
In this paper, we examined the codimension-two bifurcation analysis of a two-dimensional discrete epidemic model. More precisely, we examined the codimension-two bifurcation analysis at an endemic equilibrium state associated with 1:2, 1:3 and 1:4 strong resonances by bifurcation theory and series of affine transformations. Finally, theoretical results were carried out numerically.
Citation: Abdul Qadeer Khan, Tania Akhtar, Adil Jhangeer, Muhammad Bilal Riaz. Codimension-two bifurcation analysis at an endemic equilibrium state of a discrete epidemic model[J]. AIMS Mathematics, 2024, 9(5): 13006-13027. doi: 10.3934/math.2024634
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In this paper, we examined the codimension-two bifurcation analysis of a two-dimensional discrete epidemic model. More precisely, we examined the codimension-two bifurcation analysis at an endemic equilibrium state associated with 1:2, 1:3 and 1:4 strong resonances by bifurcation theory and series of affine transformations. Finally, theoretical results were carried out numerically.
Humans have suffered from a number of contagious diseases over the ages, including cholera, influenza, and plague [1,2,3]. Contagious diseases have long been ranked alongside conflicts and a food shortage as key threats to human advancement and existence. The transmission of contagious diseases in populations, as well as how to prevent and eradicate them, are crucial and essential topics. Mathematical analysis and modelling is an important part of infectious diseases epidemiology. Applications of mathematical models to disease surveillance data can be used to address both scientific hypotheses and disease control policy questions. The mathematical description of disease epidemics immediately leads to several useful results, including the expected size of an epidemic and the critical level that is needed for an interaction to achieve effective disease control. There are several mathematical models proposed by eminent mathematicians to investigate what transpires when a population becomes infected, and under what conditions depending on the circumstances, the disease will be eradicated. Medical professionals have created vaccinations against a variety of viruses and suggested numerous epidemic preventive strategies as part of humanity's fight against contagious diseases. Furthermore, mathematicians have significantly aided in the effort to stop the spread of disease. From the standpoint of mathematical models, it is possible to roughly determine the duration between the contagious disease's outbreak and containment, the number of individuals infected at a given point in the disease's development who require quarantine, and the number of individuals in contact with the disease at any given time. After examining the number of plague cases and patient survival days, Kermack and Mckendrick [4] developed the SIR model, a ground-breaking mathematical epidemiology model. The great majority of research that have examined contagious diseases mathematically up to this point have operated under the shadow of this model. Many academics used the contagious disease dynamics model as a foundational research tool in the study and forecasting of the COVID-19 epidemic. They also suggested a number of enhancement techniques to enhance the classical contagious disease dynamics model, allowing it to more accurately describe the real context of the epidemic's transmission and produce more plausible forecasts regarding the epidemic's development trend. Numerous practical recommendations for governance, control, and prevention have been made [5,6,7]. Dynamical system's research has gained significant interest in the last several years. Dynamical systems is an interdisciplinary area that has several applications, including predator-prey models and tumor models, in addition to the study of epidemic diseases [8,9]. We typically use mathematical models to characterize the rich dynamic behavior of epidemic diseases while some mathematical models are tailored to specific diseases, where the majority are appropriate for broad investigations into the principles underlying different epidemic diseases [10,11]. Standard differential equations are used in the majority of epidemic disease models. Nonetheless, discrete-time dynamic models are far simpler, and more computationally efficient than continuous models. In addition to being straightforward analogs of continuous epidemic models, the majority of discrete-time epidemic models also exhibit intricate dynamic characteristics that the corresponding continuous models are unable to display. Even in a one-dimensional instance, the discrete-time model can produce incredibly complex dynamics [12], and for dynamical properties of higher-dimensional epidemic models we refer the reader to work of eminent researchers [13,14,15,16,17]. Therefore, from many years, mathematical infectious models have been a popular and interesting topic [18,19,20,21,22]. From an epidemiological point of view, epidemic dynamics are a extremely transited topic of investigation. The majority of researchers investigated bifurcation phenomena when a single systemic parameter changes. Indeed, many practical models include a number of systemic parameters when more than single systemic parameter is altered simultaneously, and so it is probable that complicated bifurcation such as codimension-two bifurcations are likely to occur. Nevertheless, because of the impact of higher-order nonlinear components, codimension-two bifurcations remain extremely difficult to understand. Even elementary dynamical systems have complex dynamics that cannot be satisfactorily illustrated by theoretical analysis. A few scholars have investigated simulating the dynamics within the local parameter fluctuation using computers. As a result, the numerical techniques allow to better illustrate and comprehend the dynamics of the model in addition to validating our analytical results [23,24,25,26]. On the other hand, in recent years, many researchers have investigated the codimension-one and codimension-two bifurcations of discrete model by bifurcation theory. For instance, Ruan and Wang [27] have studied Bogdanov-Takens bifurcation for the model:
{˙I=kI21+νI2(N0−I−S)−(d+γ)I,˙S=γI−(d+v)S, | (1.1) |
where d,γ and v, respectively denote death rate, recovery rate and removed individuals rate whereas ν is a nonnegative constant. Eskandari and Alidousti [28] have examined codimension-two bifurcations of the following discrete model:
{It+1=It+h(A−dIt−λItSt),St+1=St+h(λItSt−(d+r)St), | (1.2) |
where d,λ,A and r denote the natural death rate, the bilinear incidence rate, the recruitment rate of the population, and the recovery rate of the infective individuals, respectively. Ruan et al. [29] have examined codimension-two bifurcations of the following discrete model:
{It+1=Ite1−It−Sta+I2t,St+1=Ste−d+bIta+I2t, | (1.3) |
where It and St denote infected and susceptible individuals, respectively. Abdelaziz et al. [30] have examined codimension-two bifurcations of the following discrete model:
{It+1=It+hαΓ(1+α)(rIt(1−It)−ItVt−μIt),St+1=St+hαΓ(1+α)(ItVt−δ1St),Vt+1=Vt+hαΓ(1+α)(γSt−δ2Vt), | (1.4) |
where the parameter h>0 is the time step size. Chen et al. [31] have examined codimension-two bifurcations of the following discrete model:
{It+1=It+δ(⋀−βItStIt+St−μIt+ϕSt),St+1=St+δ(βItStIt+St−(γ+μ+ϕ)St), | (1.5) |
where γ,⋀, μ, β and ϕ, respectively denote disease related death rate, recruitment rate, natural death rate, disease transmission coefficient, and rate at with individuals I return to class S, and δ>0 is a integral step size. Liu et al. [32] have examined codimension-two bifurcations of following discrete model:
{It+1=It+h(A−dIt−λItSt),St+1=St+h(λItSt−(d+r)St), | (1.6) |
where A, d, r, respectively denote recruitment rate, natural death rate, recovery rate, and finally, λ denotes bilinear incidence rate. Yi et al. [33] have examined codimension-two bifurcations of the model:
{It+1=It+δ(N0−dIt−βStIt(1+vSt)),St+1=St+δ(βStIt(1+vSt)−(d+η)St), | (1.7) |
where η, d,N0, β>0 denote effective contact rate, death rate, rate of recruitment and recovery rate, respectively. Ma and Duan [34] have explored codimension-two bifurcations of a two-dimensional discrete time Lotka-Volterra predator-prey model. Yousef et al. [35] have explored codimension-one and codimension-two bifurcations in a discrete Kolmogorov type predator-prey model. Eskandari et al. [36] have explored codimension-two bifurcations of a discrete game model. Guo et al. [37] have examined hopf bifurcations of a bioeconomic model. Inspired by the aforementioned research, in this paper, we aim to examine codimension-two bifurcations of the following discrete epidemic model with vital dynamics and vaccination [38]:
{It+1=μ1StItμ2+(1−(μ3+μ4))It,St+1=((1−μ5)μ3+μ6)μ2−μ1StItμ2+(1−(μ3+μ7+μ6))St+(μ4−μ6)It, | (1.8) |
where μ3, μ1, μ4 and μ6, respectively denote natural death rate, contact rate, cure rate, and rate of immunity loss while μ7 and μ5 are rates of vaccination in individuals St and newcomers. More precisely, our goal of this paper is to examine the existence of codimension-two bifurcation sets, and codimension-two bifurcation at endemic equilibrium state (EES) associated with 1:2, 1:3 and 1:4 strong resonances of a discrete epidemic model (1.8). Furthermore, our theoretical results are confirmed by numerical simulation.
The organization of the paper is as follows: The existence of codimension-two bifurcation sets at EES of a discrete epidemic model (1.8) are identified in Section 2 whereas Section 3 is about the study of codimension-two bifurcation at EES. In order to confirm theoretical results, simulations are presented in Section 4 whereas conclusion is given in Section 5.
In this section, we examine codimension-two bifurcation sets at EES for the discrete epidemic model (1.8). For this, the simple calculation shows that if μ6>(μ3+μ7)(μ3+μ4)−(1−μ5)μ3μ1μ1−(μ3+μ4) then model (1.8) has EES (((1−μ5)μ3+μ6)μ1μ2−(μ3+μ7+μ6)(μ3+μ4)μ2μ1(μ3+μ6),(μ3+μ4)μ2μ1) with basic reproduction number is R0:=((1−μ5)μ3+μ6)μ1(μ3+μ6+μ7)(μ3+μ4)>1. Now variation matrix V|EES of the linearized system of model (1.8) at EES is
V|EES:=(1((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)μ3+μ6−μ3−μ61+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6)), | (2.1) |
with characteristic equation is
λ2−Λ1λ+Λ2=0, | (2.2) |
where
Λ1=2+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6),Λ2=1+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6)−(μ3+μ7+μ6)(μ3+μ4)+((1−μ5)μ3+μ6)μ1. | (2.3) |
Setting
Λ1=2+G, Λ2=1+G+H, | (2.4) |
where
G=(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6),H=((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4). | (2.5) |
Finally, the roots of (2.2) are
λ1,2=2+G±√Δ2, | (2.6) |
where
Δ=(2+G)2−4(1+G+H),=(2+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6))2−4(1+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6)+((1−μ5)μ3+μ6)μ1)−(μ3+μ7+μ6)(μ3+μ4). | (2.7) |
Hereafter, following three cases are to considered in order to get codimension-two bifurcations sets:
Case 1. If H=4=−G, then from (2.6) one gets λ1,2=−1 with μ7=4(μ3+μ6)−(μ3+μ6)2−4(μ3+μ6) and μ5=μ1(μ3+μ6)2−4(μ3+μ6)(μ3+μ4)+4(μ4−μ6)μ1μ3(μ3+μ6). Therefore, at EES, model (1.8) may undergoes codimension-two bifurcation with 1:2 strong resonance where
F12|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=4(μ3+μ6)−(μ3+μ6)2−4μ3+μ6,μ5=μ1(μ3+μ6)2−4(μ3+μ6)(μ3+μ4)+4(μ4−μ6)μ1μ3(μ3+μ6)}. | (2.8) |
Case 2. If H=3=−G then from (2.6) one gets λ1,2=−1±√3ι2 with μ7=3(μ3+μ6)−(μ3+μ6)2−3μ3+μ6 and μ5=μ1(μ3+μ6)2−3(μ3+μ6)(μ3+μ4)+3(μ4−μ6)μ1μ3(μ3+μ6). Therefore, at EES, model (1.8) may undergoes codimension-two bifurcation with 1:3 strong resonance where
F13|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=3(μ3+μ6)−(μ3+μ6)2−3μ3+μ6,μ5=μ1(μ3+μ6)2−3(μ3+μ6)(μ3+μ4)+3(μ4−μ6)μ1μ3(μ3+μ6)}. | (2.9) |
Case 3. If H=2=−G then from (2.6) one gets λ1,2=±ι with μ7=2(μ3+μ6)−(μ3+μ6)2−2μ3+μ6 and μ5=μ1(μ3+μ6)2−2(μ3+μ6)(μ3+μ4)+2(μ4−μ6)μ1μ3(μ3+μ6). Therefore, at EES, model (1.8) may undergoes codimension-two bifurcation with 1:4 strong resonance where
F14|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=2(μ3+μ6)−(μ3+μ6)2−2μ3+μ6,μ5=μ1(μ3+μ6)2−2(μ3+μ6)(μ3+μ4)+2(μ4−μ6)μ1μ3(μ3+μ6)}. | (2.10) |
Remark 1. If H=0=G then λ1,2=1. However, from (2.5), it is noted that G and H are never zero and so, at EES, model (1.8) does not undergo codimension-two bifurcation with 1:1 resonance.
The codimension-two bifurcations at EES of model (1.8) will be examined in this section by bifurcation theory [39,40,41,42,43].
From (2.8), if μ7=4(μ3+μ6)−(μ3+μ6)2−4μ3+μ6 and μ5=μ1(μ3+μ6)2−4(μ3+μ6)(μ3+μ4)+4(μ4−μ6)μ1μ3(μ3+μ6) then calculation shows that λ1,2|μ7=4(μ3+μ6)−(μ3+μ6)2−4(μ3+μ6), μ5=μ1(μ3+μ6)2−4(μ3+μ6)(μ3+μ4)+4(μ4−μ6)μ1μ3(μ3+μ6)=−1 which implies that if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F12|EES then at EES model (1.8) may undergoes codimension-two bifurcation with 1:2 strong resonance, by choosing μ5 and μ7 as bifurcation parameters. Now using following transformation, EES of model (1.8) transform to (0,0):
{ut=It−I∗,vt=St−S∗, | (3.1) |
where I∗=((1−μ5)μ3+μ6)μ1μ2−(μ3+μ7+μ6)(μ3+μ4)μ2μ1(μ3+μ6) and S∗=(μ3+μ4)μ2μ1. In view of (3.1), one write the model (1.8) as follows:
{ut+1=μ1(vt+S∗)(ut+I∗)μ2+(1−(μ3+μ4))(ut+I∗)−I∗,vt+1=((1−μ5)μ3+μ6)μ2−μ1(vt+S∗)(ut+I∗)μ2+(1−(μ3+μ7+μ6))(vt+S∗)−S∗+(μ4−μ6)(ut+I∗). | (3.2) |
Now on expanding (3.2) at (0,0) up to order-2nd, one gets:
(ut+1vt+1)=(μ1μ2S∗+1−(μ3+μ4)μ1μ2I∗−μ1μ2S∗+(μ4−μ6)1−μ1μ2I∗−(μ3+μ7+μ6))(utvt)+(μ1μ2utvt−μ1μ2utvt). | (3.3) |
Now at EES, (3.3) becomes
(ut+1vt+1)=(1((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)μ3+μ6−μ3−μ61+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6))(utvt)+(f1(ut,vt)f2(ut,vt)), | (3.4) |
where
{f1(ut,vt)=μ1μ2utvt,f2(ut,vt)=−μ1μ2utvt. | (3.5) |
From (3.4), if one denotes
A(ϱ)=(1((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)μ3+μ6−μ3−μ61+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6)), | (3.6) |
then
A0=A(ϱ0)|ϱ0=(μ5,μ7)=(14μ3+μ6−μ3−μ6−3), | (3.7) |
where ϱ=(μ5,μ7) with characteristic roots are λ1,2=−1. Furthermore, eigenvector and generalized eigenvector of A0 corresponding to characteristic roots −1, respectively are q0=(1−μ3+μ62) and q1=(1−μ3+μ64). Additionally, eigenvector and generalized eigenvector of AT0 corresponding to characteristic roots −1, respectively are p1=(24μ3+μ6) and p0=(−1−4μ3+μ6), where pi, qi (i=0,1) satisfying the following relations:
{A0q0=−q0,A0q1=−q1+q0,AT0p1=−p1,AT0p0=−p0+p1,⟨q0,p0⟩=⟨q1,p1 ⟩=1,⟨q1,p0 ⟩=⟨q0,p1 ⟩=0. | (3.8) |
Now if
(utvt):=ntq0+mtq1=(11−μ3+μ62−μ3+μ64)(ntmt), | (3.9) |
with x=(utvt) then straightforward calculation yields
{nt=⟨x,p0⟩=−ut−4μ3+μ6vt,mt=⟨x,p1⟩=2ut+4μ3+μ6vt. | (3.10) |
Now in coordinates (nt,mt), the model general representation of (3.4) is
(nt+1mt+1):=(−1+a(ϱ)1+b(ϱ)c(ϱ)−1+d(ϱ))(ntmt)+(f3(nt,mt)f4(nt,mt)), | (3.11) |
where
{f3(nt,mt,ϱ)=⟨F(ntq0+mtq1,ϱ),p0⟩=a20n2t+a11ntmt+a02m2t,f4(nt,mt,ϱ)=⟨F(ntq0+mtq1,ϱ),p1⟩=b20n2t+b11ntmt+b02m2t, | (3.12) |
and
F(x,ϱ)=(μ1μ2utvt−μ1μ2utvt). | (3.13) |
From (3.9), (3.12) and (3.13), the calculation yields
{a20=μ1μ2(μ3+μ62−2),a11=3μ1μ2(μ3+μ64−1),a02=μ1μ2(μ3+μ64−1),b20=μ1μ2(2−(μ3+μ6)),b11=3μ1μ2(1−μ3+μ62),b02=μ1μ2(1−μ3+μ62), | (3.14) |
and
{a(ϱ)=⟨(A(ϱ)−A0)q0,p0⟩,=((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)2+6+2((μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6)−2(μ3+μ7+μ6),b(ϱ)=⟨(A(ϱ)−A0)q1,p0⟩,=((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)4+3+(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1μ3+μ6−(μ3+μ7+μ6),c(ϱ)=⟨(A(ϱ)−A0)q0,p1⟩,=(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ1−4+2(μ3+μ7+μ6)+2(((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)μ3+μ6),d(ϱ)=⟨(A(ϱ)−A0)q1,p1⟩,=(μ3+μ7+μ6)(μ3+μ4)−((1−μ5)μ3+μ6)μ12−2+((1−μ5)μ3+μ6)μ1−(μ3+μ7+μ6)(μ3+μ4)μ3+μ6+(μ3+μ7+μ6). | (3.15) |
Moreover, the calculation shows that a(ϱ0)=b(ϱ0)=c(ϱ0)=d(ϱ0)=0. Now denote
B(ϱ)=(1+b(ϱ)0−a(ϱ)1), | (3.16) |
with the following non-singular coordinate transformation:
(ntmt):=B(ϱ)(ptqt), | (3.17) |
model (3.17) can be writen as
(pt+1qt+1):=(−11ϵ(ϱ)−1+σ(ϱ))(ptqt)+(GH), | (3.18) |
where
{ϵ(ϱ)=c(ϱ)+b(ϱ)c(ϱ)−a(ϱ)d(ϱ),σ(ϱ)=a(ϱ)+d(ϱ), | (3.19) |
and
(GH)=B−1(ϱ)(f3((1+b(ϱ))pt,−a(ϱ)pt+qt,ϱ)f4((1+b(ϱ))pt,−a(ϱ)pt+qt,ϱ)). | (3.20) |
Now if
β=(β1β2)=(ϵ(ϱ)σ(ϱ)), | (3.21) |
then β1(ϱ0)=β2(ϱ0)=0, So, (3.18) along with (3.19)–(3.21) becomes
(pt+1qt+1)=(−11β1−1+β2)(ptqt)+(G(pt,qt,ϱ)H(pt,qt,ϱ)), | (3.22) |
where
{G(pt,qt,ϱ)=g20p2t+g11ptqt+g02q2t,H(pt,qt,ϱ)=h20p2t+h11ptqt+h02q2t, | (3.23) |
and
{g20=a20(1+b(ϱ))−a11a(ϱ)+a2(ϱ)a021+b(ϱ),g11=a11−2a(ϱ)a021+b(ϱ),g02=a021+b(ϱ),h20=(1+b(ϱ))2b20+(a20−b11)(1+b(ϱ))a(ϱ)+(b02−a11)a2(ϱ)+a3(ϱ)a021+b(ϱ),h11=a11a(ϱ)−2a2(ϱ)a021+b(ϱ)−2a(ϱ)b02+b11(1+b(ϱ)),h02=b02+a(ϱ)a021+b(ϱ), | (3.24) |
gjk=hjk=0 ∀ j,k>0 and j+k=3. Furthermore, by employing the transformation:
{pt=nt+∑2≤j+k≤3Φjk(β)njt,qt=mt+∑2≤j+k≤3Ψjk(β)njtmkt, | (3.25) |
with
{Φ03=Ψ03=0,Φ20=12g20+14h20,Φ11=12g20+12g11+12h20+14h11,Φ02=12g11+12g02+18h20+14h11+14h02,Ψ20=12h20,Ψ11=12h20+12h11,Ψ02=14h11+12h02,Φ30=Φ21=Φ12=Ψ30=Ψ21=Ψ12=0, | (3.26) |
one gets the following 1:2 resonance normal form:
(nt+1mt+1):=(−11β1−1+β2)(ntmt)+(0C(β)n3t+D(β)n2tmt), | (3.27) |
where
{C(β1(ϱ0),β2(ϱ0))=(μ1μ2)2(34(μ3+μ6)2−2(μ3+μ6)+1),D(β1(ϱ0),β2(ϱ0))=(μ1μ2)2(74×(μ3+μ6)2−3(μ3+μ6)+2). | (3.28) |
Based on above analysis, one has the following result:
Theorem 3.1. If (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F12|EES and the discriminatory quantities, which are depicted in (3.28), that is, C≠0, D+3C=(μ1μ2)2(4×(μ3+μ6)2−9(μ3+μ6)+5)≠0, then at EES, model (1.8) undergoes 1:2 strong resonance. Additionally, EES is elliptic (respectively, saddle) if C>0 (respectively, C<0), and near 1:2 point D+3C≠0 defines the following bifurcation curves:
(i) Pitchfork bifurcation curve
F15|EES:={(β1,β2):β1=0}, | (3.29) |
and furthermore, nontrivial equilibrium state exists for β1<0;
(ii) Heteroclinic bifurcation curve:
F16|EES:={(β1,β2):β1=−53β2+O((|β1|+|β2|)2), β1<0}; | (3.30) |
(iii) Non-degenerate N-S bifurcation curve:
F17|EES:={(β1,β2):β1=−β2+O((|β1|+|β2|)2), β1<0}; | (3.31) |
(iv) Homologous bifurcation curve:
F18|EES:={(β1,β2):β1=−45β2+O((|β1|+|β2|)2), β1<0}. | (3.32) |
Remark 1. The occurrence of codimension-two bifurcation associated with 1:2 strong resonance at EES of model (1.8) indicates the complex dynamical behavior if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F12|EES. An important biological consequence of the non-degenerate N-S bifurcation is the existence of periodic or quasiperiodic oscillations between individuals I and S if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F12|EES. Furthermore, periodic oscillations or the homoclinic structure may cause long-period oscillations or even chaos in the I and S individuals.
From (2.9), if μ7=3(μ3+μ6)−(μ3+μ6)2−3μ3+μ6 and μ5=μ1(μ3+μ6)2−3(μ3+μ6)(μ3+μ4)+3(μ4−μ6)μ1μ3(μ3+μ6) then from (2.1) one gets:
V|EES=A0=A(ϱ0)|ϱ0=(μ5,μ7)=(13μ3+μ6−μ3−μ6−2), | (3.33) |
with λ1,2=−1±√3ι2 which implies that if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F13|EES then at EES, model (1.8) may undergoes codimension-two bifurcation with 1:3 strong resonance. Furthermore, eigenvector and adjoint eigenvector of A0 corresponding to eigenvalues −1±√3ι2, respectively are q=(3μ3+μ63−√3ι2) and p=((μ3+μ6)(1+√3ι)6)−√3ι3) satisfying
{A0q=−1+√3ι2q,A0ˉq=−1−√3ι2ˉq,AT0p=−1−√3ι2p,AT0ˉp=−1+√3ι2ˉq,⟨p,q ⟩=1. | (3.34) |
Now if Z=(utvt)∈R2 where it can be represented by Zt=ztq+ˉztˉq then (3.4) becomes
zt+1=−1+√3ι2zt+g(zt,ˉzt,ϱ0), | (3.35) |
where
g(zt,ˉzt,ϱ0)=⟨p(ϱ),F(ztq+ˉztˉq)⟩=∑2≤j+k≤3gjkj!k!zjtˉzkt, | (3.36) |
with
F(ztq+ˉztˉq,ϱ)=(μ1μ2utvt−μ1μ2utvt), | (3.37) |
and
{g20=μ1μ2(−32(μ3+μ6)+√3(2(μ3+μ6)+3)2(μ3+μ6)ι),g11=μ1μ2(32+3√3(μ3+μ6−2)2(μ3+μ6)ι),g02=μ1μ2(3(μ3+μ6+1)2(μ3+μ6)+√3(μ3+μ6+3)2(μ3+μ6)ι). | (3.38) |
Now following transformation
zt=wt+12h20w2t+h11wt¯wt+12h20¯wt2, | (3.39) |
along with its inverse transformation is utilized in order to eliminate quadratic terms from (3.35), where it becomes
wt+1=λ1wt+∑2≤j+k≤3σjkj!k!wjtˉwkt, | (3.40) |
with
{σ20=λ1h20+g20−λ21h20,σ11=λ1h11+g11−|λ1|2h11,σ30=3(1−λ1)g20h20+3g11ˉh02+3(λ31−λ21)h220+3(λ31−|λ1|2)h11ˉh02−3λ1ˉg02h11,σ02=λ1h02+g02−¯λ12h02,σ21={2g11ˉh11+g11h20+2g20h11+g02ˉh02+2λ21(¯λ1−1)h20h11−2λ1g11h20−¯λ1g20h11+2|λ1|2(λ1−1)|h11|2−2λ1h11ˉg11+|λ1|2(λ1−1)h11h20−¯λ1g02h02+¯λ1(λ21−¯λ1)|h02|2},σ12={2g11h11+g11ˉh20+2g02ˉh11+g20h02+λ1(¯λ12−λ1)h20h02−λ1ˉg20h11−2λ1g02h20−2¯λ1g11h11+2|λ1|2(¯λ1−1)h211+|λ1|2(¯λ1−1)h11ˉh20+2¯λ12(λ1−1)h02ˉh11−2¯λ1ˉg11h02},σ03={3g11h02+3g02ˉh20+3(¯λ13−|λ1|2)h11h02−3¯λ1g02h11−3¯λ1ˉg20h02+3¯λ12(¯λ1−1)h02ˉh20}. | (3.41) |
Now, it should be noted that quadratic terms of (3.40) should be vanished if
{h20=√3ι3g20,h11=3+√3ι6g11,h02=0. | (3.42) |
Furthermore, the following transformation yields to annihilate cubic terms:
wt+1=ξt+16h30ξ3t+12h12ξtˉξ2t+12h21ξ2tˉξt+16h03ˉξ3t. | (3.43) |
Using (3.43) along with inverse transformation, from (3.40), one gets:
ξt+1=√3ι−12ξt+12g02ˉξ2t+∑2≤j+k≤3rjkj!k!ξjtˉξkt, | (3.44) |
where
{r30=√3ι−32h30+σ30,r21=σ21,r12=√3ιh12+σ12,r03=√3ι−32h03+σ03. | (3.45) |
On setting
{h30=3+√3ι6σ30,h12=√3ι3σ12,h03=3+√3ι6σ03,h21=0, | (3.46) |
the ingredients from (3.45) should be zero except r21. Therefore, the desired normal form of 1:3 resonance is
ξt+1=√3ι−12ξt+C(μ5,μ7)ˉξ2t+D(μ5,μ7)ξt|ξt|2, | (3.47) |
where
{C(μ5,μ7)=(μ12μ2)(32+32(μ3+μ6)+(√32+3√32(μ3+μ6))ι),D(μ5,μ7)=(μ1μ2)2(18(μ3+μ6)2−272(μ3+μ6)+32+(9√32(μ3+μ6)−3√32)ι). | (3.48) |
Finally, let
{C1(μ5,μ7)=3ˉλ1C(μ5,μ7),=34×μ1μ2(μ3+μ6)(3+(−2√3(μ3+μ6)−3√3)ι),D1(μ5,μ7)=−3|C(μ5,μ7)|2+3λ21D(μ5,μ7),=14×(μ1μ2)2(1(μ3+μ6)2)((−45(μ3+μ6)2+135(μ3+μ6)−135)+(54√3(μ3+μ6)−108√3)ι), | (3.49) |
then one has the following theorem for the codimension-two bifurcation with 1:3 strong resonance:
Theorem 3.2. If the discriminatory quantities, which are depicted in (3.49), that is, C1≠0 and ℜ(D1)≠0 then at EES, model (1.8) undergoes codimension-two bifurcation with 1:3 strong resonance as the parameters (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F13|EES. Additionally, if ℜ(D1)≠0 examine the bifurcation behavior then at EES, model (1.8) has the following dynamical characteristics:
(i) If ℜ(D1)>0 then invariant closed curve occurs at 1:3 resonance point is unstable;
(ii) If ℜ(D1)<0 then invariant closed curve occurs at 1:3 resonance point is stable;
(iii) At trivial equilibrium state of (3.35) one has the non-degenerate N-S bifurcation.
Remark 2. If (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F13|EES then there exists codimension-two bifurcation associated with 1:3 strong resonance. From a biological perspective, periodic or quasi-periodic oscillation may occur in individuals I and S as a result of non-degenerate N-S bifurcation.
From (2.10), if μ7=2(μ3+μ6)−(μ3+μ6)2−2(μ3+μ6) and μ5=μ1(μ3+μ6)2−2(μ3+μ6)(μ3+μ4)+2(μ4−μ6)μ1μ3(μ3+μ6) then from (2.1) one gets:
A0=A(ϱ0)|ϱ0=(μ5,μ7)=(12μ3+μ6−μ3−μ6−1), | (3.50) |
with λ1,2=±ι which implies that if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F14|EES then at EES, model (1.8) may undergoes codimension-two bifurcation with 1:4 strong resonance. Furthermore, eigenvector and adjoint eigenvector of A0 corresponding to eigenvalues ±ι, respectively are q=(−2μ3+μ61−ι) and p=((μ3+μ6)(−1−ι)4−ι2) satisfying
{Aq=ιq,Aˉq=−ιˉq,ATp=−ιp,ATˉp=ιˉq,⟨p,q ⟩=1. | (3.51) |
Now if Z=(utvt)∈R2 where it can be represented by Zt=ztq+ˉztˉq then (3.4) becomes
zt+1=ιzt+g(zt,ˉzt,ϱ0), | (3.52) |
where
g(zt,ˉzt,ϱ0)=⟨p(ϱ),F(ztq+ˉztˉq)⟩=∑2≤j+k≤3gjkj!k!zjtˉzkt, | (3.53) |
with
F(ztq+ˉztˉq,ϱ)=(μ1μ2utvt−μ1μ2utvt), | (3.54) |
and
{g20=μ1μ2(1μ3+μ6+(μ3+μ6−1)μ3+μ6ι),g11=μ1μ2(1+(μ3+μ6−2)μ3+μ6ι),g02=μ1μ2(μ3+μ6+1μ3+μ6+1μ3+μ6ι). | (3.55) |
Now the transformation, which is depicted in (3.39) along its inverse transformation, is utilized to eliminate quadratic terms from (3.52), where it becomes
wt+1=ιwt+∑2≤j+k≤3σjkj!k!wjt¯wtk, | (3.56) |
with same σ's as in (3.41). Now, it should be noted that quadratic terms of (3.56) should be vanished if
{h20=ι−12g20,h11=ι+12g11,h02=ι+12g02. | (3.57) |
Using (3.43) along with inverse transformation, from (3.56), one gets:
ξt+1=ιξt+∑2≤j+k≤3rjkj!k!ξjt¯ξtk, | (3.58) |
where
{r30=2ιh30+σ30,r21=σ21,r12=2ιh12+σ12,r03=σ03. | (3.59) |
On setting
{h30=ι2σ30,h12=ι2σ12,h03=h21=0, | (3.60) |
the ingredients from (3.59) should be zero except r21 and r03. Therefore, the desired normal form of 1:4 resonance is
ξt+1=ιξt+C(μ5,μ7)ξt|ξt|2+D(μ5,μ7)¯ξt3, | (3.61) |
where
{C(μ5,μ7)=(μ1μ2)2(52(μ3+μ6)2−32(μ3+μ6)−14+(−92(μ3+μ6)2+92(μ3+μ6)−74)ι),D(μ5,μ7)=(μ1μ2)212(μ3+μ6)2(1−(μ3+μ6)−32(μ3+μ6)2+(1−(μ3+μ6)+12(μ3+μ6)2)ι). | (3.62) |
Let
{C1(μ5,μ7)=−4ιC(μ5,μ7),=(μ1μ2)21(μ3+μ6)2(−18+18(μ3+μ6)−7(μ3+μ6)2+(−10+6(μ3+μ6)+(μ3+μ6)2)ι),D1(μ5,μ7)=−4ιD(μ5,μ7),=(μ1μ2)21(μ3+μ6)2(2−2(μ3+μ6)+(μ3+μ6)2+(−2+2(μ3+μ6)+3(μ3+μ6)2)ι). | (3.63) |
Now if D1(μ5,μ7)≠0 and
{B(μ5,μ7)=C1(μ5,μ7)|D1(μ5,μ7)|,=1√10(μ3+μ6)4+8(μ3+μ6)3−16(μ3+μ6)+8(−18+18(μ3+μ6)−7(μ3+μ6)2+(−10+6(μ3+μ6)+(μ3+μ6)2)ι), | (3.64) |
then one has the following theorem for the codimension-two bifurcation with 1:4 strong resonance:
Theorem 3.3. If the discriminatory quantity, which is depicted in (3.64), such that, ℜ(B(μ5,μ7))≠0 and ℑ(B(μ5,μ7))≠0 then at EES, model (1.8) undergoes codimension-two bifurcation with 1:4 strong resonance as (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F14|EES. Additionally, B(μ5,μ7) determine the bifurcation behavior near EES of model (1.8) and so, there are two-parameter families of equilibrium state of order four bifurcation from EES near it. Depending on the choices made for μ5 and μ7, one of these families contains unstable, attracting, or repelling invariant circles. Furthermore, in a sufficiently small neighborhood of (μ5,μ7), there exist numerous complex codimension-one bifurcation curves of (3.61).
(i) At trivial equilibrium state of (3.61) there is a N-S bifurcation. Moreover, there is an invariant circle if λ=−ι and invariant circle will disappear if λ=ι;
(ii) If |B(μ5,μ7)|>1 then there exists eight equilibrium states that disappear or appear in pairs via fold bifurcation at μ5 and μ7;
(iii) At eight equilibrium states, there exists N-S bifurcations. Furthermore, four small invariant circles bifurcate from equilibrium states, and vanish near the homoclinic loop bifurcation curve.
Remark 3. The presence of a non-degenerate N-S bifurcation is indicated by the occurrence of codeminsion-two bifurcation associated with 1:4 strong resonance. In a specific parametric region, it is also feasible to generate an invariant cycle of period-4 orbit. In biology, the non-degenerate N-S bifurcation may give rise to periodic or quasi-periodic oscillations in individuals I and S.
Example 1. In this Example, it is proved numerically that if μ1=3.9, μ2=0.5, μ3=0.28, μ4=0.22, μ6=0.9 and varying μ5∈[0.2,2.9], μ7∈[−0.6,1.9] with (I0,S0)=(0.03,0.6) then at EES, model (1.8) undergoes codimension-two bifurcation with 1:2 strong resonance. For this, in the following, first one need to prove the eigenvalues criterion for the existence of 1:2 strong resonance holds. For this, if μ1=3.9, μ2=0.5, μ3=0.28, μ4=0.22, μ6=0.9 then from (2.8) one gets: μ5=0.271900415968213,μ7=−0.5698305084745758. Therefore, if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.5,0.28,0.22,0.271900415968213,0.9,−0.5698305084745758) then model (1.8) has EES=(0.43459365493263796,0.06410256410256411) and moreover, from (2.1) one gets:
V|EES=(13.389830508474576−1.1800000000000002−3), | (4.1) |
with λ1,2=−1 and so (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.5,0.28,0.22,0.271900415968213,0.9,−0.5698305084745758)∈F12|EES=(0.43459365493263796,0.06410256410256411), and finally, from (3.28) one gets C=−19.207188000000006≠0 and D=54.555228≠0, D+3C=−3.0663359999999864≠0 which imply that model (1.8) undergoes a codimension-two bifurcation with 1:2 strong resonance. Therefore, the simulation agree with the conclusion of Theorem 3.1. Hence, codimension-two bifurcation diagrams with 1:2 strong resonance are drawn in Figure 1.
Example 2. In this Example, it is proved numerically that if μ1=3.9, μ2=0.2, μ3=0.3985, μ4=0.75, μ6=0.95 and varying μ5∈[0.612332,2.9], μ7∈[−0.6,0.855] with (I0,S0)=(0.0669333,0.0794103) then at EES, model (1.8) undergoes codimension-two bifurcation with 1:3 strong resonance. For this, first one need to prove the eigenvalues criterion for the existence of 1:3 strong resonance holds. So, if μ1=3.9, μ2=0.2, μ3=0.3985, μ4=0.75, μ6=0.95 then from (2.9) one gets: μ5=0.8806815166411702, μ7=−0.5731941045606225. Therefore, if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.2,0.3985,0.75,0.8806815166411702,0.95,−0.5731941045606225) then model (1.8) has EES=(0.11408687715695504,0.05889743589743591) and moreover, from (2.1) one gets:
V|EES=(12.224694104560623−1.3485000000000003−2), | (4.2) |
with λ1,2=−1±√3ι2 and so (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.2,0.3985,0.75,0.8806815166411702,0.95,−0.5731941045606225)∈F13|EES=(0.11408687715695504,0.05889743589743591), and finally, from (3.48) one gets C=25.470383759733036+27.228503386338268ι and D=527.560465001899+1209.8979374673806ι. From (3.49), one gets C1=32.53615127919911−107.01675321970966ι≠0 and D1=−1818.3291845871258−3185.4892003729983ι which shows that ℜ(D1)=−1818.3291845871258<0 which imply that model (1.8) undergoes a codimension-two bifurcation with 1:3 strong resonance. Therefore, the simulation agree with the conclusion of Theorem 3.2. Hence, codimension-two bifurcation diagrams with 1:3 strong resonance are drawn in Figure 2.
Example 3. Finally, it is proved numerically that if μ1=3.9, μ2=0.5, μ3=0.3985, μ4=0.45, μ6=0.95 and varying μ5∈[0.95607,2.9], μ7∈[−0.9,0.855] with (I0,S0)=(0.0446222,0.0794103) then at EES, model (1.8) undergoes codimension-two bifurcation with 1:4 strong resonance. For this, in the following, first one need to prove the eigenvalues criterion for the existence of 1:4 strong resonance holds. So, if μ1=3.9, μ2=0.5, μ3=0.3985, μ4=0.45, μ6=0.95 then from (2.10) one gets: μ5=1.8148732738022662,μ7=−0.8316294030404152. Therefore, if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.5,0.3985,0.45,1.8148732738022662,0.95,−0.8316294030404152) then model (1.8) has EES=(0.1901447952615917,0.1087820512820513) and moreover, from (2.1) one gets:
V|EES=(11.4831294030404152−1.3485−1), | (4.3) |
with λ1,2=±ι and so (μ1,μ2,μ3,μ4,μ5,μ6,μ7)=(3.9,0.5,0.3985,0.45,1.8148732738022662,0.95,−0.83162940304041)∈F14|EES=(0.1901447952615917,0.1087820512820513). On the other hand, if (μ1,μ2,μ3,μ4,μ5,μ6,μ7)∈F14|EES=(0.1901447952615917,0.1087820512820513) then from (3.62) and (3.63), one gets C=0.7573645541146341−54.00102260452535ι, D=−51.459886377274955+9.38011362272504ι, C1=−216.00409041810144−3.0294582164584933ι≠0 and D1=37.52045449090016+205.8395455090998ι≠0. Finally, from (3.64), one gets B=−1.032370271795126−0.014478997116506187ι as ℜ(B)=−1.032370271795126≠0 and ℑ(B)=−0.014478997116506187≠0 which imply that model (1.8) undergoes a codimension-two bifurcation with 1:4 strong resonance. Therefore, the simulation agree with the conclusion of Theorem 3.3. Hence, codimension-two bifurcation diagrams with 1:4 strong resonance are drawn in Figure 3.
The work is about the codimension-two bifurcation analysis of a discrete epidemic model (1.8) in the region R2+={(I,S):I, S≥0}. It is proved that if μ6>(μ3+μ7)(μ3+μ4)−(1−μ5)μ3μ1μ1−(μ3+μ4) then model (1.8) has EES (((1−μ5)μ3+μ6)μ1μ2−(μ3+μ7+μ6)(μ3+μ4)μ2μ1(μ3+μ6),(μ3+μ4)μ2μ1). At EES of model (1.8), we first identified the codimension-two bifurcations sets associated with (i) 1:2 strong resonance F12|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=4(μ3+μ6)−(μ3+μ6)2−4μ3+μ6,μ5=μ1(μ3+μ6)2−4(μ3+μ6)(μ3+μ4)+4(μ4−μ6)μ1μ3(μ3+μ6)}, (ii) 1:3 strong resonance
F13|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=3(μ3+μ6)−(μ3+μ6)2−3μ3+μ6,μ5=μ1(μ3+μ6)2−3(μ3+μ6)(μ3+μ4)+3(μ4−μ6)μ1μ3(μ3+μ6)}. |
(iii) 1:4 strong resonance
F14|EES:={(μ1,μ2,μ3,μ4,μ5,μ6,μ7):μ7=2(μ3+μ6)−(μ3+μ6)2−2μ3+μ6,μ5=μ1(μ3+μ6)2−2(μ3+μ6)(μ3+μ4)+2(μ4−μ6)μ1μ3(μ3+μ6)}, |
and then we have studied detailed codimension-two bifurcations with 1:2, 1:3, and 1:4 by bifurcation theory and series of affine transformations. Furthermore, we have also given biological interpretations of theoretical results. Finally, theoretical results are carried out numerically.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The Authors Adil Jhangeer and Muhammad Bilal Riaz are highly thankful to Ministry of Education, Youth and Sports of the Czech Republic for their support through the e-INFRA CZ (ID: 90254).
The authors declare that they have no conflict of interest regarding the publication of this paper.
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