20] on the mathematical modeling tells transmission of the virus and spread of Ebola virus on the population of human. The classical settings of mathematical studies tells about spread of EVD, such as SI model, SIR model, SEIR model [4], SEIRD model, or SEIRHD model. World medical association invented medicines for Ebola virus. Quantitative approaches and obtaining an analysis of the reproduction number of Ebola outbreak were important modeling for EVD epidemics. Demographic data on Ebola risk factors and on the transmission of virus were studied through the household structured epidemic model [4], [21]. Predications, different valuable insights, personal and genomic data for EVD was reported and discovered through mathematical models [22], [23]. In [24], the authors observed spread that follows a fading memory process and also shows crossover behaviour for the EVD. They captured this kind of spread using differential operators that posses crossover properties and fading memory using the SIRDP model in [4]. They also analyzed the Ebola disease dynamic by considering the Caputo, Caputo-Fabrizio, and Atangana-Baleanu differential operators.

    In this paper, we developed fractional order Ebola virus model by using the Caputo method of complex nonlinear differential equations. Caputo fractional derivative operator β ∈ (0,1] works to achieve the fractional differential equations. Laplace with Adomian Decomposition Methodsuccessfully solved the fractional differential equations. Ultimately, numerical simulations are also developed to evaluate the effects of the device parameter on spread of disease and effect of fractional parameter β on obtained solution which are also assessed by tabulated results.

    The classical model for Ebola virus model is given in [4], we developed the fractional order Ebola virus model in the followings equations

    Dφ1S(t)=Π(β1I+β2D+λP)SµSDφ2I(t)=(β1I+β2D+λP)S(µ+δ+γ)IDφ3R(t)=γIµRDφ4D(t)=(µ+δ)IbDDφ5P(t)=σ+ξI+αDηP
    µµµµ
    with initial conditions
    S(0)=N1,I(0)=N2,R(0)=N3,D(0)=N4,P(0)=N5
    Where S(t) represent the susceptible individuals, I(t) the individuals infected, R(t) the individuals recovered from the EVD, D(t) the individual that died with the Ebola virus and P(t) in the virus concentration in the environment. The susceptible human population is replenished by a constant recruitment at rate Pi. susceptible individuals S may acquire infection after effective contacts β1 with infectious and β2 is effective contact rate of deceased human individuals. They can also catch the infection through contact with a contaminated environment at rate λ. Infectious individuals I experience an additional death due to the disease at rate δ and they are recovered at rate γ. Deceased human individuals can be buried directly during funerals at rate b. Susceptible, infectious and recovered individuals die naturally at rate µ. η, ξ, α, represent the decay rate, shedding rate of infected, and shedding rate of deceased, respectively. The recruitment rate of the Ebola virus in the environment expressed as σ.

    Here system (2.1) is analyzed qualitatively analyzed for feasibility and numerical solution at disease free and endemic equilibrium point. For this purpose, we used

    Dφ1S(t)=Dφ2I(t)=Dφ3R(t)=Dφ4D(t)=Dφ5P(t)=0
    in system (1). For disease free equilibrium, we have E = (π/µ,0,0,0,0) and endemic equilibrium is
    E*=(S*,I*,R*,D*,P*),
    where
    S*=πµR0;I*=π(R01)(µ+δ+γ)R0;R*=πγ(R01)µ(µ+δ+γ)R0;D*=π(µ+δ)(R01)b(µ+δ+γ)R0
    µµµµµµ
    P*=π(bξ+αδ+αµ)(R01)bη(µ+δ+γ)R0
    µµ
    is endemic equilibria of the system (1). Where reproductive number is
    R0=ηπ(bβ1+β2(µ+δ))+λπ(bξ+αδ+αµ)bηµ(µ+δ+γ)
    µµµµ

    Theorem. 1 There is a unique solution for the initial value problem given in system (2.1), and the solution remains in R5, x ≥ 0.

    Proof: We need to show that the domain R5, x ≥ 0 is positively invariant. Since

    Dφ1S(t)|S=0=Π0Dφ2I(t)|I=0=(β1I+β2D+λP)S0Dφ3R(t)|R=0=γI0Dφ4D(t)|D=0=(µ+δ)I0Dφ5P(t)|P=0=σ+ξI+αD0
    µ
    Hence the solution lies in feasible domain, so the uniqueness and solution of the system exists.

    Consider the fractional-order Ebola virus model (2.1), by using Caputo definition with Laplace transform, we have

    {Dφ1S(t)}=Π{1}β1{IS}β2{DS}λ{PS}µ{S}{Dφ2I(t)}=β1{IS}+β2{DS}+λ{PS}(µ+δ+γ){I}{Dφ3R(t)}=γ{I}µ{R}{Dφ4D(t)}=(µ+δ){I}b{D}{Dφ5P(t)}=σ{1}+ξ{I}+α{D}η{P}
    Sφ1{S(t)}Sφ11S(0)=Π{1}β1{IS}β2{DS}λ{PS}µ{S}Sφ2{I(t)}Sφ21I(0)=β1{IS}+β2{DS}+λ{PS}(µ+δ+γ){I}Sφ3{R(t)}Sφ31R(0)=γ{I}µ{R}Sφ4{D(t)}Sφ41D(0)={µ+δ}{I}b{D}Sφ5{P(t)}Sφ51P(0)=σ{1}+ξ{I}+α{D}η{P}
    by using the initial conditions (2.2), we get
    {S(t)}=N1S+ΠSφ1+1β1Sφ1{IS}β2Sφ1{DS}λSφ1{PS}µSφ1{S}{I(t)}=N2S+β1Sφ2{IS}+β2Sφ2{DS}+λSφ2{PS}µ+δ+γSφ2{I}{R(t)}=N3S+γSφ3{I}µSφ3{R}{D(t)}=N4S+µ+δSφ4{I}bSφ4{D}{P(t)}=N5S+σSφ5+1+ξSφ5{I}+αSφ5{D}ηSφ5{P}
    We have followings infinite series solution
    S=k=0Sk,I=k=0Ik,R=k=0Rk,D=k=0Dk,P=k=0Pk
    The nonlinearity IS, DS and PS can be written as
    IS=k=0Ak,DS=k=0Bk,PS=k=0Ck
    where Ak, Bk and Ck is called the Adomian polynomials. We have the followings results
    {S0}=N1S+ΠSφ1+1,{I0}=N2S,{R0}=N3S,{D0}=N4S,{P0}=N5S+σSφ5+1
    Similarly, we have
    {S1}=β1Sφ1{A0}β2Sφ1{B0}λSφ1{C0}µSφ1{S0},...{Sk+1}=β1Sφ1{Ak}β2Sφ1{Bk}λSφ1{Ck}µSφ1{Sk}
    {I1}=β1Sφ2{A0}+β2Sφ2{B0}+λSφ2{C0}µ+δ+γSφ2{I0},...{Ik+1}=β1Sφ2{Ak}+β2Sφ2{Bk}+λSφ2{Ck}µ+δ+γSφ2{Ik}
    {R1}=γSφ3{I0}µSφ3{R0},...{Rk+1}=γSφ3{Ik}µSφ3{Rk}
    {D1}=µ+δSφ4{I0}bSφ4{D0},...{Dk+1}=µ+δSφ4{Ik}bSφ4{Dk}
    {P1}=ξSφ5{I0}+αSφ5{D0}ηSφ5{P0},...{Pk+1}=ξSφ5{Ik}+αSφ5{Dk}ηSφ5{Pk}

    We get the followings generalized form for analysis and numerical solution.

    {Sk+1}=β1Sφ1{Ak}β2Sφ1{Bk}λSφ1{Ck}µSφ1{Sk}
    {Ik+1}=β1Sφ2{Ak}+β2Sφ2{Bk}+λSφ2{Ck}µ+δ+γSφ2{Ik}
    {Rk+1}=γSφ3{Ik}µSφ3{Rk}
    {Dk+1}=µ+δSφ4{Ik}bSφ4{Dk}
    {Pk+1}=ξSφ5{Ik}+αSφ5{Dk}ηSφ5{Pk}

    The results of fractional order model (2.1) is represented in followings tables and graphs.

    Table 1.  Numerical solution of S(t) with at different fractional values ϕ.
    t ϕ = 1 ϕ = 0.9 ϕ = 0.8 ϕ = 0.5
    1 39.6912 39.5848 39.51 39.4383
    1.5 39.3299 39.1117 39.0235 39.1181
    3 37.2519 36.2268 36.264 37.5584
    4.5 32.0652 29.2568 30.0105 34.514
    6 19.0984 14.0068 17.2162 29.3074

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical solution of I(t) with at different fractional values ϕ.
    t ϕ = 1 ϕ = 0.9 ϕ = 0.8 ϕ = 0.7
    1 10.4879 10.6026 10.6542 10.7698
    2 12.052 11.9378 11.8579 11.5689
    4 13.2768 12.5387 12.2499 11.8581
    6 8.7256 9.97166 10.5704 12.7973
    8 5.0464 11.4457 13.7013 19.4353

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical solution of R(t) with at different fractional values ϕ.
    t ϕ = 1 ϕ = 0.9 ϕ = 0.8 ϕ = 0.7
    2 20.504 20.5195 20.5331 20.5414
    4 21.952 21.8288 21.6815 21.509
    6 25.448 24.6081 23.8154 23.076
    8 32.096 29.4003 27.1489 25.2921
    10 43 36.6889 31.8489 28.1877

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical solution of D(t) with at different fractional values ϕ.
    t ϕ = 1 ϕ = 0.9 ϕ = 0.8 ϕ = 0.7
    0.5 10.5931 10.6768 10.7803 10.9131
    1 11.3486 11.5116 11.7127 11.9551
    1.5 12.531 12.8083 13.1141 13.4308
    2 14.4048 14.7773 15.124 15.4064

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical solution of P(t) with at different fractional values ϕ.
    t ϕ = 1 ϕ = 0.95 ϕ = 0.9 ϕ = 0.85
    1 5.67835 5.6959 5.68235 5.7302
    2 6.4746 6.46707 6.45834 6.44629
    4 8.788 8.62982 8.47227 8.30553
    6 12.6746 12.1038 11.562 11.0225
    8 18.8688 17.417 16.0958 14.8435
    10 28.105 25.0658 22.3972 19.9683

     | Show Table
    DownLoad: CSV
    Figure 1.  Simulation of S(t) at different fractional values in time t.
    Figure 2.  Simulation of I(t) at different fractional values in time t.
    Figure 3.  Simulation of R(t) at different fractional values in time t.
    Figure 4.  Simulation of D(t) at different fractional values in time t.
    Figure 5.  Simulation of P(t) at different fractional values in time t.
    Figure 6.  Simulation of S(t) at different fractional values in time t.
    Figure 7.  Simulation of I(t) at different fractional values in time t.
    Figure 8.  Simulation of R(t) at different fractional values in time t.
    Figure 9.  Simulation of D(t) at different fractional values in time t.
    Figure 10.  Simulation of P(t) at different fractional values in time t.

    The objective of our work is to develop a scheme of epidemic fractional Ebola virus model with Caputo fractional derivative also numerical solutions have been obtained by using the Laplace with the Adomian Decomposition Method. The results of fractional order Ebola virus model is presented and convergence results of fractional-order model are also presented to demonstrate the efficacy of the process. The analytical solution of the fractional-order Ebola virus model consisting of the non-linear system of the fractional differential equation has been presented by using the Caputo derivative. To observe the effects of the fractional parameter on the dynamics of the fractional-order model (2.1), we conclude several numerical simulations varying the values of parameter given in [4]. These simulations reveal that a change in the value affects the dynamics of the model. The numerical solutions at classical as well as different fractional values by using Caputo fractional derivative can be seen in Figures 15 for disease free equilibrium. The rate of susceptible individuals and pathogens decreases by reducing the fractional values to acquire the desired value, whereas the other compartment starts decreasing by increasing the fractional values. The fractional-order model shows the convergence with theoretical contribution and numerical results. The fractional-order parameter values show the impact of increasing or decreasing the disease. Also, we can fix the parameter values where the rate of infection is decrease and the recover rate will increase for some values which are representing in figures and tables. These results can be used for disease outbreak treatment and analysis without defining the control parameters in the model based on fractional values. In general, approaches to fractional-order modeling in situations with large refined data sets and good numerical algorithms may be worth it. The simulation and numerical solutions at classical as well as different fractional values by using Caputo fractional derivative can be seen in Figures 610 for endemic equilibrium as well as in Tables 15. Results in both cases are reliable at fractional values to overcome the outbreak of this epidemic and meet our desired accuracy. Results discuss in [1], [5] for classical model, but our results are on fractional order model, fractional parameters easily use to adjust the control strategy without defining others parameters in the model. Another important feature that plays a critical role in the 2014 EVD outbreaks is traditional/cultural belief systems and customs. For instance, while some individuals in the three Ebola-stricken nations believe that there is no Ebola, control the population or harvest human organs. We conclude that depending on the specific data set, the fractional order model either converges to the ordinary differential equation model and fits data similarly, or fits the data better and outperforms the ODE model.

    We develop a scheme of epidemic fractional Ebola virus model with Caputo fractional derivative for numerical solutions that have been obtained by using the Laplace with the Adomian Decomposition Method. In [24] the use of three different fractional operators on the Ebola disease model suggests that the fractional-order parameter greatly affects disease elimination for the non-integer case when decreasing α. We constructed a numerical solution for the Ebola virus model to show a good agreement to control the bad impact of the Ebola virus for the different period for diseases free and endemic equilibrium point as well. However, in this work, we introduced the qualitative properties for solutions as well as the non-negative unique solution for a fractional-order nonlinear system. It is important to note that the Laplace Adomian Decomposition Method is used for the Ebola virus fractional-order model differential equation framework is a more efficient approach to computing convergent solutions that are represented through figures and tables for endemic and disease-free equilibrium point. Convergence results of the fractional-order model are also presented to demonstrate the efficacy of the process. The techniques developed to provide good results which are useful for understanding the Zika Virus outbreak in our community. It is worthy to observe that fractional derivative shows significant changes and memory effects as compare to ordinary derivatives. This model will assist the public health planar in framing an Ebola virus disease control policy. Also, we will expand the model incorporating determinist and stochastic model comparisons with fractional technique, as well as using optimal control theory for new outcomes.



    [1] Traveling waves in spatial SIRS models. J. Dynam. Differential Equations (2014) 26: 143-164.
    [2] Harnack's inequality for cooperative weakly coupled elliptic systems: Harnack's inequality. Comm. Partial Differential Equations (1999) 24: 1555-1571.
    [3] Travelling wave solutions in multigroup age-structured epidemic models. Arch. Ration. Mech. Anal. (2010) 195: 311-331.
    [4] Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. (2014) 46: 3678-3704.
    [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 2008.
    [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.
    [7] Hyperbolic travelling fronts. Proc. Edinburgh Math. Soc. (1988) 31: 89-97.
    [8] Travelling fronts for correlated random walks. Canad. Appl. Math. Quart. (1994) 2: 27-43.
    [9] Stability and bifurcation for an SEIS epidemic model with the impact of media. Phys. A (2018) 490: 702-720.
    [10] Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network. J. Franklin Inst. (2019) 356: 7411-7443.
    [11] J. S. Jia, X. Lu, Y. Yuan, G. Xu, J. Jia and N. A. Christakis, Population flow drives spatio-temporal distribution of COVID-19 in China, Nature, 1–5.
    [12] S.-L. Jing, H.-F. Huo and H. Xiang, Modeling the effects of meteorological factors and unreported cases on seasonal influenza outbreaks in Gansu province, China, Bull. Math. Biol., 82 (2020), Paper No. 73, 36 pp. doi: 10.1007/s11538-020-00747-6
    [13] Effect of non-pharmaceutical interventions to contain COVID-19 in China. Nature (2020) 585: 410-413.
    [14] Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl. Math. Comput. (2014) 247: 723-740.
    [15] Global stability of an epidemic model with latent stage and vaccination. Nonlinear Anal. Real World Appl. (2011) 12: 2163-2173.
    [16] Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. (2007) 60: 1-40.
    [17] Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China. Science (2020) 368: 742-746.
    [18] J. D. Murray, Mathematical Biology, Springer, 1989. doi: 10.1007/978-3-662-08539-4
    [19] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, 2012.
    [20] L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, American Mathematical Society, 2003. doi: 10.1090/surv/102
    [21] Large-scale spatial-transmission models of infectious disease. Science (2007) 316: 1298-1301.
    [22] Global Hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease. J. Math. Biol. (2018) 76: 1249-1267.
    [23] Analysis of an epidemic system with two response delays in media impact function. Bull. Math. Biol. (2019) 81: 1582-1612.
    [24] Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. (2002) 180: 29-48.
    [25] J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Analysis: Real World Applications, 58 (2021), 103208, 19 pp. doi: 10.1016/j.nonrwa.2020.103208
    [26] Traveling waves of the spread of avian influenza. Proc. Amer. Math. Soc. (2012) 140: 3931-3946.
    [27] Traveling waves in a nonlocal anisotropic dispersal Kermack-Mckendrick epidemic model. Discrete Contin. Dyn. Syst. Ser. B (2013) 18: 1969-1993.
    [28] Minimal wave speed for a class of non-cooperative reaction–diffusion systems of three equations. J. Differential Equations (2017) 262: 4724-4770.
    [29] Existence of traveling wave solutions for influenza model with treatment. J. Math. Anal. Appl. (2014) 419: 469-495.
    [30] Minimal wave speed for a class of non-cooperative diffusion–reaction system. J. Differential Equations (2016) 260: 2763-2791.
    [31] Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages. IMA J. Appl. Math. (2016) 81: 795-823.
    [32] Traveling wave solutions in a two-group epidemic model with latent period. Nonlinearity (2017) 30: 1287-1325.
    [33] Traveling wave solutions in a two-group SIR epidemic model with constant recruitment. J. Math. Biol. (2018) 77: 1871-1915.
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