Abstract
In this paper we study a two infection SIRSIR compartmental model, considering biological features described in dengue fever epidemiology. Due to a progressive loss of protective antibodies there is waning immunity in the first infection stage and disease enhancement or protection effects by the second infection stage. Bifurcation analysis reveals two codim2 bifurcations as organizing centers. The unfolding of a cusp bifurcation describes the transition of the diseasefree equilibrium into an endemic equilibrium by varying a parameter. These equilibria allow an analytical solution with explicit expressions which allow for a full geometrical interpretation of the occurring bifurcations related to stationary dynamics. A BogdanovTakens point is the starting point in the parameter space where oscillatory endemic dynamics occurs including a homoclinic connection. These findings bring additional insights on biological mechanisms able to generate rich and complicated dynamical behavior in simple epidemic models that are, so far, largely unexplored.
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1 Introduction
With relevant biological applications, mathematical models applied to infectious disease transmission are dynamical systems of interest, often providing a framework to help to understand the behaviour of disease spreading and control.
The simplest mathematical models for infectious disease dynamics are based on the principle of the mass action contact model between infected and susceptible individuals and recovery of the infected. However, the incorporation of several aspects of the disease can imply rich dynamical behaviour. Study of existence, boundedness, phaseplane, stability and bifurcation analysis are useful tools to investigate qualitatively and quantitatively the behavior of such complex dynamical systems.
In epidemiology, the SIR (SusceptibleInfectedRecovered) modeling framework is often used to describe the dynamics of infectious disease transmitted from person to person. This translates into a compartmental model described by the following system of ordinary differential equations (ODEs)
for the temporal evolution of the number of individuals in each of the three compartments, where N represents the population size.
In a system of ODEs, a bifurcation occurs when the parameter value varies causing an immediate qualitative change in its dynamic behaviour [15, 22]. The basic SIR model has only two coexisting equilibrium as possible stationary solutions. By plotting the solutions of the system as function of a parameter, considering, for example, the prevalence of the population I with respect to the infection rate \(\beta \), a oneparameter bifurcation diagram is obtained. A bifurcation occurs at the specific threshold value \(\beta =\gamma \) (in epidemiology known as forward bifurcation) [11, 27], where by increasing the infection rate \(\beta \) one system eigenvalue passes through zero and thereby exchanging the stability of the Disease Free Equilibrium (DFE) and the Disease Endemic Equilibrium (DEE). One is necessarily unstable when the other is stable. This explains the importance of the threshold value of a parameter (often \(\beta \)) where the supercritical transcritical bifurcation occurs.
Dengue fever is the most important viral mosquitoborne disease worldwide, with approximately 3.9 billion people at risk of infection. Caused by any of the antigenically related but distinct viruses, DENV1 to DENV4, it is estimated that around 400 million dengue infections occur every year [6, 38]. Primary natural dengue infections often go unnoticed as they are asymptomatic, but the main risk factor for developing a severe form of the disease is a secondary dengue infection with a different serotype [17, 30, 32, 34]. Infection with one serotype results in lifelong protective immunity, while the high antibody titters attained after primary infection appear to generate a degree of crossprotection to all serotypes for a while, the so called temporary crossimmunity (TCI) period [37]. When the temporary crossprotection fades, patients who experience a secondary heterologous dengue virus infection face an increased risk of severe disease, primarily due to the AntibodyDependent Enhancement process (ADE) [9, 16, 19, 29].
Dengue fever epidemiological dynamics shows large fluctuations in disease incidence, and several mathematical models describing the transmission of dengue viruses have been proposed to explain the irregular behavior of dengue epidemics [1, 2]. The combination of two realistic immunological aspects of dengue in a minimalistic hosttohost 2serotype model lead to rich dynamical behavior, including to a new chaotic window in an unexpected and much wider biological parameter region [3], able to qualitatively describe well the available empirical incidence data in Thailand [5].
Motivated by the wellknown complex dynamics observed in dengue fever epidemiological models, in this paper, we analyse a two infection SIRSIR compartmental model. The proposed model describes differences between primary and secondary infections. Without strain structure of pathogens, we consider temporary immunity after a primary infection (analogous to the TCI period) and disease enhancement in a subsequent infection (analogous to the ADE effect).
In this paper, our primary focus is on the mathematical analysis of the system, specifically exploring the theoretical aspects of the nonlinear dynamical system and bifurcation theory. Our research builds upon a recent study ([35]) that investigates complex dynamics resulting from the widely recognized biological features of dengue epidemiology, namely TCI and ADE. The model counts six epidemiological compartments, and because the population size is constant, the dimension of the model is reduced to five.
The analysis of our model reveals two codim2 bifurcations: the cusp CPbifurcation and BogdanovTakens BTbifurcation. These bifurcation structures act as organising centers from which a number of codim1 bifurcation curves originate. A Tbifurcation emerges from the CPbifurcation point lying on a transcritical TCbifurcation curve. From the BTbifurcation point a tangent Tbifurcation, a Hopf Hbifurcation and a global homoclinic Gbifurcation originate. Important is that the unfolding of the two codim2 bifurcations is different from those found the literature dealing with epidemiological models.
In the literature [23, 25, 28, 31] other types of extensions of the simple epidemiological models are discussed where besides introduction of additional compartments (such as Exposed and Treated) reformulation of epidemiological mechanisms by more complex are included. For instance, the linear incidence rate in the SIS or SIR model replaced by the saturating incidence rate, or the proportional susceptible recruitment rate by the logistic growth, and further, the saturated instead of proportional treatment rate, and finally, the number of hospital beds. These models are, sometimes after reduction, 2dimensional and this makes a rigorous analysis of these model possible. Often the TCbifurcation is not analysed in detail, but use is made of the next generation matrix approach to find the expression for \(\mathcal{{R}}_{0}\). In these models bifurcations with the same names used in this paper are analysed, but their unfolding into the underlying phaseplane dynamics is different. Generally the Tbifurcation, often part of a backward bifurcation and called a saddlenode bifurcation, occurs when one system eigenvalue is zero while all other have negative real parts. However, in the SIRSIR model case also at the Tbifurcation one system eigenvalue is zero, but there is in addition one eigenvalue positive and therefore the Tpoint cannot be considered as saddlenode bifurcation: saddlesaddle bifurcation would be more appropriate.
The paper is organized as follows. Section 2 presents the SIRSIR model including temporary immunity after a primary infection and disease enhancement or protection in a subsequent infection. Based on explicit expressions for the equilibrium solutions, Sect. 3 starts with a detailed analysis of the unfolding of the CPbifurcation. After the description of the equilibria, their stability is studied in the second part of Sect. 3. In Sect. 4 a detailed numerical bifurcation analysis is performed whereby the baseline parameter values are taken from dengue epidemiology studies by Aguiar and coworkers. [3,4,5, 21]. The effects of temporary immunity and enhancement or protection of the secondary infection is discussed in Sect. 5. Section 6 concludes this work with a discussion on modeling approaches and results.
2 Description of the SIRSIR model
As an extended version of the Kermack and McKendrick SIR model [11, 20], we study a 5dimensional two infection epidemiological SIRSIR model including a temporary immunity after a primary infection, and disease enhancement/protection in secondary infections. These additional mechanisms are likely to be present in various infectious diseases acting as a paradigmatic study system of much wider interest.
In Fig. 1 the disease epidemiological transmission processes occur with the infection rate \(\beta \), the recovery rate \(\gamma \) and the waning immunity \(\alpha \). The demographic transition processes by the birth and death rate is parameterized by \(\mu \).
While a first infection confers a period of immunity, parameterised by \(\alpha \), referring to the progressive loss of protective antibodies (analogous to the so called temporary crossimmunity period in dengue fever epidemiology), recurrent infection can evolve to a severe disease manifestation via disease enhancement/protection, parameterized by \(\phi \).
The parameter \(\phi \) is the ratio of secondary infection contribution to the overall frequencydependent force of infection, \(\beta (I_P+\phi I_S)/N\), where \(\phi < 1\) acts decreasing the infectivity of secondary infection, due to the immunity level created by the previous infection. However, when \(\phi > 1\), infectivity of secondary infection is larger than in the primary infection. If \(\phi =1\), there is no protection nor enhancement, thus the secondary infected individuals transmit the disease as much as individuals experiencing a first infection.
The system of ordinary differential equations for the twoinfection model is 6dimensional, given by
under the initial conditions
The set of ordinary differential equations describing the temporary evolution of the number of individuals in each of the model compartments reads
The total population size \(N=S+I_P+R_P+S_P+I_S+R\) is constant, with the demographic birth rate of new susceptible individuals been equal to the death rate that is assumed to be equal for individuals in all compartments.
This model is a simplified version of the minimalistic multistrain dengue model capturing differences between primary and secondary infection proposed in [5], and can be reduced by one dimension by replacing the Eq. (3f) to the constraint \(R=NSI_PR_PS_PI_S\). Dimension reduction of the system to a four equations system is not possible only when \(\alpha =0\).
Note that the compartmental state variables are scaled with N, and for the sake of simplicity, we assume \(N=100\), with the results interpreted as percentages of the total population size. In this study, the parameters \(\beta \) and \(\gamma \) are named the primary parameters, while the two parameters, \(\alpha \) and \(\phi \) the secondary parameters.
3 Analysis of the epidemiological model
The model (2) defined by Eq. (3) is positive invariant, with the solutions lying in the nonnegative first quadrant because for each zero state variable the time derivative is positive. The total nonnegative sizes of the model compartments equals the total population size N, with all state variables in the interval [0, N] and therefore, the system is bounded.
3.1 Equilibria of the dynamical system
The equilibrium equations for the reduced system, given by Eq. (3), are
After some algebraic manipulations the following expressions for \(I^*_P\), \(R^*_P\), \(S_P^*\) and \(I^*_S\) depending on the single variable \(S^*\), can be derived:
Substituting these expressions, Eq. (4a) yields a third order polynomial in \(S^*\). This expression can be factorised in a linear factor \(NS^*\) and a quadratic factor (secondorder polynomial in \(S^*\)). The linear factor is the trivial DFE solution \(E_0=(N,0,0,0,0)^T\). Solving the quadratic factor gives the two solutions for \(S^*\):
and with Eq. (5) the explicit values for possibly coexisting DEE, \(E_1=(S^*,I_P^*,R_P^*,S_P^*,I_S^*)^T\). Two DEE exist when the discriminant (the expression under squareroot) is nonnegative and one DEE exists when the discriminant is zero.
The equilibrium curve in the equilibrium diagram for parameter \(\beta \) is a 90 degrees clockwise rotated parabola. One of its branches intersect the horizontal axis where \(\beta =\gamma +\mu \). From Eq. (6) and with \(S^*=N\) we derive that the DFE and DEE curves intersect at a point where \(\beta =\gamma +\mu \).
Both (DEE) branches coincide when the discriminant of the quadratic factor in Eq. (6) is zero:
This occurs at the minimum \(\beta _T\) value of the tangent Tbifurcation point. A bifurcation point fixed as a minimum parameter point is also mentioned in [22, page 527].
In order to derive a criterion that all three equilibria coincide with \(S^*=N\) we take \(\gamma \) as an addition free parameter. The \((\beta , \gamma )\) combination where this occurs is fixed by the condition:
This gives the critical value for \(\gamma \):
Figure 2 shows for three \(\gamma \) values the equilibrium curves for the primary infected \(I_P\). When Eq. (9) is satisfied the equilibrium curve is symmetric with respect to the horizontal axis and the three solutions are all zero at \(\beta =\gamma _{CCP}+\mu \). Using the epidemiological literature terminology we conclude: when \(\gamma <\gamma _{CCP}\) the bifurcation is called a forward and when \(\gamma >\gamma _{CCP}\) a backward bifurcation.
These results ware obtained solely based on geometrical aspects, no stability criteria.
3.2 Stability of the equilibria and the TCbifurcation when \(\beta \) is varied
The Jacobian matrix evaluated at a given point \(E=(S, I_P,R_P,S_P,I_P)\) reads
and, evaluated at the DFE \(E_0=(N,0,0,0,0)\), reads
giving the following eigenvalues:
Four eigenvalues are negative and only one is zero when \(\beta = \gamma +\mu \), where the equilibrium \(E_0\) changes stability. Therefore, a TCbifurcation where it exchanges stability with \(E_1\) or a Tbifurcation can occur, see [15, Theorem 3.4.1]. Note that the \(\beta \)value at the bifurcation point depends exclusively on the other primary parameter \(\gamma \), and is not directly dependent of the secondary parameters: the temporary immunity period \(\alpha \) and the disease enhancement/protection factor \(\phi \).
In order to identify these bifurcations transversality conditions have to be satisfied. They depend on the righteigenvector \(V_0\) and lefteigenvector \(W_0\) of the Jacobian matrix \(J_0\) corresponding to the simple zero eigenvalue (where all other eigenvalues have negative real parts) given by
As described in [15, p 148], the transversality conditions for the TCbifurcation occurs at \(\beta _{TC}=\gamma + \mu \) read:
and the transversality conditions for the Tbifurcation at \(\beta _{T}=\gamma + \mu \) are:
The expressions on the left hand side of the system Eq. (3) are evaluated at the TCbifurcation point occurring at \(\beta =\gamma +\mu \), and are
While the first condition, Eq. (14a), and the second condition, Eq. (14b), are satisfied for the TCbifurcation, the first condition for nondegeneracy for the Tbifurcation in Eq. (15a) is not satisfied along the \(\beta =\gamma + \mu \) curve. Hence, the third condition for the TCbifurcation, Eq. (14c), determines whether the TCbifurcation is not degenerated. The right hand side of Eq. (16c), denoted by CCP, is the critical point of the TCbifurcation curve where a cusp point CP is located, i.e. the TC is degenerated when \({CCP}=0\). The evaluation of this criteria for the primary parameter \(\gamma \) gives
This expression is equals to Eq. (9) where it was derived based on geometrically stationary shape of the equilibrium curve.
3.3 Cusp bifurcation in the threeparameter space
In a similar way, as for the parameter \(\gamma \), we can derive the conditions for the CPbifurcation where the secondary parameters \(\alpha \) and \(\phi \) are varied. For the immunity period parameter, \(\alpha \), we have:
depending on \(\phi \). At this critical point (where \(\beta =\gamma +\mu \)) the TCbifurcation changes criticality, from supercritical to subcritical. The TCbifurcation is called supercritical when CCP is negative and subcritical when it is positive. Similarly, as described above, the critical point for the disease enhancement/protection parameter \(\phi \) is obtained:
Figure 3 shows the curve for \(\beta =\gamma +\mu \) and \(\alpha _{CCP}=0\) in the 3dimensional parameter space (\(\beta ,\gamma ,\alpha \)). This curve is in the vertical TCbifurcation sheet where above the \(\alpha _{CCP}\)curve, the TCbifurcation is subcritical and below supercritical.
The results discussed in this section are summarised in the following remarks. Remarks:

1.
System Eq. (3) has always a trivial equilibrium \(E_0= (N, 0,0,0,0)\), with at most two endemic equilibria in the positive invariant (biological) region \(\Omega = \{ (S, I_P, R_P, S_P, I_S, R) \in \textbf{R}^6 : S > 0, I_P \ge 0, R_P \ge 0, S_P \ge 0, I_S\ge 0 , S+ I_P+ R_P+ S_p+ I_S+ R=N \}\).

2.
The trivial equilibrium \(E_0= (N, 0,0,0,0)\) of the system Eq. (3) is locally asymptotically stable if \(\beta < \gamma + \mu \), and unstable otherwise.

3.
The TCbifurcation occurs at \(\beta = \gamma + \mu \). The TC is supercritical, for \( \phi < \phi _{CCP}\), given by Eq. (19), \(\phi _{CCP}= \frac{(\gamma +\mu )(\alpha +\mu )}{\gamma \alpha }\). On the other hand, if \(\phi > \phi _{CCP}\), the TC is subcritical.

4.
At \(\beta = \gamma + \mu \), the transcritical TC is supercritical for \( \alpha < \alpha _{CCP}\), given by Eq. (18), \(\alpha _{CCP} = \frac{\mu (\gamma +\mu )}{(\gamma \phi (\gamma +\mu ))}\). On the other hand, if \( \alpha > \alpha _{CCP} \), the TC is subcritical.

5.
If a supercritical TC occurs, System Eq. (3) has only one trivial equilibrium when \(\beta < \gamma + \mu \). If subcritical TC occurs, System Eq. (3) has two DEE for \(\beta < \gamma + \mu \).
Using computer packages such as Maple [26] it is often not possible to obtain explicit expressions where the TCbifurcation (of DFE) and/or Tbifurcation (of DEE) occur or the expressions can be unmanageable. In those cases a (numerical) bifurcation analysis can be performed which yielding results for bifurcations of equilibria and limit cycles, and further local and global bifurcations.
4 Numerical bifurcation analysis
The numerical bifurcation analysis is performed using AUTO [14] and Matcont [10] software. A combination of one, two and threeparameter diagrams are discussed. The analysis of the oneparameter diagrams is extended with the analyses of state space dynamics. The baseline parameter values are given in Table 1.
The list of bifurcation structures, and its definitions, can be found in Table 2.
The bifurcation structures are studied by continuation and the numerical results are presented as follows. We start varying the two primary parameters \(\beta \) and \(\gamma \) separately and then simultaneously, with results shown for all possible combinations, \((\beta ,\gamma )\), \((\beta ,\alpha )\) and \((\alpha ,\gamma )\) where \((\alpha ,\phi )\) are fixed. In the next sections secondary parameters \((\alpha ,\phi )\) are fixed.
4.1 Cases where the two secondary parameters are zero.
With \(\alpha =0\) the SIRSIR model Eq. (3) reduces to the SIR model Eq. (20) with three compartments, \(S,I_P,R_P\) and the plane \(S+I_P+R_P\) is attracting and invariant. The ordinary differential equations of the 3dimensional system are
The bifurcation diagram with \(I_P\) as a function of the parameter \(\beta \) is shown in Fig. 4a. Here, the transcritical TCbifurcation is supercritical since for \(\alpha =0\) it is smaller than \(\alpha _{CCP}=0.00962\).
With positive \(\beta \), either the stable trivial boundary equilibrium DFE, where all individuals are susceptible \(S=N\), or the stable endemic equilibrium DEE, where the state variables \(I_P, R_P\) are positive, exist. At the TC at \(\beta _{TC}=\gamma +\mu \) one eigenvalue of the Jacobian matrix is zero and the two equilibria, DFE and DEE, interchange stability.
On the other hand, Fig. 4b shows the case when the temporary immunity period is equal to the baseline value \(\alpha =52\), and with no disease enhancement (\(\phi =0\)), i.e., no contribution of the secondary infections to the force of infection. In addition to the variable \(I_P\), the variable \(I_S\) is also shown as a function of the parameter \(\beta \). The TC occurs as in the SIR model at \(\beta _{TC}=\gamma +\mu \), where the equilibria DFE and DEE interchange stability. With \(\phi =0\), the SIRSIR model reduces to the SIR model, and \(\phi =0<\phi _{CCP}=1.0006\). This shows that for the SIRSIR model with \(\phi =0\) the TCbifurcation is supercritical.
4.2 Effects of the two primary parameters \(\beta \) and \(\gamma \) varied simultaneously
The twoparameter bifurcation diagram with the two primary parameters \(\beta \) and \(\gamma \) as free parameters is shown in Fig. 5. The two secondary parameters \(\alpha \) and \(\phi \) are fixed as shown in Table 1. Note in Fig. 5 the TC bifurcation curve \(\beta =\gamma +\mu \) in red, and the Tbifurcation curve in blue. The Tbifurcation occurs when two equilibria collide in one endemic equilibrium. The DEE become unstable at a Hbifurcation. For a detailed list of the bifurcation structures identified in this study we refer to Table 2.
Figure 5b shows part of the diagram depicted in Fig. 5a, for smaller parameter values, highlighting the results for the region of \(0<\beta <0.8\) and \(0<\gamma <0.5\). In this region of the diagram, two codim2 bifurcation points are identified as the organising centers of bifucartions, the CP and the BT points, from which the codim1 bifurcation curves originate.
At the BTbifurcation point, two eigenvalues of the Jacobian in the DEE (\(E_1\)), are zero. From this point, three bifurcation curves emerge, the Tbifurcation curve, Hbifurcation curve, and also a global homoclinic Gbifurcation curve (see [15, 22]).
In Fig. 5b it is possible to identified the bifurcation structures around the cusp CP point. From this point on, of the TCcurve of the DFE, the Tbifurcation curve of the DEE emerges. It marks also where the TC changes from supercritical to subcritical.
By increasing \(\beta \), the transition steps in Fig. 5b are observed as follows. The TCbifurcation of the DFE that is supercritical below the CPpoint, becomes subcritical, between CP and BT points, and emerges a Tbifurcation of the DEE. In order to get more insight in this process, we count the number of positive, \(n_{+}\), zero, \(n_{0}\) and negative real parts \(n_{}\) of the total 5 eigenvalues, identified by \(n=(n_{+},n_{0},n_{})\).
At point CP and also between the CP and BT points we have \(n=(1,0,4)\). At the BT point we have \(n=(0,2,3)\) and and above the BT on the Tbifurcation curve \(n=(1,1,2)\) and this does not change until \(\gamma =52\) (see also Fig. 5). From the BTpoint at the Hbifurcation curve we have \(n=(0,2,3)\).
4.2.1 Oneparameter diagrams for the infection rate parameter \(\beta \)
The oneparameter diagram where \(\beta \) is varied for the baseline parameter set given in Table 1, hence fixed \(\gamma \)value 52, is shown in Fig. 6. The asymptotic solutions of the model for variables \(I_P\) and \(I_S\) are plotted as \(\beta \) changes in the interval [0, 104]. Below the TCpoint there is only the stable DFE. The TCpoint is subcritical. This means that for lower \(\beta \)values DEEs exist. However, these DEEs are unstable in the \(\beta \) interval above the Tbifurcation. Starting from the TCpoint where, the system eigenvalues count is \(n=(0,1,4)\) following the lower branch of the DEE we have \(n=(1,0,4)\), that is one eigenvalue is positive, and at the T point, \(n=(1,1,3)\). For the upper branch of the DEE, \(n=(2,0,3)\), with two eigenvalues positive in the whole interval, till \(\beta =104\).
The equilibrium values for the single unstable DEE read:
with the eigenvalues count \(n=(2,0,3)\). Starting simulations close to this saddlefocus point the solution spirals to a stable limit cycle with a large period shown in the phaseplane in Fig. 7 and its one period time dependency for the two infected populations \(I_P\) and \(I_S\), in Fig. 8. There is convergence to a homoclinic connection of the DFE (\(E_0\)) point at a homoclinic global Gbifurcation. Because there are time intervals where the state variables \(I_P\), \(R_P\)and \(I_S\) are small we call the dynamics pulsed.
To get more insight in this dynamics we take the state variables \(I_P,I_S,R_P\) in the model equations of order \(\mathcal {O}(\epsilon )\) with \(\epsilon \) small. The system becomes:
Hence, during one cycle when the solutions for \(I_P,I_S,R_P\) are small also their rates \(\dot{I}_P,\dot{I}_S,\dot{R}_P\) are small and this explains the pulsed dynamics shown in Fig. 8.
In the twoparameter diagram \((\beta ,\gamma )\) Fig. 5a the Gbifurcation originates from the BT point which is not a DFE. Therefore we study the dynamics for \(\gamma =0.35\) instead of \(\gamma 52\). The oneparameter diagram is shown in Fig. 9
The projections of the cycles and the onecycle time simulations for the infected stages are studied in the phaseplane in Fig. 10, and its time dependency in Fig. 11.
From the results in Fig. 10 it is clear that the homoclinic orbit DEL does not approach the DFE, instead it approaches the unstable lower branch of the DEE shown in Fig. 9.
The complicated dynamics as a result of the pulsed limit cycles gave numerical difficulties using continuation software. When parameters \(\beta \) and \(\gamma \) are smaller than 1, in the oneparameter diagrams in Fig.9, with \(\gamma =0.35\), the software AUTO was used. But for the oneparameter diagrams in Fig. 6, with \(\gamma =52\), we used the software Matlab, solver “ode45” adding in the “odeset” function the option “NonNegative”. This is necessary, to avoid negative numbers of the state variables knowing that the model solutions are in the nonnegative quadrant of the 5dimensional state space.
4.2.2 One parameter bifurcation diagram for the recovery rate \(\gamma \)
The equilibria for the two infected compartments as a function of the parameter \(\gamma \) are depicted in Fig. 12. Below the supercritical Hbifurcation the DEEs are stable. Observe that, for very low recovery rate \(\gamma \) values, \(I_P\) converges to a point where the population is almost completely infected, that is where:
5 The role of temporary immunity
In this section we analyze the twoparameter diagrams (\(\beta ,\alpha \)) and (\(\gamma ,\alpha \)) and continue with a one parameter bifurcation diagram varying the temporary immunity parameter \(\alpha \).
5.1 Twoparameter \((\beta ,\alpha )\) and \((\gamma ,\alpha )\) diagrams
Figure 13 show the twoparameter diagram where \(\beta \) and \(\alpha \) are varied simultaneously. The bifurcations curves and points detailed in the diagrams for the parameters \(\beta \) and \(\gamma \) in Fig. 5 are also seen in this figure. However, in this case, a subcritical Hbifurcation is occurring very close to the \(T_c\)bifurcation of the emerging limit cycle.
In Fig. 14a the unfolding of the CP and BTpoints are shown. The \(T_c\)bifurcation curve terminates at the Gbifurcation curve that is, as depicted, very close to the TC birfurcation curve.
Now, analysing the twoparameters \((\gamma ,\alpha )\) diagram, there is a Hbifurcation curve having \(n=(0,2,3)\) as showed in Fig. 14. Below the Hbifurcation curve a stable DEE \((E_1)\) having \(n=(0,0,5)\), and above the curve an unstable DEE having \(n=(2,0,3)\). In addition, a stable DEL, \((L_1)\) appears (see Fig. 14b). A \(T_c\)bifurcation curve very close to the Hbifurcation is studied in more detail in the next section with the description of the oneparameter diagram for parameter \(\alpha \).
5.2 Oneparameter diagrams of the secondary parameter \(\alpha \)
Figure 15 gives the diagram where \(\alpha \) is varied in a region around the Hbifurcation, for the values of the parameters fixed in the Table 1. This point occurs for the \(\alpha \) value depicted in Fig. 13, for \(\beta =104\) and for \(\gamma =52\).
Between the supercritical Hbifurcation and the \(T_c\)bifurcation of the emerging stable, there is a coexistence of the stable DEE (\(E_1\)) and two stable DELs (\(L_1\)) of which the maximum and minimum values are shown. At the \(T_c\)bifurcation one Floquet multiplier is equal to 1 and the other are inside the complex unit circle, that is, \(n=(0,1,3)\) (the one multiplier equal to 1, because it is a limit cycle, is not counted). The stable DEE and the two DELs for the infected stages \(I_P\) and \(I_S\), are show in the phaseplane in Fig. 16. In addition, below the \(T_c\)bifurcation point the DEE is stable and above H point the DEL is stable.
5.3 Three parameter \((\beta ,\gamma ,\alpha )\) diagram
Unfortunately, the free available computer packages in AUTO [14] and MatCont [10] do not support continuation of the codim2 bifurcations in three parameters. Therefore, to study the two codim2 CP and BTbifurcation, we conciliate these curves in the three parameter diagram space for the parameters \((\beta ,\gamma ,\alpha )\) in Fig. 17, by continuation of the Tbifurcation and detection of the BT and CPpoints.
We start at a point on the Tbifurcation curve for a specific value of parameter \(\alpha \). By continuation, in the two (\(\beta ,\gamma \)) plane, both the BT and CP points were found. The procedure is repeated for a different initial \(\alpha \)value. Finally, connecting those critical bifurcations points found, for the different \(\alpha \)values, it will give the curves. Observe that the CP curve is the same as the degenerated TC curve in Fig. 3, that is, curve \(\alpha _{CCP}(\beta ,\gamma )\).
On the positive cube faces (see Fig. 17) the twoparameter diagram resulted are depicted as follows: \((\beta ,\gamma )\) from Fig. 5, \((\gamma ,\alpha )\) from Fig. 14 and \((\beta ,\alpha )\) from Fig. 13. In this way, the codim1 bifurcation curves form connected curves in the 3 faces.
To finalize this analysis, Fig. 18 shows the codim2 curves for a range closer to the origin and also highlighting part of the cube space in Fig. 17. In Fig. 18a the TC curve for \(\alpha =0\) in the bottomface is shown. In that region of the parameter space the cusp is just above this curve in the vertical plane where \(\beta =\gamma +\mu \), except for the region close to the origin. There the CP curve bends sharply upwards. Figure 18b shows that the two codim2 curves leave the cube very close to the bottomface.
6 The role of disease enhancement/protection
We follow the same analysis sequence as in the previous section to study the effects of the secondary infection contribution to the force of infection by taking \(\phi \) as bifurcation parameter together with \(\beta \) and \(\gamma \).
6.1 Two parameter \((\beta ,\phi )\) and \((\gamma ,\phi )\) diagrams
The twoparameter \((\beta ,\phi )\) diagram in Fig. 19 shows the same codim1 and codim2 bifurcation curves and points as found for the parameter \(\alpha \) in the previous section. However, in this diagram, the occurrence of a codim2 Bautin Bbifurcation point for the Hbifurcation, where it changes criticality from supercritical \(H^+\) to subcritical \(H^\) is showed. The parameter value where this point precisely occur is very hard to obtain, as well as, the \(T_c\)bifurcation of emerging limit cycles are also not displayed. Moreover, the BTbifurcation is the terminal point of the Hbifurcation curve, and the CPbifurcation is the terminal point of the Tbifurcation curve.
On the left side of the TCbifurcation curve the DFE is stable and the solutions of the system will converge to the stable DFE. In the right half plane and below the Hbifurcation curve the DEE is stable and above of the curve there is a stable (possibly pulsed with long episodes of infection states close to zero) DELs.
The twoparameter diagram in Fig. 20a shows the Bbifurcation point of the Hbifurcation, where it changes criticality from subcritical to supercritical and visaversa, where DELs emerges (not shown). Moreover, the BTbifurcation as the terminal point of the Hbifurcation, and the CPbifurcation as terminal point of the Tbifurcation are shown.
In the two parameter (\(\gamma ,\phi \)) diagram (see Fig. 20b), the Hbifurcation divided the parameter space into two regions. Below the Hbifurcation curve, the DEE are stable and above, a stable DEL exists.
6.2 Oneparameter \(\phi \) diagram
Figure 21 gives the diagram where parameter \(\phi \) is varied in a region around the Hbifurcation point, at \(\phi = 1.75\). This Hbifurcation point occurs for the baseline parameters, that is, \(\alpha =52\), \(\gamma = 52\) and \(\beta =104\), as depicted, respectively, in the Figs. 19 and Fig. 20. The size of the two infected states \(I_P\) and \(I_S\) of the stable DEE, below the Hbifurcation point is remarkably small. Furthermore, the amplitude of the stable DEL just above the Hbifurcation increases very fast and the minimum values of the infected states DEL become very small.
6.3 Three parameter \((\beta ,\gamma ,\phi )\) diagram
The threeparameter bifurcation diagram for the parameters \((\beta , \gamma ,\phi )\) is showed in Fig. 22 and combines the three twoparameter diagrams \((\beta ,\gamma )\) (see Fig. 5a), \((\phi ,\beta )\) (see Fig. 19), and \((\gamma ,\phi )\) (see Fig. 20b). The codim1 bifurcation curves form connected curves in the 3 planes (top, back and right, respectively). The 2dimensional sheets of the codim1 bifurcations within the cube are not shown. For instance, below and before this sheet of the Hbifurcation a stable endemic region exists in the 3dimensional parameter space.
In this figure the CP and BTbifurcations are shown. It is striking that below the plane \(\phi =1\) we have only the vertical plane above the TCcurve in the bottomface where \(\beta =\gamma +\mu \).
7 Discussion and conclusions
Modeling insights for epidemiological scenarios characterized by complex dynamics have been largely unexplored. The development of mathematical guiding tools taking into account intrinsic dynamical behaviour of each modeling framework and its limitations for different epidemiological scenarios is of major importance.
In the epidemiological literature the basic reproduction number \(\mathcal{{R}}_{0}\) is considered as the most important quantity. In [12] it was introduced as the spectral radius of the next generation matrix. With applications in compartmental models this matrix is related to the Jacobian matrix evaluated for the DFE [11, 13, 36]. Is important to observe that the expressions for the elements of the decomposition of the Jacobian matrix are derived strictly by the epidemiological status of each compartment, for instance, the transmission is separated from transition, such as, the removal rate by death rate of individuals.
In this paper we do not use the method of the next generation matrix and calculation of the basic reproduction number \(\mathcal{{R}}_{0}\). The explicitly calculation of \(\mathcal{{R}}_{0}\) using this method can be found in [35]. Instead we use the nonlinear dynamical system theory approach for the continuous time ODE compartmental model. In this context the threshold condition \(\mathcal{{R}}_{0}=1\) is fixed by parameter values where the ODE system possesses a TCbifurcation, or stated otherwise, where one eigenvalue of the Jacobian matrix is equals to zero. This is related to a zero natural rate of increase (Malthusian parameter) a fact not explicitly used here. The TC bifurcation can be either supercritical (forward) or subcritical (backward).
If a subcritical TC bifurcation occurs, in the SIRSIR model the equilibrium curve is a 90 degrees clockwise rotated parabola of which the lower branch intersects the horizontal axis at the TCbifurcation point and also a Tbifurcation occurs at the minimum \(\beta \) value of that equilibrium curve.
The exchange from forward to backward bifurcation, occurs when the equilibrium curve leaves the TC point on the horizontal axis perpendicularly where now the three equilibria coincide. This gives the conditions for the occurrence of the change of criticality of the TCbifurcation. This simple procedure does not yield the results related to stability and unfolding of the involved codim1 bifurcations and the CPbifurcation itself. This criticality depends on transversality conditions [15, 33, 36] formulated at the TC point. The application of bifurcation theory gives a mathematically solid derivation of the system in the neighborhood of the cusp bifurcation.
In the SIRSIR model we found the CPbifurcation as one of the most important bifurcation. It fixes the point in the parameter space where the TC changes its criticality and the emergence of a Tbifurcation. Generally at a CPbifurcation, two Tbifurcations emerge, but here a Tbifurcation of the DEE emerges at the CP point from a point on the TCbifurcation curve. Hence, the complex dynamics is due to the larger dimension of the system (higher than two dimensional as in the case of the SIR model).
In order to illustrate that the CPbifurcation is the point where the TCbifurcation changes its criticality, we mention the numerical results discussed in [28] where a SIR model with logistic growth, nonlinear incidence rate and saturated treatment was analysed. The dynamics of that system shows some similarities with that of the model considered in this paper. In [28, Fig. 7] two cusp bifurcation points were calculated by continuation, but the fact that at these points the criticality of the TCbifurcation changes is not mentioned.
The BTbifurcation marks the emergence of a Hbifurcation and a Gbifurcation. This make more complex dynamics possible for instance Fig. 16 shows a near a subcritical Hbifurcation where one stable DEE coexists with two stable DELs.
In Fig. 23 the function \(\beta _{CPP}(\alpha ,\phi )\) derived from Eq. (19) whereby \(\beta =\gamma +\mu \), so that \(\beta _{CPP}\) is given by
For this we conclude that with moderately infection rates \(\beta \) the CP and also the BTbifurcations occur as organizing centers leading to complicated dynamics for \(\alpha >0\) and \(\phi >1\).
The role of secondary infection contribution to the force of infection can be rephrased observing that the feedback from the effects of the secondary infection is negative when \(\phi <1\) and positive when \(\phi >1\), (see also diagram Fig. 1). Complex dynamics is possible in a large region of the parameter space when the feedback of the secondary infection is positive.
The CP and BTbifurcations were also found earlier in the epidemiological literature dealing with various other extensions of the classical SIS or SIR model [7, 8, 18, 23,24,25, 31, 33]). These models are described by 2dimensional ODE systems allowing a more general analysis of the unfolding. Bifurcations of higher dimensional system as discussed in this work where also these bifurcations occur, may differ from those for 2dimensional systems. As an example, the Tbifurcations for equilibria and \(T_c\)bifurcations for the limit cycles found in this work are of a different type than the classical once. Of the unfolding bifurcations the number of the positive/zero/negative system eigenvalues differ. As a result, even with the existence of a “backward” bifurcation there is no hysteresis phenomenon and the system can be unconditional diseasefree in the parameter range below the TCbifurcation point.
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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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This research is supported by the Basque Government
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M.A. acknowledges the financial support by the Ministerio de Ciencia e Innovacion (MICINN) of the Spanish Government through the Ramon y Cajal Grant RYC2021031380I funded by MICIU/AEI/10.13039/501100011033 and by the European Union extGenerationEU/ PRTR. A.K.S. acknowledged the financial support by the ministerio de ciencia e innovación (MICINN) of the Spanish Government through the juan de la cierva grant FJC2021046826I /MICIU/AEI /10.13039/501100011033 y por la Unión Europea NextGenerationEU/ PRTR. This research is supported by the Basque Government through the “Mathematical Modeling Applied to Health” Project, BERC 2022 2025 program and by the Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation CEX2021001142/MICIN/AEI/10.13039/501100011033.
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Aguiar, M., Steindorf, V., Srivastav, A.K. et al. Bifurcation analysis of a two infection SIRSIR epidemic model with temporary immunity and disease enhancement. Nonlinear Dyn 112, 13621–13639 (2024). https://doi.org/10.1007/s11071024097109
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DOI: https://doi.org/10.1007/s11071024097109