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Stability and Bifurcation Analysis of a Reaction–Diffusion SIRS Epidemic Model with the General Saturated Incidence Rate

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Abstract

In this paper, we are concerned with the dynamics of a reaction–diffusion SIRS epidemic model with the general saturated nonlinear incidence rates. Firstly, we show the global existence and boundedness of the in-time solutions for the parabolic system. Secondly, for the ODEs system, we analyze the existence and stability of the disease-free equilibrium solution, the endemic equilibrium solutions as well as the bifurcating periodic solution. In particular, in the language of the basic reproduction number, we are able to address the existence of the saddle-node-like bifurcation and the secondary bifurcation (Hopf bifurcation). Our results also suggest that the ODEs system has a Allee effect, i.e., one can expect either the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium solution, or the coexistence of a stable disease-free equilibrium solution and a stable periodic solution. Finally, for the PDEs system, we are capable of deriving the Turing instability criteria in terms of the diffusion rates for both the endemic equilibrium solutions and the Hopf bifurcating periodic solution. The onset of Turing instability can bring out multi-level bifurcations and manifest itself as the appearance of new spatiotemporal patterns. It seems also interesting to note that p and k, appearing in the saturated incidence rate \(kSI^p/(1+\alpha I^p)\), tend to play far reaching roles in the spatiotemporal pattern formations.

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References

  • Agnew, P., Koella, J.: Life-history interactions with environmental conditions in a host-parasite relationship and the parasite’s mode of transmission. Evol. Ecol. 13, 67–89 (1999)

    Google Scholar 

  • Andral, L., Artois, M., Aubert, M., Blancou, J.: Radiotracking of rabid foxes. Comp. Immun. Microbiol. Infec. Dis 5, 285–291 (1982)

    Google Scholar 

  • Berestycki, H., Roquejoffre, J., Rossi, L.: Propagation of epidemic along lines with fast diffusion. Bull. Math. Biol. 83(2), 1–34 (2021)

    MathSciNet  Google Scholar 

  • Brauer, F., van den Driessche, P., Wu, J.: Mathematical Epidemiology. Lecture Notes in Mathematics, Springer, Berlin (2008)

    Google Scholar 

  • Britton, N.: Essential Mathematical Biology. Springer, London (2003)

    Google Scholar 

  • Capasso, V., Serio, G.: A generalization of the Kermack–McKendrick determinist epidemic model. Math. Biosci. 42, 43–61 (1978)

    MathSciNet  Google Scholar 

  • Capone, F., Cataldis, V., Luca, R.: Influence of diffusion on the stability of equilibria in a reaction–diffusion system modeling cholera dynamic. J. Math. Biol. 71(5), 1107–1131 (2015)

    MathSciNet  Google Scholar 

  • Chen, S., Huang, J.: Destabilization of synchronous periodic solutions for patch models. J. Differ. Eq. 364, 378–411 (2023)

    MathSciNet  Google Scholar 

  • Coddington, E., Levinsion, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    Google Scholar 

  • Conway, E., Hoff, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35(1), 1–16 (1978)

    MathSciNet  Google Scholar 

  • Du, Z., Peng, R.: A priori \(L^\infty \) estimates for solutions of a class of reaction–diffusion systems. J. Math. Biol. 72, 1429–1439 (2016)

    MathSciNet  Google Scholar 

  • Ebert, D., Zschokke-Rohringer, C., Carius, H.: Dose effects and density-dependent regulation of two microparasites of Daphnia magna. Oecologia 122, 200–209 (2000)

    Google Scholar 

  • Grassly, N., Fraser, C., Wenger, J., Deshpande, J., Sutter, R., Heymann, D., Aylward, R.: New strategies for the elimination of polio from India. Science 314, 1150–1153 (2006)

    Google Scholar 

  • Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    Google Scholar 

  • Hethcote, H., van den Driessche, P.: Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287 (1991)

    MathSciNet  Google Scholar 

  • Hu, Z., Bi, P., Ma, W., Ruan, S.: Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete Cont. Dyn. B 15, 93–122 (2011)

    MathSciNet  Google Scholar 

  • Le, D.: Dissipativity and global attractors for a class of quasilinear parabolic systems. Commun. Part. Differ. Equ. 22, 413–433 (1997)

    MathSciNet  Google Scholar 

  • Li, G., Wang, W.: Bifurcation analysis of an epidemic model with nonlinear incidence. Appl. Math. Comp. 214, 411–423 (2009)

    MathSciNet  Google Scholar 

  • Little, T., Ebert, D.: The cause of parasitic infection in natural populations of Daphnia (Crustacea: Cladocera): the role of host genetics. Proc. R. Soc. Lond. B 267, 2037–2042 (2000)

    Google Scholar 

  • Liu, W.: Criterion of Hopf bifurcation without using eigenvalues. J. Math. Anal. Appl. 182, 250–256 (1994)

    MathSciNet  Google Scholar 

  • Liu, W., Levin, S., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986)

    MathSciNet  Google Scholar 

  • Lizana, M., Rivero, J.: Multiparametric bifurcations for a model in epidemiology. J. Math. Biol. 35, 21–36 (1996)

    MathSciNet  Google Scholar 

  • Lou, Y., Zhao, X.: A reaction–diffusion malaria model with incubation period in the vector population. J. Math. Biol. 62, 543–568 (2011)

    MathSciNet  Google Scholar 

  • Lu, M., Huang, J., Ruan, S., Yu, P.: Bifurcation analysis of an SIRS epidemic model with a generalized monotone and saturated incidence rate. J. Differ. Equ. 267, 1859–1898 (2019)

    Google Scholar 

  • Maginu, K.: Stability of spatially homogeneous periodic solutions of reaction–diffusion equations. J. Differ. Equ. 31(1), 130–138 (1979)

    MathSciNet  Google Scholar 

  • Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Applications. Applied Mathematics Series A, vol. 19. Springer, New York (1970)

    Google Scholar 

  • McLean, A., Bostock, C.: Scrapie infections initiated at varying doses: an analysis of 117 titration experiments. Philos. Trans. R. Soc. Lond. B 355, 1043–1050 (2000)

    Google Scholar 

  • Morita, Y.: Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions, Japan. J. Appl. Math. 1, 39–65 (1984)

    MathSciNet  Google Scholar 

  • Murray, J., Seward, W.: On the spatial spread of rabies among foxes with immunity. J. Theor. Biol. 156, 327–348 (1992)

    Google Scholar 

  • Murray, J., Stanley, E., Brown, D.: On the spatial spread of rabies among foxes. Proc. Roy. Soc. Lond. B 229, 111–150 (1986)

    Google Scholar 

  • Okubo, A., Levin, S.: Diffusion and Ecological Problems, 2nd edn. Springer, New York (2001)

    Google Scholar 

  • Oliveira, L.: Instability of homogeneous periodic solutions of parabolic-delay equations. J. Differ. Equ. 109, 42–76 (1994)

    MathSciNet  Google Scholar 

  • Regoes, R., Ebert, D., Bonhoeffer, S.: Dose-dependent infection rates of parasites produce the Allee effect in epidemiology. Proc. Roy. Soc. Lond. B 269, 271–279 (2002)

    Google Scholar 

  • Ruan, S.: Diffusion-driven instability in the Gierer–Meinhardt model of morphogenesis. Nat. Resour. Model. 11, 131–142 (1998)

    MathSciNet  Google Scholar 

  • Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135–163 (2003)

    MathSciNet  Google Scholar 

  • She, G., Yi, F.: Dynamics and bifurcations in a non-degenerate homogeneous diffusive SIR rabies model. SIAM J. Appl. Math. 84(2), 632–660 (2024)

    MathSciNet  Google Scholar 

  • Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997)

    Google Scholar 

  • Song, Q., Yi, F.: Spatiotemporal patterns and bifurcations in a delayed diffusive predator–prey system with fear effects. J. Differ. Equ. 388, 151–187 (2024)

    Google Scholar 

  • Tang, Y., Huang, D., Ruan, S., Zhang, W.: Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate. SIAM J. Appl. Math. 69(2), 621–639 (2008)

    MathSciNet  Google Scholar 

  • Turing, A.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. Lond. B237, 37–72 (1952)

    MathSciNet  Google Scholar 

  • van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    MathSciNet  Google Scholar 

  • Wang, W., Zhao, X.: A nonlocal and time-delayed reaction–diffusion model of dengue transmission. SIAM J. Appl. Math. 71, 147–168 (2011)

    MathSciNet  Google Scholar 

  • Wang, W., Zhao, X.: Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11(4), 1652–1673 (2013)

    MathSciNet  Google Scholar 

  • Wang, M., Yi, F.: On the dynamics of the diffusive Field-Noyes model for the Belousov–Zhabotinskii reaction. J. Differ. Equ. 318, 443–479 (2022)

    MathSciNet  Google Scholar 

  • Wang, W., Wu, G., Wang, X., Feng, Z.: Dynamics of a reaction-advection–diffusion model for cholera transmission with human behavior change. J. Math. Biol. 373, 176–215 (2023)

    MathSciNet  Google Scholar 

  • Williams, O., Gouws, E., Hankins, C., Getz, W., Hargrove, J., do Zoysa, I., Dye, C., Auvert, B.: The potential impact of male circumcision on HIV in sub-Saharor Africa. Plos Med. 3, 1032–1040 (2006)

    Google Scholar 

  • Xiao, D., Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208, 419–429 (2007)

    MathSciNet  Google Scholar 

  • Yi, F.: Turing instability of the periodic solutions for reaction–diffusion systems with cross-diffusion and the patch model with cross-diffusion-like coupling. J. Differ. Equ. 281, 379–410 (2021)

    MathSciNet  Google Scholar 

  • Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differ. Equ. 246(5), 1944–1977 (2009)

    MathSciNet  Google Scholar 

  • Zhang, R., Wang, J.: On the global attractivity for a reaction–diffusion malaria model with incubation period in the vector population. J. Math. Biol. 84(6), 1–12 (2022)

    MathSciNet  Google Scholar 

  • Zhang, F., Cui, W., Dai, Y., Zhao, Y.: Bifurcations of an SIRS epidemic model with a general saturated incidence rate. Math. Biosci. Eng. 19(11), 10710–10730 (2022)

    MathSciNet  Google Scholar 

Download references

Funding

This work is financially supported by National Key R & D Program of China (2023YFA1009200), the National Natural Science Foundation of China (11971088, 12371160, 12271144) and the Fundamental Research Funds for the Central Universities (DUT18RC(3)070, DUT23LAB304).

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Authors and Affiliations

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, Heilongjiang Province, China

    Gaoyang She & Fengqi Yi

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning Province, China

    Fengqi Yi

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  1. Gaoyang She
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Correspondence to Fengqi Yi.

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Communicated by Kevin Painter.

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The authors were partially supported by National Natural Science Foundation of China (12371160, 12271144) and the Fundamental Research Funds for the Central Universities (DUT23LAB304).

Appendix: The Calculation of the First Lyapunov Coefficient

Appendix: The Calculation of the First Lyapunov Coefficient

First, let \({u}=S-S_2, v=I-I_2, w=R-R_2\) and \(F(I)=\frac{kI^p}{1+\alpha I^p}\). Then, system (1.4) becomes

$$\begin{aligned} \begin{aligned} \dfrac{\textrm{d}u}{\textrm{d}t}&=-du-F(I_2)u+\delta w-S_2F'(I_2)v-F'(I_2)uv \\ &\quad -\frac{1}{2}S_2F''(I_2)v^2-\frac{1}{2}F''(I_2)uv^2-\frac{1}{6}S_2F'''(I_2)v^3+O(|u||v|^3,|v|^4),\\ \dfrac{\textrm{d}v}{\textrm{d}t}&=F(I_2)u+S_2F'(I_2)v-(d+\mu )v+F'(I_2)uv \\ &\quad +\frac{1}{2}S_2F''(I_2)v^2+\frac{1}{2}F''(I_2)uv^2+\frac{1}{6}S_2F'''(I_2)v^3+O(|u||v|^3,|v|^4),\\ \dfrac{\textrm{d}w}{\textrm{d}t}&=\mu v-(d+\delta )w. \end{aligned} \end{aligned}$$
(6.1)

At \(\lambda =\overline{\lambda }(p)\), system (6.1) can be rewritten in the following vector form

$$\begin{aligned} \begin{aligned} \begin{bmatrix} u'\\ v'\\ w'\\ \end{bmatrix}=L_0(\overline{\lambda }(p))\begin{bmatrix} u\\ v\\ w \end{bmatrix}+\begin{bmatrix} -F_{1}( u, v, w) \\ F_{1}(u, v, w) \\ 0\\ \end{bmatrix}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} F_{1}(u, v, w)&:=b_{11}uv+b_{02}v^2+b_{12}uv^2+b_{03}v^3+O(|u||v|^3,|v|^4), \end{aligned} \end{aligned}$$

where \(L_0({\overline{\lambda }}(p))=L_0(I_2^p)\) for \(I_2^p={\overline{\lambda }}(p)\) and

$$\begin{aligned} \begin{aligned} b_{11}&=\frac{kp[\overline{\lambda }(p)]^{\frac{p-1}{p}}}{(1 +\alpha \overline{\lambda }(p))^2},\; b_{02}=-\frac{p(d+\mu ) \big [1 + \alpha \overline{\lambda }(p) + p (\alpha \overline{\lambda }(p)-1 )\big ]}{2[\overline{\lambda }(p)]^{\frac{1}{p}}(1 +\alpha \overline{\lambda }(p))^2},\\ b_{12}&=-\frac{kp[\overline{\lambda }(p)]^{\frac{p-2}{p}} \big [1 + \alpha \overline{\lambda }(p) + p (\alpha \overline{\lambda }(p)-1 )\big ]}{2(1 +\alpha \overline{\lambda }(p))^3},\\ b_{03}&=\frac{p(d+\mu )\big [2 (1 + \alpha \overline{\lambda }(p))^2 + 3 p (\alpha ^2{[\overline{\lambda }(p)]}^{2}-1 ) + p^2 (1 - 4\alpha \overline{\lambda }(p) + \alpha ^2{[\overline{\lambda }(p)]}^{2 } )\big ]}{6[\overline{\lambda }(p)]^{\frac{2}{p}}(1 +\alpha \overline{\lambda }(p))^3}.\\ \end{aligned} \end{aligned}$$

Then, \(L_0({\overline{\lambda }}(p))\) has \(\pm \omega _0i\) and \(-a_2({\overline{\lambda }}(p))\) as its eigenvalues, where \(\omega _0:=\sqrt{\varrho \mu -(d+\delta )^2}\) and \(a_2({\overline{\lambda }}(p))=d\), where \(\varrho \) is defined by (4.26).

Let \({\kappa _1}\) and \(\kappa _2\) be the eigenvectors corresponding to the eigenvalues \(i\omega _0\) and \(-a_2({\overline{\lambda }}(p))\) of \(L_0({\overline{\lambda }}(p))\), respectively. More precisely,

$$\begin{aligned} \begin{aligned} {\kappa _1}&:=\big (-d-\mu -\delta -i\omega _0, d+\delta +i\omega _0, \mu \big )^T, \\ {\kappa _2}&:= \big (\delta (L_{11}(\overline{\lambda }(p))+L_{33}(\overline{\lambda }(p))), \delta L_{21}(\overline{\lambda }(p)), \mu L_{21}(\overline{\lambda }(p))\big )^T. \end{aligned} \end{aligned}$$

Choose

$$\begin{aligned} Y:=\big ({\text {Re}}({\kappa _1}),-{\text {Im}}({\kappa _1}), {\kappa _2}\big )=\begin{bmatrix} -(d+\mu +\delta )& \omega _0 & -\delta (2d+\delta +\varrho )\\ d+\delta & -\omega _0 & \delta \varrho \\ \mu & 0 & \mu \varrho \\ \end{bmatrix}. \end{aligned}$$

Then, \(Y^{-1}\) takes in the form of

$$\begin{aligned} \frac{1}{|Y|}\begin{bmatrix} -\omega _0\mu \varrho & \quad -\omega _0\mu \varrho & \quad -\omega _0\delta (2d+\delta )\\ -\textrm{d}\mu \varrho & \quad \mu \delta (2d+\delta )-\mu \varrho (d+\mu ) & \quad \mu \delta \varrho -\delta (d+\delta )(2d+\delta ) \\ \mu \omega _0& \quad \mu \omega _0 & \quad \mu \omega _0 \\ \end{bmatrix}, \end{aligned}$$

where \(|Y|:=\mu \omega _0\big (\mu \varrho -\delta (2d+\delta )\big )\).

Therefore, we have

$$\begin{aligned} Y^{-1}L_0({\overline{\lambda }}(p))Y=\begin{bmatrix} 0& \quad -\omega _0 & \quad 0 \\ \omega _0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad -d \end{bmatrix}. \end{aligned}$$

Let \((u, v, w)^T=Y(x,y,z)^T\). Then, we have

$$\begin{aligned} \begin{aligned} \begin{bmatrix} x'\\ y'\\ z'\\ \end{bmatrix}=\begin{bmatrix} 0& \quad -\omega _0 & \quad 0 \\ \omega _0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad -a_2({\overline{\lambda }}(p)) \end{bmatrix}\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}+Y^{-1}\begin{bmatrix} -H( x, y, z) \\ H( x, y, z) \\ 0\end{bmatrix}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} H(x, y, z)&:=c_{200}x^2+c_{110}xy+c_{020}y^2+c_{101}xz+c_{011}yz+c_{002}z^2+c_{300}x^3+^2y\\&\quad +c_{120}xy^2 +c_{030}y^3+c_{201}x^2z+c_{111}xyz\\&\quad +c_{021}y^2z+c_{102}xz^2+c_{012}yz^2+c_{003}z^3, \end{aligned} \end{aligned}$$

in which

$$\begin{aligned} \begin{aligned} c_{200}&=(d + \delta ) \big (b_{02} (d + \delta ) - b_{11} (d + \delta +\mu )\big ),\\ c_{110}&=w_0 \big ( b_{11} (2 d + 2 \delta + \mu )-2 b_{02} (d + \delta ) \big ),\; c_{020}=(b_{02} - b_{11}) w_0^2,\\ c_{101}&=2 \varrho b_{02} (d + \delta )-b_{11} (d + \delta )(2d+\delta +\varrho ) - b_{11} \varrho (d + \delta + \mu ),\\ c_{011}&=-w_0\delta \big [2\varrho b_{02} - b_{11}(2d+\delta +2\varrho )\big ],\\ c_{002}&=\delta ^2\varrho \big [\varrho b_{02} -b_{11}(2d+\delta +\varrho ) \big ],\\ c_{300}&=(d + \delta )^2 \big [b_{03} (d + \delta ) - b_{12} (d + \delta +\mu )\big ],\\ c_{210}&=-w_0 (d + \delta ) \big [3 b_{03 }(d +\delta ) - b_{12} (3 d + 3 \delta + 2 \mu )\big ],\\ c_{120}&=w_0^2 \big [3 b_{03 }(d + \delta ) - b_{12} (3 d + 3 \delta +\mu )\big ],\; c_{030}= ( b_{12}-b_{03}) w_0^3,\\ c_{201}&= \delta (d + \delta ) \big [3\varrho b_{03} (d +\delta ) - b_{12} (d + \delta )(2d+\delta +\varrho ) - 2 b_{12} \varrho (d + \delta + \mu )\big ],\\ c_{111}&=2 w_0\delta \big [-3 \varrho b_{03} (d + \delta )+b_{12} (d + \delta )(2d+\delta +\varrho ) + b_{12}\varrho (2 d + 2 \delta +\mu )\big ], \\ c_{021}&=\delta w_0^2\big [3\varrho b_{03} - 2\varrho b_{12} - b_{12}(2d+\delta +\varrho )\big ],\\ c_{102}&=\delta ^2\varrho \big [3\varrho b_{03} (d + \delta ) -2 b_{12} (d + \delta )(2d+\delta +\varrho - b_{12} \varrho (d + \delta + \mu ))\big ],\\ c_{012}&=w_0 \delta ^2\varrho \big [2 b_{12}(2d+\delta ) + 3\varrho (b_{12}- b_{03})\big ], \\ c_{003}&=\varrho ^2\delta ^3 \big [\varrho b_{03} - b_{12}(2d+\delta +\varrho )\big ]. \end{aligned} \end{aligned}$$

Let \((f_1, f_2, f_3)^T:=Y^{-1}(-H,H,0)^T\). Then, we have

$$\begin{aligned} f_1(x, y, z)=0,\;\;f_2(x, y, z)=-\dfrac{1}{\omega _0}H(x, y, z),\;\;f_3(x, y, z)=0. \end{aligned}$$

Then, the normal form of system (6.1) can be given by

$$\begin{aligned} \begin{aligned} \begin{bmatrix} x^{\prime } \\ y^{\prime } \\ z^{\prime } \\ \end{bmatrix}=\begin{bmatrix} 0& \quad -\omega _0 & \quad 0 \\ \omega _0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad -d \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}+\begin{bmatrix} f_{1}( x, y, z) \\ f_{2}( x, y, z) \\ f_{3}(x, y, z) \\ \end{bmatrix}. \end{aligned} \end{aligned}$$

By Hassard et al. (1981), She and Yi (2024), and Wang and Yi (2022), the first Lyapunov coefficient \(c_{1}({\overline{\lambda }}(p))\) is defined by

$$\begin{aligned} c_{1}({\overline{\lambda }}(p))=\dfrac{i}{2\omega _0}\big [g_{11}g_{20}-2|g_{11}|^{2}-\frac{1}{3}|g_{02}|^{2}\big ]+\dfrac{g_{21}}{2}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} g_{11}&:=\frac{1}{4}\big [(f_1)_{xx}+(f_1)_{yy}+i\big [(f_2)_{xx}+(f_2)_{yy}\big ]\big ],\\ g_{02}&:=\frac{1}{4}\big [(f_1)_{xx}-(f_1)_{yy}-2(f_2)_{xy}+i\big [(f_2)_{xx}-(f_2)_{yy}+2(f_1)_{xy}\big ]\big ],\\ g_{20}&:=\frac{1}{4}\big [(f_1)_{xx}-(f_1)_{yy}+2(f_2)_{xy}+i\big [(f_2)_{xx}-(f_2)_{yy}-2(f_1)_{xy}\big ]\big ],\\ h_{11}&:=\frac{1}{4}\big [(f_3)_{xx}+(f_3)_{yy}\big ],\;\;h_{20}:=\frac{1}{4}\big [(f_3)_{xx}-(f_3)_{yy}-2 i(f_3)_{xy}\big ],\\ G_{110}&:=\frac{1}{2}\big [(f_1)_{xz}+(f_2)_{yz}+i\big [(f_2)_{xz}-(f_1)_{yz}\big ]\big ],\\ G_{101}&:=\frac{1}{2}\big [(f_1)_{xz}-(f_2)_{yz}+i\big [(f_2)_{xz}+(f_1)_{yz}\big ]\big ],\\ G_{21}&:=\frac{1}{8}\big [(f_1)_{xxx}+(f_1)_{xyy}+(f_2)_{xxy}+(f_2)_{yyy}\big ]\\&\quad +\frac{i}{8}\big [(f_2)_{xxx}+(f_2)_{xyy}-(f_1)_{xxy}-(f_1)_{yyy}\big ],\\ \omega _{11}&:=\frac{h_{11}}{\tau _2({\overline{\lambda }}(p))},\;\omega _{20}:=\frac{h_{20}}{\tau _2({\overline{\lambda }}(p))+2\omega _0i},\; g_{21}:=G_{21}+2G_{110} w_{11}+G_{101}w_{20}, \end{aligned} \end{aligned}$$

where all the quantities are evaluated at \(\lambda ={\overline{\lambda }}(p)\) and \((x, y, z)=(0,0,0)\).

In particular, we have

$$\begin{aligned} \begin{aligned}&{\text {Re}}\big (c_{1}({\overline{\lambda }}(p))\big ) =-\frac{1}{8\omega _0}\big (c_{210}+3c_{030}\big )-\frac{1}{8\omega _0^3}c_{110}\big (c_{200}+c_{020}\big ),\\&{\text {Im}}(c_{1}({\overline{\lambda }}(p))) =-\frac{1}{8\omega _0}\big (c_{120}+3c_{300}\big )-\frac{1}{24\omega _0^3}\big (10c_{200}^2+c_{110}^2+4c_{020}^2+10c_{200}c_{020}\big ). \end{aligned}\nonumber \\ \end{aligned}$$
(6.2)

By (3.42), \(a_1({\overline{\lambda }}(p))a_2'({\overline{\lambda }}(p))+a_1'({\overline{\lambda }}(p))a_2({\overline{\lambda }}(p))-a_0'({\overline{\lambda }}(p))>0\). Then, by Lemma 3.1, if \({\text {Re}}\big (c_{1}({\overline{\lambda }}(p))\big )<0\) (\(\text {resp}., >0\)), the Hopf bifurcation occurring at \(I_2^p=\overline{\lambda }(p)\) is supercritical and backward (resp., subcritical and forward).

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She, G., Yi, F. Stability and Bifurcation Analysis of a Reaction–Diffusion SIRS Epidemic Model with the General Saturated Incidence Rate. J Nonlinear Sci 34, 101 (2024). https://doi.org/10.1007/s00332-024-10081-z

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