Abstract
In this article, we study the dynamics of a new proposed age-structured population mathematical model driven by a seasonal forcing function that takes into account the variability of the climate. We introduce a generalized force of infection function to study different potential disease outcomes. Using nonlinear analysis tools and differential inequalities theorems, we obtain sufficient conditions that guarantee the existence of a positive periodic solution. Moreover, we provide sufficient conditions that assure the global attractivity of the positive periodic solution. Numerical results corroborate the theoretical results in the sense that the solutions are positive and the periodic solution is a global attractor. This type of models are important, since they take into account the variability of the weather and the impact on some epidemics such as the current COVID-19 pandemic.
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1 Introduction
Mathematical modeling of dynamical systems is a useful tool to analyze a variety of infectious diseases, to understand their dynamical behavior, to predict their impact on different populations, and to study how different factors can impact the evolution of diseases. One preferred technique to model the behavior of communicable diseases in the population is using systems of ordinary differential equations, where the state variables represent different sub-populations. These systems could be autonomous or nonautonomous depending on the nature of the parameters of the equations. For some infectious diseases and for some particular situations, it makes sense to consider parameters varying over the time, and then, a nonautonomous system is obtained. For instance, there are many diseases that vary depending on the environment conditions. Thus, if we take into account that in many regions around the world, there are four seasons, it seems logical to include the variability of the environment (Shobugawa et al. 2017; Vega et al. 2017; Paiva et al. 2012; Ferrero et al. 2016; Noyola and Mandeville 2008; Traoré and Sangaré 2017; Corberán-Vallet et al. 2018). Moreover, several researchers have remarked and studied the effects of climate change on diseases and ecology (Donaldson 2006; Fagan et al. 2014; Sengupta and Das 2019; Sekerci and Petrovskii 2015; Hurtado et al. 2014; Okuneye et al. 2018; Posny and Wang 2014; Codeço 2001). Then, one approach to include the effect of the weather in the aforementioned mathematical models is to allow time varying parameters. For instance, one disease that shows seasonal behavior is the respiratory syncytial virus (RSV) (Shobugawa et al. 2017; Vega et al. 2017; Paiva et al. 2012; Ferrero et al. 2016; Noyola and Mandeville 2008; Arenas et al. 2009; Jornet-Sanz et al. 2017).
In this paper, we study the dynamics of a developed age-structured population mathematical model driven by a seasonal forcing function to take into account the variability of the climate. In particular, the proposed model is an SEIR (susceptible-exposed-infected-recovered) type, which is suitable to model the dynamics of several diseases such as influenza, cholera, dengue, RSV, and others (Brauer 2017; Hethcote and van den Driessche 1991). For instance in Hogan et al. (2016b), the authors used an SEIRS model, to explore how the birth rate, varying immunity and seasonality, influences the different dynamics of RSV. Moreover, some authors have pointed out the relation between weather and the current COVID-19 pandemic (Azuma et al. 2020; Bashir et al. 2020; Malki et al. 2020).
Another important aspect of using this type of mathematical models is that once time the model is constructed, it is possible to fit the mathematical model to real data related to infected individuals to estimate the values of the unknown parameters to study different future outcomes or to study the impact of some health policies. It has been argued that nonautonomous models are in some cases more suitable, since the coefficients of the system are varying with the time; for example, the coefficients could be changed with the seasonal factors (White et al. 2007; Arenas et al. 2009; Rogovchenko and Rogovchenko 2009; Rosa and Torres 2018; González-Parra et al. 2009; Bezruchko and Smirnov 2000).
After Kermack-McKendrick published a deterministic model to represent disease transmission, many papers have been published in the direction of generalizing the rate of infection. In Capasso and Serio (1978), a generalization of the Kermack model was proposed, introducing an interaction term in which the dependence on the number of infectious occurs through a nonlinear bounded function that takes into account the saturation phenomena for a large number of infectious. This approach is different than the classical mass action and standard incidence functions where the function is not bounded. Using an interaction term that tends to a saturation level is more realistic to model many diseases (Hethcote and van den Driessche 1991; Capasso and Serio 1978; Brauer 2017).
One way to see the interaction term is by means of the functional form Sg(I), where the function g(I) is the force of infection. There are many functions that can be used for the function g(I). For instance, \(g(I)=\beta I\) would give a mass action transmission, which implies that the transmission rate increases with the size of the population, due to an increase of contacts with crowding. However, for some cases like diseases related to sexual contact, this assumption seems not realistic. Another situation that needs to be considered is that when there is a dangerous epidemic the people would react to reduce the number of contacts or even go to a self-quarantine to decrease the probability of infection (Capasso and Serio 1978; Hethcote and van den Driessche 1991). Thus, in some cases, it makes sense to assume a function g(I) with a saturate level or even a function that decreases for large number of infectives due to an awareness in the population (Capasso and Serio 1978). Mathematical models have been published using different types of infection force (Hethcote and van den Driessche 1991)
In Guerrero-Flores et al. (2019), the authors present a family of epidemiological models of the SIRS form, which model the transmission of a fatal disease, which has seasonal variations and has a periodic contagion rate. In these models, the force of infection is given by a function \(\beta (t)f(I)\) where I represents the number of infectious people. Using Leray–Schauder degree theory, they showed that the solutions are periodic depending on the value of the threshold parameter \(R_0\). Other interesting works related to periodic solutions include seasonally forced SIR and SEIRS models with impact of media coverage are presented in Zu and Wang (2015), Ávila-Vales et al. (2017).
When an epidemic model includes an incidence function that is nonlinear, usually, it presents complex structures which somehow reflect the effects of saturation. A typical functional form is the one based on the Holling type (Safi and Garba 2012), which is given by:
where \(\beta \) includes the product of the contact rate and the probability of transmission given a contact. In addition, the parameter \(\eta \) represents the effect of reducing the transmission of the disease by educated individuals compared to the class of the uneducated infectives. Thus, a minimum value of \(\eta =0\) means that educated individuals would not be able to spread the disease, and a maximum value of \(\eta =1\) means that both types of infectives transmit the disease with the same rate. Along these lines, a type of infection force has been defined as \(g(S,I)=kS^{h}I/(S^h+\alpha I^h)\) (Ren and Zhang 2017; Ponciano and Capistran 2011).
For many diseases, there is an incubation period after the individual has an effective contact with an infected individual. That is, the action of the pathogen is not immediate, but after a period of time has elapsed. The symptoms of the disease appear after the incubation period, and then, latent individuals enter the infectious class.
One step further beside the introduction of the force of infection functions with saturation effect is the use of functions varying on time. For instance, in Mateus and Silva (2017), the authors presented a force of infection given by a function of the form \(\beta (t)\phi (S,N,I)\). The authors presented the existence and stability analysis of endemic periodic solutions for a family of SEIRS models. Similar interesting models with some variants are also presented in Derrick and Van den Driessche (1993), Gao et al. (2018).
The force of infection is an inherent characteristic of each model, it has been generalized through different studies. In Luca et al. (2018), the authors proposed a force of infection for a susceptible individual with an age i, and given by the following expression:
where j runs on age classes, \(I^{p}_j(t)\) and \(N^{p}(t)\) count the total number of infectious individuals of age group j and the total population size of patch p at time t, respectively. Other interesting works along these lines can be seen in Korobeinikov (2009), Valle et al. (2013), Blackwood and Childs (2018) and Gölgeli and Atay (2020).
In this paper, we generalize different mathematical models presented previously and mentioned above. In particular, we study the existence and the analytical behavior of a \(S_i,E_i,I_i,R_i\) type model. This model is based on a deterministic ordinary differential equation system, where the force of infection is given by a function \(\beta _i(t)f(I_1(t),I_2(t))),\) which generalizes the models presented in Hogan et al. (2016b), White et al. (2005), White et al. (2007), Weber et al. (2001), Hogan et al. (2016a) and Hogan et al. (2017). We employ some interesting ideas used in related works in mathematical modeling in epidemiology (Luca et al. 2018; Capasso and Serio 1978; Guerrero-Flores et al. 2019; Mateus and Silva 2017; Valle et al. 2013; Blackwood and Childs 2018; Gölgeli and Atay 2020; Safi and Garba 2012; Jia and Zhang 2014). The generalized model that we propose is given by the following nonautonomous system:
where \(N_i(t)=S_i(t)+E_i(t)+I_i(t)+R_i(t)\) for \(i=1,2\) and total population is given by \(N(t)=N_1(t)+N_2(t)\). Moreover, the parameter \(\mu \) is the birth rate, \(\eta \) is the aging rate, \(\beta _1(t), \beta _2(t)\) are T-periodic functions related to the transmission of the disease, \(1/\delta \) is the mean value of the latent period, \(1/\gamma \) is the mean value of the infectious period, and \(1/\nu \) is the mean value of the immunity period. The seasonal force of infection is given in terms of \(\beta _1(t)f(I_1,I_2), \beta _2(t)f(I_1,I_2)\) where \(\beta _2(t)=\alpha \beta _1(t)\) with \(0<\alpha <1,\) and \(\alpha \) is a scale parameter for reduced susceptibility for the age class \(i=2\) (older class). This type of infection force was presented in previous works (Blackwood and Childs 2018; Valle et al. 2013; Gölgeli and Atay 2020).
One important problem related to population dynamics is the study of epidemiological models in a seasonal environment. Thus, in this paper, we are interested in the analysis of the existence and global stability of positive periodic solutions. These last characteristics play a similar role as the globally stable equilibrium does in the autonomous models. Therefore, it is of paramount importance to find out the necessary conditions under which the nonautonomous system underlying the proposed mathematical model would present a positive periodic solution and to show that this particular solution is globally asymptotically stable.
We will start with some preliminary steps that will help to find the aforementioned conditions. First, since the model (3) is nonhomogeneous of degree one, and for the sake of clarity, we make the following change of variable:
which that (3) can be written as:
This paper is organized in the following form. In Sect. 2, we present some notation and preliminary results for the sake of clarity. In the next Sect. 3, we prove the positivity and boundedness of the solutions. Our main results are presented in the next Sect. 4; we prove the existence of periodic solutions in terms of the related parameters of the model in a common interval period. In Sect. 5, we provide sufficient conditions that assure the global attractivity of the positive periodic solution. In Sect. 6, we conduct numerical simulations that support the theoretical results obtained. Finally, in Sect. 7, we draw the conclusions.
2 Notation and preliminary results
In this section, we will define some mathematical concepts. In addition, we will show some assumptions and establish some preliminary results. We define the following conventions:
where h(t) is a continuous T-periodic function. Next, the following hypotheses are assumed:
-
\(\left[ \mathrm{H}_1\right] \) The parameters: \(\mu , \eta , \alpha , \delta , \gamma , \nu \) belong to \((0,M_T],\) where \(M_T>0.\)
-
\(\left[ \mathrm{H}_2\right] \) There is \(T>0\), such that \(\beta _1(t), \beta _2(t)\) are positive T-periodic continuous functions in [0, T] and there are positive numbers \(\beta _1^{u},\beta _1^{l}\) with \(\beta _1^{u}\ge \beta _1(t)\ge \beta _1^{l}\), and \(\beta _2^{u} \ge \beta _2(t)\ge \beta _2^{l}.\)
-
\(\left[ \mathrm{H}_3\right] \) \( f :[0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) is a differentiable function. \(f(0,0)=0\), and \(f(x,y)>0,\) for all \(x,y>0.\) \(\dfrac{\partial f(x,y)}{\partial x},\dfrac{\partial f(x,y)}{\partial y}>0,\) for all \(x,y\ge 0.\)
-
\(\left[ \mathrm{H}_4\right] \) \(x\dfrac{\partial f(x,y)}{\partial x}+y\dfrac{\partial f(x,y)}{\partial y}-f(x,y)\le 0,\) for all \(x,y>0.\)
-
\(\left[ \mathrm{H}_5\right] \) There exists a \(\zeta _2>0,\) such that for all \(x,y>0\), it holds that: \(0<f(x,y)\le \zeta _2\).
From system (3), it can be seen that the hypothesis \(\left[ \mathrm{H}_1\right] \), is biologically correct, since these represent time lapses and contact rates. Moreover, the presumption for \(\beta _1(t)\) and \(\beta _2(t)\) in \(\left[ \mathrm{H}_2\right] \) is a natural way to introduce seasonality in these types of models (White et al. 2005; Hogan et al. 2016a; Arenas et al. 2008). Now, from condition \(\left[ \mathrm{H}_3\right] ,\) if there are no infectives, then there is no contact between susceptible and infectives, which is biologically natural for the this mathematical model (3).
After some calculations, it is verified that the class of functions \(f(w_3, w_7)\) that represent the incidence rate, and that satisfy the above assumptions, are given by: \(f(w_3, w_7) = aw_3 + bw_7, (a, b> 0)\), which is considered in Mitchell and Kribs (2019), Blackwood and Childs (2018), Arenas et al. (2008), Hogan et al. (2016a), Hogan et al. (2017), Gölgeli and Atay (2020), and in the same way those presented in Eq. (1) and in Eq. (2). Indeed, for the case equation (1), we have:
-
1.
\(f(0,0)=0,\,f(I_{u},I_{e})>0,\, \text { for all } I_{u}>0,\,I_{e}>0,\)
-
2.
\(\dfrac{\partial f}{\partial I_{u}}(I_{u},I_{e})=\dfrac{\beta }{\left( 1+\alpha _1I_{u}\right) ^{2} }>0,\,\dfrac{\partial f}{\partial I_{e}}(I_{u},I_{e})=\dfrac{\beta \eta }{\left( 1+\alpha _2I_{e}\right) ^{2} }>0,\,\text { for all } I_{u},I_{e}\ge 0,\)
-
3.
\(\dfrac{I_{u}\beta }{\left( 1+\alpha _1I_{u}\right) ^{2} }+\dfrac{I_{e}\beta \eta }{\left( 1+\alpha _2I_{e}\right) ^{2} }-\dfrac{I_{u}\beta }{\left( 1+\alpha _1I_{u}\right) }-\dfrac{I_{e}\beta \eta }{\left( 1+\alpha _2I_{e}\right) }\le 0,\,\text { for all } I_{u},I_{e}\ge 0,\)
-
4.
\(0<f(I_{u},I_{e})=\dfrac{\beta \alpha _1I_{u}}{\alpha _1\left( 1+\alpha _1I_{u}\right) }+\dfrac{\beta \eta \alpha _2I_{e}}{\alpha _2\left( 1+\alpha _2I_{e}\right) }<\dfrac{\beta }{\alpha _1}+\dfrac{\beta \eta }{\alpha _2}=\zeta _2,\,\text { for all } I_{u},I_{e}>0.\)
Another type of incidence functions less general than the bi-linear one that we use and present in Eq. (1) are functions of the form \(f(w_3)=\beta w_3/(1 + kw_3)\). These particular forms have been used to represent psychological or media effects for the infected population, and can be seen in Safi and Garba (2012), Guerrero-Flores et al. (2019).
3 Positivity and boundedness of the solutions
The model (4) must be consistent with the characteristics of a system of differential equations that represent a biologic population; in this case, the solution must be positive and bounded for all time. The following result shows these two characteristics.
Lemma 1
Assume \(w(t)=(w_1(t),\ldots ,w_8(t))\) is a solution of system (4), such that \(w_i(0)>0\) for all \(i=1,\ldots ,8\), and \(\left[ \mathrm{H}_1\right] -\left[ \mathrm{H}_4\right] \) hold. Then, \((w_1(t),\ldots ,w_8(t))\) is positive for all \(t>0\) and ultimately bounded.
Proof
Using the fact that the solution of system (4) is continuous, we can define:
with \(\mathfrak {M}(0)>0\). Now, if there exists a \(t_1>0\), such that \(\mathfrak {M}(t_1)=0,\) \(\mathfrak {M}(t)>0\) for \(t\in [0,t_1)\), and we suppose that \(\mathfrak {M}(t_1)=w_1(t_1)=0\) and \(w_4(t)>0\) for \(t\in [0,t_1).\) From the first equation of system (4), one gets:
for \(t\in [0,t_1].\) Therefore, applying the Comparison Principle in ordinary differential equation and using the continuity of \(w_1(t)\):
which is a contradiction. Considering the cases for \(\mathfrak {M}(t_1)=w_2(t_1)=\cdots =w_8(t_1),\) respectively, similar contradictions are deduced (Caraballo and Han 2017; Calatayud and Jornet 2020; Li et al. 2017). Thus, it follows that \(\mathfrak {M}(t)>0\) for all \(t>0\), i.e., \(x_i(t)>0\) for all \(t>0\) and \(i=1,\ldots ,8\).
Next, to prove boundedness of system (4), first, we define:
Then:
Thus:
Second, using the following function
it follows that:
By the theorem of differential inequality (Garrett Birkhoff 1989), one gets that:
as \(t \longrightarrow \infty .\) Thus, all the solutions of system (4) are ultimately bounded. \(\square \)
Remark 1
Moreover, from a biologically realistic point of view for sub-populations, for any \(\epsilon >0\), there is \(t_\epsilon > 0\), such that for \(t\ge t_\epsilon ,\) the set
is a positively invariant set for system (4).
4 Existence of a positive periodic solutions
The following results will be used to show the periodicity of the solutions of the mathematical model (4).
Proposition 1
If \(\left[ \mathrm{H}_1\right] -\left[ \mathrm{H}_5\right] \) hold, then the system of algebraic equations:
has a unique solution \(e^{\mathfrak {p_1}}>0,e^{\mathfrak {p_2}}>0,\ldots ,e^{\mathfrak {p_8}}>0.\)
Proof
It can be seen that the solutions are given by:
For \(e^\mathfrak {p_1}\), the function \(g:[0,\frac{\mu }{\eta }]\longrightarrow R,\) given by:
is continuous and satisfies:
and
By Bolzano’s theorem, there is \(\zeta \in ]0,\frac{\mu }{\eta }[\), such that \(g(\zeta )=0.\) Moreover:
for all \(t\in ]0,\frac{\mu }{\eta }[.\) Thus, \(\zeta \) is unique. Consequently, \(e^{\mathfrak {p_1}}\) exists and, therefore, satisfies the second equation that combining with the first equation yields that \(e^{\mathfrak {p_2}}=\dfrac{\mu -\eta e^{\mathfrak {p_1}} }{\delta +\eta }>0.\) Thus:
where \(e^{\mathfrak {p_6}}\) results from combining equations five and six. \(\square \)
Lemma 2
Let f be a function, such that \(f(x)\ge 0\), integrable, and uniformly continuous on \([0,+\infty ]\), and then, \(\lim \nolimits _{x \longrightarrow \infty } f(x)=0\), see Samanta et al. (2015), Gopalsamy (2013).
One of the topological tools used as an alternative for the demonstration of the existence of periodic solutions is the well-known continuation theorem of Mawhin or degree of coincidence published by Jean Mawhin in the decade of the seventies.
In Mawhin (2005), it is shown how it was the origin of the degree of coincidence. The strength and usefulness of this theorem for seasonal epidemiological models is based on using the parameters of the model in a common periodical interval, which can be seen in the following references (Zhang et al. 2012; Gaines and Mawhin 2006; Jódar et al. 2008; Li et al. 2017; Wang 2015; Chen 2005; Li and Xiong 2010; Li and Qin 2018). On the other hand, using a novel similar continuation theorem of coincidence degree theory, some authors proved the existence of anti-periodic solutions (Li et al. 2019).
Consider \(\mathscr {X}\) and \(\mathscr {Z}\) two normed and real spaces of infinite dimension. The linear operator \(\mathscr {L}:Dom\mathscr {L}\subset \mathscr {X}\longrightarrow \mathscr {Z}\); it is said of Fredholm of index zero if:
-
1.
Dimension of \(\mathrm {Ker} \mathscr {L}\) is finite, and
-
2.
\(\mathrm {Im} \mathscr {L}\) is closed in \(\mathscr {Z}\) and has a finite co-dimension:
-
3.
\(\mathrm {Index}\mathscr {L}=\mathrm {dim} \mathrm {Ker}\mathscr {L}- \mathrm {codim}\mathrm {Im}\mathscr {L}=0\).
This implies that there are continuous projectors \( \mathscr {P}: \mathscr {X}\longrightarrow \mathscr {X} \) and \( \mathscr {Q}: \mathscr {Z}\longrightarrow \mathscr {Z}\) with \(\mathrm {Im}\mathscr {P} = \mathrm {Ker}\mathscr {L}\), \(\mathrm {Im}\mathscr {L} = \mathrm {Ker}\mathscr {Q} = \mathrm {Im} (\mathscr {I}-\mathscr {Q})\). In addition, \(\mathscr {X} = \mathrm {Ker}\mathscr {L}\oplus \mathrm {Ker}\mathscr {P}\) and \(\mathscr {Z} = \mathrm {Im}\mathscr {L}\oplus \mathrm {Im}\mathscr {Q}\). Thus, the mapping \(\mathscr {O}=\mathscr {L}\mid _{Dom\mathscr {L} \bigcap \mathrm {Ker} \mathscr {P}}: (\mathscr {I}-\mathscr {P})\mathscr {X}\longrightarrow \mathrm {Im} \mathscr {L}\) is invertible. Let \(\mathscr {C}_{P}\) the inverse of \(\mathscr {O}\). Let \(\varOmega _0\subset \mathscr {X}\) an open and bounded, the mapping \(\mathscr {N}\) is called \(\mathscr {L}\)-compact on \(\varOmega _0\) if \(\mathscr {Q}\mathscr {N}(\varOmega _0)\) is bounded, and \(\mathscr {C}_P(1-\mathscr {Q})\mathscr {N} : \varOmega _0 \longrightarrow \mathscr {X}\) is compact.
Now, the following Mawhin’s theorem will allow us to find sufficient conditions to prove the existence of at least one periodic positive solution of the model (4).
Theorem 1
Let \(\varOmega _0 \subset \mathscr {X}\) be an open-bounded set. Let \(\mathscr {L}\) be a Fredholm mapping of index zero and \(\mathscr {N}\) be \(\mathscr {L}\)-compact on \(\bar{\varOmega }_0\). Then, the equation \(\mathscr {L}\mathrm {x}=\mathscr {N}\mathrm {x}\) has at least one solution in \(Dom \mathscr {L}\cap \bar{\varOmega }_0 \) if it holds:
-
1.
\( \mathscr {L}\mathrm {x} \ne \lambda \mathscr {N}\mathrm {x},\) with \(\lambda \in (0,1)\), and \(\mathrm {x}\in \partial \varOmega _0 \cap Dom\mathscr { L}\),
-
2.
\(\mathscr {Q}\mathscr {N}\mathrm {x}\ne 0,\) \(\mathrm {x}\in \partial \varOmega _0\cap \mathrm {Ker} \mathscr {L}\),
-
3.
\(\mathrm {deg}\{\mathscr {\mathscr {J}}\mathscr {Q}\mathscr {N},\varOmega _0 \cap \mathrm {Ker} \mathscr {L},0\} \ne 0. \)
One of the important results in this paper is the following:
Theorem 2
Suppose that \(\left[ \mathrm{H}_1\right] {-}\left[ \mathrm{H}_5\right] \) hold, and then, there exists a positive T-periodic solution of model (4).
Proof
Let us use the following change of variables:
Thus, (4) is equivalent to:
Now, suppose that one solution \(\bigl (\mathfrak {u_1}^{*}(t),\ldots ,\mathfrak {u_8}^{*}(t)\bigr )^{T},\) of (8) is positive T-periodic, and then, \(\bigl (w_{1}^{*}(t),\ldots ,w_{8}^{*}(t)\bigr )^{T},\) is a positive T-periodic solution of the system (4). This implies that there exists one solution of system (3). Therefore, it is sufficient to show that the system (8) has at least one positive T-periodic solution. To verify the hypotheses of Theorem 1, we define the following spaces:
with norm
It is easy to verify that \((\mathscr {X},||\cdot ||)\) is a Banach space, Ding and Zhao (2012), Rui et al. (2004).
We define \(\mathscr {L}:Dom \mathscr {L} \cap \mathscr {X}\longrightarrow \mathscr {X},\) by \(\mathscr {L}(\mathfrak {u}(t))=\dot{\mathfrak {u}}(t)=\frac{\mathrm{d}\mathfrak {u}(t)}{\mathrm{d}t},\) with:
and \(\mathscr {N}:\mathscr {X}\longrightarrow \mathscr {X}\) is:
Put the projectors \(\mathscr {P}:\mathscr {X} \longrightarrow \mathscr {X}\) and \(\mathscr {Q}:\mathscr {Y}\longrightarrow \mathscr {Y}\) defined by:
It follows that \(\mathrm {Ker} \mathscr {L}=\mathbb {R}^{8} \), and:
is clearly closed in \(\mathscr {X}\). In addition, \(\mathrm {Index} \mathscr {L}=0\), since \(\mathrm {\mathrm {dim}}\mathrm {Ker}\mathscr {L}= \mathrm {\mathrm {codim}}\mathrm {Im}\mathscr {L} = 8\). Accordingly, \(\mathscr {L}\) is a Fredholm mapping of index zero.
Now, after some calculations, we can see that:
is invertible, and its inverse (to \(\mathscr {O}_P\)), \(\mathscr {C}_{P}: \mathrm {Im} \mathscr {L}\longrightarrow Dom \mathscr {L} \cap \mathrm {Ker} \mathscr {P}\), is given by:
Thus, \(\mathscr {Q}\mathscr {N}:\mathscr {X}\longrightarrow \mathscr {X}\) is:
The mapping \(\mathscr {C}_{P}(\mathscr {I}-\mathscr {Q})\mathscr {N}:\mathscr {X}\longrightarrow \mathscr {X}\) has the form:
Moreover, this mapping \(\mathscr {Q}\mathscr {N}\) and \(\mathscr {C}_{P}(\mathscr {I}-\mathscr {Q})\mathscr {N}\) are continuous by the Lebesgue’s convergence theorem, Chen (2005). If \(\varOmega _0 \subset \mathscr {X}\) is a open-bounded set by Arzela–Ascoli’s theorem, \(\overline{\mathscr {C}_{P}(\mathscr {I}-\mathscr {Q})\mathscr {N}(\overline{\varOmega _0})}\) are relatively compact. Moreover, \(\mathscr {Q}\mathscr {N}(\overline{\varOmega _0})\) is bounded. Thus, \(\mathscr {N}\) is \(\mathscr {L}-\)compact under \(\overline{\varOmega _0}\) with any open-bounded set \(\varOmega _0 \subset \mathscr {X}\). The isomorphism \(\mathscr {J}\) from \(\mathrm {Im} \mathscr {Q}\) under \(\mathrm {Ker} \mathscr {L}\) can be the identity mapping, since that \(\mathrm {Im} \mathscr {Q} = \mathrm {Ker} \mathscr {L}\).
To verify the hypothesis of Theorem 1, an open-bounded subset \(\varOmega _0\) must be constructed in the Banach \(\mathscr {X}\) space. Thus, for the operator equation, \(\mathscr {L}\mathfrak {u}=\lambda \mathscr {N}\mathfrak {u}\) with \(\lambda \in (0,1)\), we have:
Suppose that \(\mathfrak {u}(t)=(\mathfrak {u_1}(t),\ldots ,\mathfrak {u_8}(t))^{T}\in \mathscr {X}\) is any solution of system (9) for a certain \(\lambda \in (0,1)\). Integrating (9) over the interval [0, T] leads to:
It follows from (10)–(17) that:
We are considering the solution \(\mathfrak {u}(t)\in \mathscr {X}\) in the interval [0, T]. Then, there exist \(\xi _{i}, \eta _{i}\in [0,T]\) for \(i=1,\ldots ,8\), such that:
To obtain upper and lower bounds of \(\mathfrak {u_1}(t),\ldots , \mathfrak {u_8}(t)\), we multiply by \(e^{\mathfrak {u_i}(t)}\) the ith equation of system (9) (for \(i=1,\ldots ,8)\) and integrate over [0, T]. Thus:
Then, from Eq. (26a), we have that:
Therefore:
Now, from Eq. (26b), there is \(\theta _1^{*}\in [0,T]\), such that:
Thus:
From (26c), there exists \(\theta _{2}^{*}\in [0,T]\), such that:
In this way:
Next, there exists a \(\theta _{3}^{*}\in [0,T]\) in the Eq. (26d), such that:
Then:
Now, from Eq. (26e), there exists \(\theta _{1}^{\circ }\in [0,T]\), such that:
Therefore, one gets:
In Eq. (26f), there exists \(\theta _{2}^{\circ }\in [0,T]\), such that:
It follows that:
From Eq. (26g), one gets:
It deduces that:
And finally from Eq. (26h), there exists \(\theta _{4}^{*}\in [0,T]\), such that:
It follows that:
Now, from (26a)–(26d), it can been seen:
and from (26e) to (26f), it follows that:
Adding (35) and (36), it follows that:
Thus, we can derive:
Therefore, for \(t\in [0,T]\), from (18) to (25),(27) to (34), and (37), one gets:
From (38), it deduces that:
Clearly, each \(\mathscr {R}_{i}\) \((i=1,\ldots ,8)\) does not depend on \(\lambda \). Let \(0\le \mu _0\le 1,\) and the algebraic equations:
where \((\mathfrak {u_1},\ldots ,\mathfrak {u_8})^{T}\) in \(\mathbb {R}^{8}\). Now, from (40a) to (40h), we can deduce:
Thus, summing from (41a) to (41c), one gets that:
Moreover, from Eq. (40a), it follows that:
and using (42), we obtain:
Choosing,
we can see that:
Now, from (41d) to (41f), one gets that:
Using Eqs. (40e) and (43), we obtain that:
Therefore, \(e^{u_i}<2M_0+C_2:=M_M\), for \(i=1,\ldots ,8.\) Next, Eq. (40a) implies that:
and using (48), we derive that:
Thus:
Choosing
we can see that (49) is holds. Now, from (41c) and (41b), it follows that:
and
that is:
and
We can see that \(e^{\mathfrak {u_6}}>\dfrac{m_1\eta }{(\delta +\eta )}:=m_6,\) holds with (54). Finally, from (40g) and (40h), one gets that:
If we select \(\mathscr {R}=\max \limits _{i\in \{1,\ldots ,8\}}\left\{ |\ln (M_M)|,|\ln (m_{i})| \right\} \), then:
Thus, with \(\mathscr {R}_{T}=\sum \limits _{i=0}^{8}\mathscr {R}_{i}+\mathscr {R}_{0},\) we define:
Therefore, \( \mathscr {L}\mathfrak {u} \ne \lambda \mathscr {N}\mathfrak {u},\) with \(\lambda \in (0,1)\), and \(\mathfrak {u}\in \partial \varOmega _0 \cap Dom\mathscr {L}\), and thus, \(\varOmega _0\) holds condition (1) of Theorem 1.
For \(\mathfrak {u}=(\mathfrak {u_1},\ldots ,\mathfrak {u_8})^{T}\in \partial \varOmega _0 \cap \mathrm {Ker} \mathscr {L}=\partial \varOmega _0 \cap \mathbb {R}^{8}\), \(\mathfrak {u}\) is a constant vector in \(\mathbb {R}^{8}\) with \(\Vert \mathfrak {u}\Vert =\mathscr {R}_{T}\). If \(\mathscr {Q}\mathscr {N}\mathfrak {u}=0\), then \((\mathfrak {u_1},\ldots ,\mathfrak {u_8})^{T}\) is a constant solution of system (40a–40h) with \(\mu _{0}=1\). From (57), we have that \(\Vert \mathfrak {u}\Vert <\mathscr {R}_{0}\) which is contradictory to \(\Vert \mathfrak {u}\Vert =\mathscr {R}_{T}\). It follows that for each \(\mathfrak {u}\in \partial \varOmega _0 \cap \mathrm {Ker} \mathscr {L}\), \(\mathscr {Q}\mathscr {N}\mathfrak {u}\ne 0\). This shows that condition (2) of Theorem 1 is satisfied.
To verify the condition (3) of the Continuation Theorem, we define
\(\phi : Dom\mathscr {L}\cap \mathrm {Ker} \mathscr {L}\times [0,1]\longrightarrow \mathscr {X}\) by:
where \(\mu _{0} \in [0,1]\). If \(\mathfrak {u}=(\mathfrak {u_1},\ldots ,\mathfrak {u_8})^{T}\in \partial \varOmega _0 \cap \mathrm {Ker} \mathscr {L}=\partial \varOmega _0 \cap \mathbb {R}^{8}\), then \(\mathfrak {u}\) is constant, such that \(\mathfrak {u}\in \mathbb {R}^{8}\), \(\Vert \mathfrak {u}\Vert =\mathscr {R}_{T}\) and \(\phi (\mathfrak {u_1},\ldots ,\mathfrak {u_8},\mu _{0})\ne 0\). Therefore:
Using Proposition (1), we obtain that:
Consequently, by Theorem 1, the system (4) has at least one periodic solution in \(Dom \mathscr {L}\cap \overline{\varOmega _0}\). \(\square \)
Remark 2
\(f:=f(e^{\mathfrak {u_3^{*}}},e^{\mathfrak {u_7^{*}}})\) and \(\frac{\partial f}{\partial w_3}:=\frac{\partial f(e^{\mathfrak {u_3^{*}}},e^{\mathfrak {u_7^{*}}})}{\partial w_3},\,\,\frac{\partial f}{\partial w_7}:=\frac{\partial f(e^{\mathfrak {u_3^{*}}},e^{\mathfrak {u_7^{*}}})}{\partial w_7},\) and \(e^{\mathfrak {u_i^{*}}}\) are the solutions of system given in Proposition 1.
5 Global attractivity
Next, we analyze the conditions for which the solution of system (4) is globally attractive.
Definition 1
System (4) is said to be globally attractive if there exists a positive T-periodic solution \((w_1(t),\ldots ,w_8(t))\), such that for any \((y_1(t),\ldots ,y_8(t))\) arbitrary positive solution of system (4), it holds:
and the solution \((w_1(t),\ldots ,w_8(t))\) is called globally attractive, see (Guo and Chen 2011; Yan and Sugie 2020; Zhu and Wang 2011).
Without loss of generality, we will use the following space:
with norm
From now on, we use the notation:
with \(K=\max \limits _{t\in [0,T]}\left\{ \dfrac{\partial f(w_3,w_7)}{\partial w_3}, \dfrac{\partial f(w_3,w_7)}{\partial w_7}\right\} ,\) and we also assume that:
-
\(\left[ \mathrm{H}_6\right] \) \(I_1=\dfrac{p+\sqrt{ p^2-4q\frac{\mu }{d_1}}}{2q}>0, \)
-
\(\left[ \mathrm{H}_7\right] \) \(I_2=\dfrac{p-\sqrt{ p^2-4q\frac{\mu }{d_1}}}{2q}, \) where \(p^2-4q\frac{\mu }{d_1}>0.\)
Theorem 3
If conditions \(\left[ \mathrm{H}_1\right] -\left[ \mathrm{H}_7\right] \) hold, then there exists a positive constant \(\mathfrak {B} \in [I_1,I_2],\) such that for all w(t) solution of (4), it holds:
Proof
If (59) is not true, then there exists \(t_0>0,\) such that:
Now, for \(0\le t<t_0\) and from (4), we have that:
Thus:
Therefore:
Accordingly:
which contradicts \(\mathrm{H}_7.\) Thus, the proof is completed. \(\square \)
Now, let \((w_1(t),\ldots ,w_8(t))\) and \((y_1(t),\ldots ,y_8(t))\) be two positive T-periodic solutions of system (4), and define:
then, system (4) is written as:
Thus, the solution \((w_1(t),\ldots ,w_8(t))\) is globally attractive to the system (4) if and only if the zero solution of system (62) is globally attractive.
Theorem 4
If \(\left[ \mathrm{H}_1\right] {-}\left[ \mathrm{H}_7\right] \) hold, then the solution \((0_1(t),\ldots ,0_8(t))\) to the system (62) is global attractive.
Proof
From Theorem 3, we can see that there exists a non-negative constant \(\rho \), such that:
Thus, for \(\epsilon >0\) small enough, there is a \(t_1>0\), such that:
From system (62), when \(t\ge t_1\), one gets:
Solving the system (65), one obtains that:
When \(t \longrightarrow \infty \) and \( \epsilon \longrightarrow 0,\) it follows that:
Now, if \(\rho \ne 0,\) then:
which means that:
On the other hand, from Theorem 3 for all \(t>0\), we have that:
Thus:
Therefore, \(\rho =0.\) This implies that \( \limsup \nolimits _{t \longrightarrow \infty }||z(t)||=0.\) Then, \( \lim \nolimits _{t \longrightarrow \infty }|w_i(t)-y_i(t)|=0,\) for \(i=1,\ldots ,8.\) This completes the proof of Theorem 4, which implies that system (4) is globally attractive. \(\square \)
6 Numerical simulations
To exemplify the behavior of the analytical solution of system (4) and corroborate the previous theoretical results, we perform numerical simulations using the Matlab routine ode45 for solving systems of differential equations. We use a particular set of biologically feasible parameter values for the RSV transmission for children. These values are given in Table 1, and were taken from Hogan et al. (2016b). Parameters \(\mu \), \(b_{0}\), \(b_{1}\), \(\phi \) correspond, respectively, to the birth (death) rate, average of the transmission rate, and amplitude of the seasonal fluctuation, and \(0\le \varphi \le 1 \) is the phase angle normalized. The parameters \(\gamma \) and \(\nu \) are the rates of immunity lost and recover of the transmission of virus RSV, respectively.
We use a canonical periodic positive solution [see Coleman et al. (1979)] to compare with solutions of the system (3), and thus corroborate the previous theoretical results obtained in the previous sections. We use two different initial conditions to generate two solutions that approach to the aforementioned canonical periodic solution. The initial conditions for the canonical and the other two solutions are presented in Table 2.
In Fig. 1, it can be observed three numerical solutions of the system (4), where the canonical periodic positive solution is the reference to compare the attractive behavior that it produces on the two other positive solutions with different initial conditions. Moreover, all the numerical solutions are positive and have a periodic behavior (using Definition 1). This numerical result has a good agreement with our previous theoretical results. Furthermore, in Fig. 1, it can be observed the numerical solution of the proportion of susceptible children group 1 (\(S_1\)), exposed children group 1 (\(E_1\)), infected children group 1 (\(I_1\)), recovered children group 1 (\(R_1\)), susceptible children group 2 (\(S_2\)), exposed children group 2 (\(E_2\)), infected children group 2 (\(I_2\)), and recovered children group 2 (\(R_2\)), for the system (4) using parameter values corresponding to Australia. As it can be seen, all the numerical solutions are positive and have a periodic behavior, and these results have a good agreement with our theoretical results. From a biological point of view, the two age groups can be applied to different diseases, where the individuals in each age group have different characteristics in regard to the disease.
Globally attractive behaviors of numerical solutions of system (4), using parameter values related to RSV in Australia. The canonical solution (blue) and two different solutions (red and black) corresponding to two initial conditions (color figure online)
7 Conclusions
In this article, we studied the dynamics of a proposed age-structured population mathematical model driven by a seasonal forcing function. We proposed a generalized force of infection function to study different potential epidemic outcomes. We established sufficient criteria for the existence of positive periodic solutions in an age-structured ordinary differential equation SEIR model with a force of infection \(\beta _i(t)f(I_1(t),I_2(t)).\) We defined a convenient norm function, to provide sufficient conditions that assure the global attractivity of the positive periodic solution.
The proposed model includes two age groups that can be used for different diseases. We included two age groups, since there are many diseases that can be described using two age groups. This situation happens due to different conditions in each age group that would be unrealistic to average them to use only one age group. For instance, critical infections are usually presented in children, but not in adults. Thus, the infection outcomes are different for these two age groups.
We presented numerical simulations to illustrate and corroborate the theoretical results obtained in this research work. In particular, we used as an example for the generalized force of infection the following form \(f(I_1(t),I_2(t))=\beta (t)\left( I_1(t)+ I_2(t)\right) \), where \(\beta (t)\) is a continuous T-periodic function. In regard to specific parameter values, we used reported RSV data from Australia, and performed numerical simulations where it can be seen the periodic behavior of the solution. Future directions of research include the introduction of noise in the harmonic seasonal forcing function to take into account uncertainty. In addition, modeling control policies such as treatment or vaccines are interesting topics and much more under climate variability. This type of studies can provide efficient public policies to reduce the prevalence of different diseases around the globe.
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Acknowledgements
The authors are grateful to the reviewers for their careful reading of this manuscript and their useful comments to improve the content of this paper. The first author gratefully acknowledges the partial funding from Universidad de Córdoba, Colombia, for this research.
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Arenas, A.J., González-Parra, G. & De La Espriella, N. Nonlinear dynamics of a new seasonal epidemiological model with age-structure and nonlinear incidence rate. Comp. Appl. Math. 40, 46 (2021). https://doi.org/10.1007/s40314-021-01430-9
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DOI: https://doi.org/10.1007/s40314-021-01430-9
Keywords
- Seasonal epidemic model
- Age-structured
- Mathematical modeling
- Global attraction
- Positive periodic solution