Forward-backward and period doubling bifurcations in a discrete epidemic model with vaccination and limited medical resources

Abstract

A discrete epidemic model with vaccination and limited medical resources is proposed to understand its underlying dynamics. The model induces a nonsmooth two dimensional map that exhibits a surprising array of dynamical behavior including the phenomena of the forward-backward bifurcation and period doubling route to chaos with feasible parameters in an invariant region. We demonstrate, among other things, that the model generates the above described phenomena as the transmission rate or the basic reproduction number of the disease gradually increases provided that the immunization rate is low, the vaccine failure rate is high and the medical resources are limited. Finally, the numerical simulations are provided to illustrate our main results.

Introduction

Mathematical modeling is a fundamental tool to investigate the asymptotic behavior of epidemic models. The classical results (Anderson and May 1991; Diekmann and Heesterbeek 2000; Hethcote 2000) of epidemic models are such that the disease is persistent provided that the basic reproduction number is greater than one. Otherwise, the disease dies out. Moreover, the portion of infected population depends continuously on the parameters. On the other hand, the backward bifurcation has been found by more and more researchers in their papers (Feng et al. 2000; Brauer 2004, 2011; Castillo-Chavez et al. 1989a, b; Dushoff et al. 1998; Sharomi et al. 2007; Elbasha and Gumel 2006; Kribs-Zaleta and Valesco-Hernández 2000; Gumel and Song 2008; Hadeler and Castillo-Chavez 1995; Reluga and Medlock 2007; Gómez-Acevedo and Li 2005; Julien et al. 2003). The typical features of such bifurcations are that, in the early stages of disease spreading when the infected population is small, the models exhibit a discontinuous outbreak transition from the disease free state to the high prevalence state, as the basic reproduction number $R_0$ crosses an outbreak threshold ${\overline{R}}_o$, which is equal to one. Furthermore, in the reverse scenario, when the outbreak of the disease occurs, they exhibit discontinuous eradication transition from the high prevalence state to the disease free state, as the basic reproduction number $R_0$ reaches an eradication threshold $R_e(<{\overline{R}}_o)$ from the above. The above described phenomena are called a backward bifurcation. The difference in $R_e$ and ${\overline{R}}_o$ highlights that once the outbreak of the disease occurs, driving down the basic reproduction number to one is not enough to eradicate the diseases. In fact, we need to drive the rate further down to $R_e$ for the epidemic to die out, which requires more effort and results in greater economic costs. We also remark that the backward bifurcation has been classified into three types (Song et al. 2013). Another similar type of phenomena is termed the forward-backward bifurcation (Wang 2006; Rodriguez et al. 2018). One difference between these two types of bifurcations lies in that for the latter its corresponding outbreak threshold ${\overline{R}}_o$ is greater than one. That is to say that the corresponding model exhibits discontinuous jump from the low prevalence state to the high prevalence state. Furthermore, the corresponding eradication threshold $R_e(<{\overline{R}}_o)$ could be either no greater than one or greater than one. This implies that the model exhibits either the eradication transition from the high prevalence state to the disease free state or the low prevalence state, depending on the value of $R_e$. It is also worth mentioning that seasonal influenza models [see e.g., Roberts et al. (2019), Huang et al. (2022a), Yakubu and Franke (2006) and the works cited therein] are capable of generating complicated and unpredictable dynamics such as period states, bistable periodic states, or chaotic attractors.

The treatment and vaccination are important methods (Brauer 2011; Arino et al. 2008; Feng and Thieme 1995; Hyman and Li 1998; Wu and Feng 2000) to prevent the spread of the infectious diseases. In classical epidemic models, the treatment rate of infectives is assumed to be proportional to the number of infectives. However, hospitals can be overwhelmed by high volumes of infected patients. Indeed, surges in Covid-19 cases have stressed hospital systems in many regions of the world. To cope with this situation, it is then essential to have limited medical resources placed on disease spreading models. We shall adopt the concept first proposed in Wang (2006) by assuming that the treatment rate is proportional to the percentage of the number of the infectives up to a certain maximal capacity. Moreover, the vaccination is an effective tool to fight against the spread of epidemic diseases, e.g., pertussis, measles, influenza, or covid-pandemic. Therefore, prevention and intervention measures are essential to control and eliminate disease. Inclusion of vaccination in mathematical models also aids in deciding on an optional vaccination strategy (Huang et al. 2022b; Klepac et al. 2011).

In this work, we consider a discrete epidemic spreading model consisting of three states: susceptible (S), infectious (I), and vaccinated (V), for which the changes of states between S and I or S and V take into account the limited medical resources. The continuous $S-(I,V)-S$ models (Kribs-Zaleta and Valesco-Hernández 2000; Gumel and Moghadas 2003; Knipl et al. 2015; Peng et al. 2013, 2016; Lv et al. 2020) with limited medical resources have been extensively studied partly because there are more tractable mathematically. However, a good reason for studying discrete models is that data are collected at discrete times and hence it may be easier to compare data with the output of a discrete model. Moreover, the discrete model even in lower dimension, is capable of generating complicated dynamics. Indeed, our model exhibits both the forward-backward bifurcation and period doubling route to chaos in a feasible invariant region. Specifically, our main results contain the following. First, we are able to obtain a sufficient condition on parameters so that our model is invariant in a region with feasible parameters. Second, the existence and stability/instability of equilibria can be completely characterized. As a result, we are able to obtain two types of forward-backward bifurcations. Third, it is demonstrated numerically that as the transmission rate $\beta$ or the basic reproduction number $R_0$ becomes larger the model exhibits period doubling route to chaos in the invariant region provided that the immunization rate is low, the vaccine failure rate is high and the the medical resources are limited. This, in turn, makes the another valid point that vaccination and medical resources are important tools to combat the highly transmissible diseases. Finally, the numerical simulations are provided to illustrate our main results.

We will conclude the introductory section by mentioning the organization of the paper. The derivation of the model and its invariant region under suitable parameter conditions are presented in Sect. 2. The main results containing, among other things, the forward-backward bifurcation and period doubling route to chaos are given in Sect. 3. In Sect. 4, we numerically illustrated our main results. Some concluding remarks are given in Sect. 5.

Model and its invariance

An $S-(I,V)-S$ model of N individuals with treatment and vaccination is considered, based on the microscopic Markov-chain approximation (Wang et al. 2003; Gómez et al. 2010; Granell et al. 2013). Specifically, every individual i has a certain probability of being in one of the three states, susceptible, infected and vaccinated, at time n, denoted by $S_i(n)$, $I_i(n)$ and $V_i(n)$, respectively. Moreover, it is assumed that $S_i(n)+I_i(n)+V_i(n)=1$ for all time n. The equation can then be simplified as follows.

$$\begin{array}{cc}\hfill I_i(n+1)=& (1-{\zeta }_i)(1-\delta )(1-I_i(n)-V_i(n))+f(I_i(n)),\hfill \end{array}$$ 1a

$$\begin{array}{cc}\hfill V_i(n+1)=& \delta (1-I_i(n)-V_i(n))+(1-\alpha )V_i(n),\hfill \end{array}$$ 1b

where $1\le i\le N$,

$$\begin{array}{c}\hfill {\zeta }_i=\prod _{j\in N_i}(1-\tilde{\beta }{I}_j(n)),\end{array}$$ 1c

and

$$\begin{array}{c}\hfill f(I_i(n))=\left\{\begin{array}{cc}(1-(r_1+r_2))I_i(n),& \frac{{\sum }_{i=1}^N{I}_i(n)}{N}\le I_0,\\ (1-(r_1+r_2))I_0+(1-r_1)(I_i(n)-I_0),& \frac{{\sum }_{i=1}^N{I}_i(n)}{N}>I_0.\end{array}\right)\end{array}$$ 1d

The amount of $I_i(n+1)$ comes from two contributions. The first one is the probability of individual i who is in the susceptible state at time n and becomes infected at time $n+1$. In particular, $1-{\zeta }_i$ is the infection rate for an individual in the susceptible state becoming the infected state, where $\tilde{\beta }$ is the contact rate for an individual in an infected state passing a virus to person in the susceptible state. $N_i$ is the neighborhood of individual i. The second contribution, $f(I_i(n))$, is the probability of individual i in the infected state at time n and remains infected at the next time step. Here $r_1$ and $r_1+r_2$, $r_1$, $r_2>0$, denote, respectively, the natural recovery rate and treatment rate. Moreover, $I_0$ denotes the ratio of the maximum capacity of medical resources versus the total population N. The parameters $\delta$ and $\alpha$ in (1b) are, respectively, the immunization rate and vaccine failure rate. For a homogeneous society, we may assume that everyone has the same number of the neighborhoods, say $|N_i|=k,k\le N-1$. For such a society, we seek the uniform solutions, independent of the index i, of (1a)-(1d). Upon using the first order approximation on $1-{\zeta }_i$, we have that (1a)-(1d) reduced to

$$\begin{array}{cc}\hfill I(n+1)& =(1-{(1-\tilde{\beta }I(n))}^k)(1-\delta )(1-I(n)-V(n))+f(I(n))\hfill \\ \hfill & \approx k\tilde{\beta }(1-\delta )I(n)(1-I(n)-V(n))+f(I(n))\hfill \\ \hfill & =:\beta (1-\delta )I(n)(1-I(n)-V(n))+f(I(n))\hfill \end{array}$$ 2a

$$\begin{array}{cc}& =:{\beta }^{\prime }I(n)(1-I(n)-V(n))+f(I(n))=:g_1(I(n),V(n)),\hfill \\ \hfill V(n+1)& =(1-\alpha )V(n)+\delta (1-I(n)-V(n))=:g_2(I(n),V(n)).\hfill \end{array}$$ 2b

Such discrete model is characterized by a nonsmooth two-dimensional map F of the following form

$$\begin{array}{c}\hfill F(I,V)=({\beta }^{\prime }I(1-I-V)+f(I),(1-\alpha )V+\delta (1-I-V)).\end{array}$$ 3

Note that all the parameters in (2a) and (2b) are assumed to be in between zero and one except that $\beta$, to be termed transmission rate, is allowed to be greater than one.

We next derive conditions for which the following feasible region

$$\begin{array}{c}\hfill △:=\{(I,V):I,V\ge 0\text{and}I+V\le 1\}\end{array}$$ 4

is invariant with respect to the map F defined in (3). For discrete model, unlike its continuous counterpart, finding an effective parameter region for which the region $△$ to be invariant with respect to F is a nontrivial matter. Since $g_1$ and $g_2$ are greater than or equal to zero, it suffices to show that $g_1(I,V)+g_2(I,V)\le 1$ whenever $(I,V)\in △$. To verify the above inequality, we need to only prove that $g(I):=g_1(I,0)+g_2(I,0)=({\beta }^{\prime }I+\delta )(1-I)+f(I)\le 1$ for $0\le I\le 1$. Note that g(I) has the following form.

$$\begin{array}{c}\hfill g(I)=\left\{\begin{array}{cc}-{\beta }^{\prime }{(I-a_1)}^2+C_1({\beta }^{\prime },\delta ),\hfill & 0\le I\le I_0,\\ \hfill & \\ -{\beta }^{\prime }{(I-a_2)}^2+C_2({\beta }^{\prime },\delta ),\hfill & I_0\le I\le 1.\end{array}\right)\end{array}$$ 5

Here $a_1=\frac{1-r_1-r_2+{\beta }^{\prime }-\delta }{2{\beta }^{\prime }}$, $C_1({\beta }^{\prime },\delta )=\delta +\frac{{(1-r_1-r_2+{\beta }^{\prime }-\delta )}^2}{4{\beta }^{\prime }}$, $a_2=\frac{1-r_1+{\beta }^{\prime }-\delta }{2{\beta }^{\prime }}$ and $C_2({\beta }^{\prime },\delta )=\delta -r_2{I}_0+\frac{{(1-r_1+{\beta }^{\prime }-\delta )}^2}{4{\beta }^{\prime }}$. To show that $g(I)\le 1$ for $0\le I\le 1$, we first need the following lemma.

Lemma 2.1

If $a_1\notin [0,I_0]$ and $a_2\notin [I_0,1]$, then $g(I)\le 1$ for $0\le I\le 1$.

Proof

Suppose $a_1$ and $a_2$ are as defined. Hence, $a_2\ge a_1$. Then the maximum of g(I) for $0\le I\le 1$ occurs at the endpoints 0 or 1. Some direct calculations would yield that max$\{g(0),g(1)\}\le 1$. $\square$

Note that $a_1\in [0,I_0]$ is equivalent to

$$\begin{array}{c}\hfill -{\beta }^{\prime }+\delta \le 1-r_1-r_2,\end{array}$$ 6a

and

$$\begin{array}{c}\hfill -(1-2I_0){\beta }^{\prime }+\delta \ge 1-r_1-r_2.\end{array}$$ 6b

Similarly, $a_2\in [I_0,1]$ is equivalent to

$$\begin{array}{c}\hfill {\beta }^{\prime }+\delta \ge 1-r_1,\end{array}$$ 6c

and

$$\begin{array}{c}\hfill -(1-2I_0){\beta }^{\prime }+\delta \le 1-r_1.\end{array}$$ 6d

Denote by equalities in (6a)-(6d), representing the straight lines, ${\ell }_1$, ${\ell }_2$, ${\ell }_3$ and ${\ell }_4$, respectively, with $I_0$, $r_1$ and $r_2$ arbitrarily fixed in the ${\beta }^{\prime }-\delta$ plane.

$$\begin{array}{c}\hfill \text{Let}{\mathrm{\Gamma }}_1(\text{resp.,}{\mathrm{\Gamma }}_2)\text{denote the region satisfying}(6a)\text{and}(6b)(\text{resp.,}(6c)\text{and}(6d))\end{array}$$ 7

Lemma 2.1 amounts to saying that, for those parameter pairs $({\beta }^{\prime },\delta )$ in ${\mathrm{\Gamma }}_1^c\cap {\mathrm{\Gamma }}_2^c$, the region $△$ is invariant with respect to the map F, as defined in (3). Here ${\mathrm{\Gamma }}^c$ denotes the complement of $\mathrm{\Gamma }$. Our goal next is to find a sufficient condition on parameters so that if the parameter pairs $({\beta }^{\prime },\delta )$ in ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2$, then the corresponding model is invariant on $△$.

Theorem 2.1

Let $0\le \alpha ,\delta ,r_1+r_2:=r,I_0\le 1,r_1,r_2\ge 0\text{and}{\beta }^{\prime }\ge 0$. The region $△$ is invariant with respect to map F provided that

• (i)$({\beta }^{\prime },\delta )\in {\mathrm{\Gamma }}_1^c\cap {\mathrm{\Gamma }}_2^c$, where ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are defined in (7), or
• (ii)
  If $({\beta }^{\prime },\delta )\in {\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2$ and $({\beta }^{\prime },\delta )$ satisfies the following inequality

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\beta }^{\prime }& \le min\{{(\sqrt{1-\delta }+\sqrt{r})}^2,{(\sqrt{1-\delta +r_2{I}_0}+\sqrt{r_1+r_2{I}_0})}^2\}\hfill \\ \hfill & =:{\overline{{\beta }^{\prime }}}_u,\hfill \end{array}\end{array}$$ 8

  or
• (iii)
  $$\begin{array}{c}\hfill \beta \le min\{{(1+\sqrt{r})}^2,{(\sqrt{1+r_2{I}_0}+\sqrt{r_1+r_2{I}_0})}^2\}=:{\overline{\beta }}_u.\end{array}$$ 9

Consequently, (8) or (9) is a sufficient condition for $△$ to be invariant.

Proof

It follows from Lemma 2.1 that if $({\beta }^{\prime },\delta )\in {\mathrm{\Gamma }}_1^c\cap {\mathrm{\Gamma }}_2^c$, then the region $△$ is invariant. To prove the assertion in (ii), we first note that $C_1({\beta }^{\prime },\delta )\le 1$ and $C_2({\beta }^{\prime },\delta )\le 1$, respectively, are equivalent to

$$\begin{array}{c}\hfill 1+r-2\sqrt{(1-\delta )r}\le {\beta }^{\prime }+\delta \le 1+r+2\sqrt{(1-\delta )r}\end{array}$$ 10a

and

$$\begin{array}{c}\hfill 1+r_1+2r_2{I}_0-2\sqrt{(1-\delta +r_2{I}_0)(r_1+r_2{I}_0)}:=A-2\sqrt{B}\le {\beta }^{\prime }+\delta \le A+2\sqrt{B}.\end{array}$$ 10b

To get (8), it will be helpful if we are able to visualize the region represented by the inequalities $C_1({\beta }^{\prime },\delta )\le 1$ and $C_2({\beta }^{\prime },\delta )\le 1$. To this end, we shall momentarily treat ${\beta }^{\prime }$ and $\delta$ as two independent variables. So, the equation $C_1({\beta }^{\prime },\delta )=1$ represents a parabola for which its axis is ${\beta }^{\prime }+\delta =1$. The equality of the right (resp., left) hand side of the inequalities in (10a) is upper (resp., lower) part of the parabola $C_1({\beta }^{\prime },\delta )=1$, i.e., the one portion of the parabola above (resp., below) its axis is ${\beta }^{\prime }+\delta =1$, see Fig. 1. Assume that $({\beta }^{\prime },\delta )\in {\mathrm{\Gamma }}_1$. The inequality in (8) is equivalent to both right hand side of the inequalities in (10a) and (10b) being satisfied. Let $A_1$ (resp., $B_1$) be the intersection of the equality in (6a) (resp., (6b)) and the line $\delta =1$ (resp., the $\delta$-axis). Using the assumption that the inequality (8) is satisfied, we conclude that inequalities in (10a) must be satisfied. And, so, $C_1({\beta }^{\prime },\delta )\le 1$. Similarly, if $({\beta }^{\prime },\delta )\in {\mathrm{\Gamma }}_2$, and the inequality in (8) is satisfied, then inequalities in (10b) must be satisfied. And so $C_2({\beta }^{\prime },\delta )\le 1$. Thus, if statement in (ii) is fulfilled, then the region $△$ is invariant.

Fig. 1.

The light green (resp., light pink) region is ${\mathrm{\Gamma }}_1^c\cap {\mathrm{\Gamma }}_2^c$ (resp., $({\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2)\cap {\mathrm{\Gamma }}_3$). Here ${\mathrm{\Gamma }}_3$ is the region determined by the inequality in (8) and $(r_1,r_2,I_0)=(0.6,0.3,0.5)$ (colour figure online)

To proof the assertion in (iii), we first verify that $(1-\delta ){(1+\sqrt{r})}^2\le {(\sqrt{1-\delta }+\sqrt{r})}^2$ and $(1-\delta ){(\sqrt{1+r_2{I}_0}+\sqrt{r_1+r_2{I}_0})}^2\le {(\sqrt{1-\delta +r_2{I}_0}+\sqrt{r_1+r_2{I}_0})}^2$. It then follows from (8) that (9) holds as claimed. We have just completed the proof of the theorem. $\square$

It should be remarked that the sufficient condition (9) is independent of vaccination parameters $\alpha$ and $\delta$.

Main results

The dynamics of model (2a) and (2b) is to be investigated in this section. We begin with the existence of the equilibria of (2a) and (2b). The equation that the I-coordinates of the equilibria of (2a) and (2b) satisfy is to be expressed as a function of I and to be displayed in the $\beta -I$ plane. The results are summarized in Proposition 3.1.

Proposition 3.1

• (i)The disease free equilibrium $(I,V)=(0,\frac{\delta }{\alpha +\delta })$ exists for all feasible parameters.
• (ii)
  For $0<I\le I_0$, let the endemic equilibrium of (2a) and (2b) be (I, V), then

  $$\begin{array}{c}\hfill \beta =\frac{{\beta }_c}{1-I}:=h_1(I),\text{where}{\beta }_c=\frac{(\alpha +\delta )r}{\alpha (1-\delta )}.\end{array}$$ 11a
• (iii)
  For $I\ge I_0$, let the endemic equilibrium of (2a) and (2b) be (I, V), then

  $$\begin{array}{c}\hfill \beta =(\frac{\alpha +\delta }{\alpha (1-\delta )})\frac{r_{1}I+r_2{I}_0}{I(1-I)}=:h_2(I).\end{array}$$ 11b

  Moreover, let

  $$\begin{array}{c}\hfill I_e=\frac{r_2{I}_0}{r_2{I}_0+\sqrt{r_2^2{I}_0^2+r_1{r}_2{I}_0}}.\end{array}$$ 11c

  Then

  $$\begin{array}{c}\hfill {\beta }_e:=h_2(I_e)=\frac{\alpha +\delta }{\alpha (1-\delta )}\overline{r},\end{array}$$ 11d

  where

  $$\begin{array}{c}\hfill \overline{r}={(\sqrt{r_1+r_2{I}_0}+\sqrt{r_2{I}_0})}^2,\end{array}$$ 11e

  and

  $$\begin{array}{c}\hfill \frac{d\beta }{\mathit{dI}}\ge 0\text{(resp.,}\le 0)\text{according as}I\ge I_e\text{(resp.,}\le I_e)\text{on}I\in [0,1].\end{array}$$ 11f
• (iv)$I_e\ge I_0$ according as ${\displaystyle \frac{r_2}{r_1+2r_2}\ge I_0}$.
• (v)${\beta }_e\ge {\beta }_c$, or equivalently, $\overline{r}\ge r$, according as ${\displaystyle I_0\ge \frac{r_2}{4(r_1+r_2)}}$.

Proof

We skip the proof of the first two assertions of the proposition. To see the assertion in (iii), we first note that I satisfies the following algebraic equation $\frac{\alpha \beta }{\alpha +\delta }{I}^2+(r_1-\frac{\alpha \beta }{\alpha +\delta })I+r_2{I}_0=0$, or equivalently, $\beta =h_2(I)$. Some direct calculation would yield that the critical points satisfy the following equation

$$\begin{array}{c}\hfill r_1{I}^2+2r_2{I}_{0}I-r_2{I}_0=0.\end{array}$$ 12

The only positive critical point of h(I) occurs at $I_e$, which is in between zero and one. To see ${\beta }_e=\frac{\alpha +\delta }{\alpha (1-\delta )}\overline{r}$, it suffices to show that $\frac{r_1{I}_e+r_2{I}_0}{I_e(1-I_e)}=\overline{r}$. To this end, we first note, via (12), that $I_e-{(I_e)}^2=(\frac{2r_2{I}_0}{r_1}+1)I_e-\frac{r_2}{r_1}{I}_0$. Consequently,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{r_1{I}_e+r_2{I}_0}{I_e(1-I_e)}& =\frac{C}{(\frac{2r_2{I}_0}{r_1}+1)(-\frac{r_2}{r_1}{I}_0+\frac{C}{r_1})-\frac{r_2}{r_1}{I}_0}\hfill \\ \hfill & =\frac{r_1^{2}C}{(2r_2{I}_0+r_1)(-r_2{I}_0+C)-(C^2-r_2^2{I}_0^2)}\hfill \\ \hfill & =\frac{r_1^{2}C}{-2C^2+r_{1}C+2r_2{I}_{0}C}=\frac{r_1^2}{r_1+2r_2{I}_0-2C}\hfill \\ \hfill & =\frac{r_1^2}{{(\sqrt{r_1+r_2{I}_0}-\sqrt{r_2{I}_0})}^2}=\overline{r},\hfill \end{array}\end{array}$$

where $C=\sqrt{r_2^2{I}_0^2+r_1{r}_2{I}_0}$. The last two assertions of the Proposition 3.1 can be easily verified. $\square$

The main points of Proposition 3.1 can be summarized in the following. Let (I, V) be endemic equilibrium of (2a) and (2b). Then the corresponding graphs of $\beta =h(I)$, a continuous function, are displayed in Figs. 2, 3 and 4. Here,

$$\begin{array}{c}\hfill h(I)=\left\{\begin{array}{cc}{h}_1(I),& 0<I\le I_0,\hfill \\ h_2(I),& I_0\le I\le 1.\hfill \end{array}\right)\end{array}$$

In particular, we have the following.

• (i)$I=0$ exists for all $\beta$, see Figs. 2, 3 and 4.
• (ii)For $I_0\ge \frac{r_2}{r_1+2r_2}$, h(I) is increasing in I and $h(0)={\beta }_c$. A generic graph of such h(I) is given in Fig. 2.
• (iii)For $I_0<\frac{r_2}{r_1+2r_2}$, the function $\beta =h(I)$ has a local maximum $h(I_0):={\beta }_o$ and local minimum ${\beta }_e$, respectively. Moreover, ${\beta }_e\ge {\beta }_c$ according as $I_0\ge \frac{r_2}{4(r_1+r_2)}$, a generic graph of such h(I) is displayed in Figs. 3 and 4. In fact, ${\beta }_o$ and ${\beta }_e$ are to be termed the outbreak threshold and the eradication threshold, respectively, with respect to the transmission rate.

It should be noted that the function $\beta =h(I)$ can be equivalently expressed as a function of $R_0=\overline{h}(I)$. Specifically, it follows from (11a) that

$$\begin{array}{c}\hfill R_0=\frac{\alpha (1-\delta )\beta }{(\alpha +\delta )r}\end{array}$$ 13

and, so,

$$\begin{array}{c}\hfill \overline{h}(I)=\left\{\begin{array}{cc}{\overline{h}}_1(I),& 0<I\le I_0,\hfill \\ {\overline{h}}_2(I),& I_0\le I\le 1.\hfill \end{array}\right)\end{array}$$

Here ${\overline{h}}_1(I)=\frac{1}{1-I}$ and ${\overline{h}}_2(I)=\frac{r_{1}I+r_2{I}_0}{rI(1-I)}$. It follows from (13) that the graphs of h(I) and $\overline{h}(I)$ are similar. In fact, the graph of $\overline{h}(I)$ can be obtained from that of h(I) in the horizontal direction by a scale factor of $\frac{\alpha (1-\delta )}{(\alpha +\delta )r}$. The graphs of $\overline{h}(I)$, the counterparts to Figs. 3 and 4, having two turning points of $\overline{h}(I)$ also occur at $I=I_0$ and $I_e$. Likewise, we define the outbreak threshold ${\overline{R}}_o$ and eradication threshold $R_e$ with respect to the basic reproduction number to be ${\overline{h}}_1(I_0)=\frac{1}{1-I_0}$ and ${\overline{h}}_2(I_e)=\frac{\overline{r}}{r}$, respectively. Furthermore, $\beta ={\beta }_c$ is equivalent to $R_0=1$.

Fig. 2.

The graphs of $h_1(I)$ and $h_2(I)$ with $I_0\ge \frac{r_2}{r_1+2r_2}$. Here ${\beta }_c=\frac{\alpha +\delta }{\alpha (1-\delta )}r$ and ${\beta }_o=h(I_0)$

Fig. 3.

The graphs of $h_1(I)$ and $h_2(I)$ with $\frac{r_2}{4(r_1+r_2)}\le I_0<\frac{r_2}{r_1+2r_2}$. Here ${\beta }_e=\frac{\alpha +\delta }{\alpha (1-\delta )}\overline{r}\ge {\beta }_c$ and $I_e>I_0$

Fig. 4.

The graphs of $h_1(I)$ and $h_2(I)$ with $I_0<\frac{r_2}{4(r_1+r_2)}$. Here ${\beta }_e<{\beta }_c$ and $I_e>I_0$

We next investigate the stability of equilibrium (I, V) of model (2a) and (2b). To this end, we see that its Jacobian matrix with respect to model (2a) and (2b) at (I, V) is of the the following form.

$$\begin{array}{c}\hfill J(I,V)=\left[\begin{array}{cc}1+a_I-{\beta }^{\prime }I& -{\beta }^{\prime }I\\ -\delta & 1-\alpha -\delta \end{array}\right].\end{array}$$ 14a

Here

$$\begin{array}{c}\hfill a_I=\left\{\begin{array}{cc}-r+\frac{\alpha {\beta }^{\prime }}{\alpha +\delta }(1-I)=0,& \text{if}I<I_0,\hfill \\ -r_1+\frac{\alpha {\beta }^{\prime }}{\alpha +\delta }(1-I)=r_2\frac{I_0}{I},& \text{if}I>I_0.\hfill \end{array}\right)\end{array}$$ 14b

We have used (11a) and (11b) to justify the equalities in (14b).

Proposition 3.2

• (i)For any equilibrium (I, V) of model (2a) and (2b), both eigenvalues of J(I, V) are real.
• (ii)
  Equilibrium (I, V) is stable provided that

  $$\begin{array}{c}\hfill \text{det}(J\pm I_2)>0,\text{tr}(J-I_2)<0\text{and tr}(J+I_2)>0,\end{array}$$

  where $I_2$ is the $2\times 2$ identity matrix.

Proof

Let $J=J(I,V)$. It is clear that

$$\begin{array}{c}\hfill \text{tr}(J)=(1-\alpha +1+a_I-{\beta }^{\prime }I)-\delta \end{array}$$ 15a

and

$$\begin{array}{c}\hfill \text{det}(J)=(1-\alpha )(1+a_I-{\beta }^{\prime }I)-\delta (1+a_I).\end{array}$$ 15b

Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(\text{tr}(J))}^2-4\text{det}(J)& ={[(1-\alpha )+(1+a_I-{\beta }^{\prime }I)]}^2-2\delta (2+a_I-\alpha -{\beta }^{\prime }I)+{\delta }^2\hfill \\ \hfill & -4(1-\alpha )(1+a_I-{\beta }^{\prime }I)+4\delta (1+a_I)\hfill \\ \hfill & ={(a_I+\alpha -{\beta }^{\prime }I)}^2+2\delta (a_I+\alpha +{\beta }^{\prime }I)+{\delta }^2\ge 0.\hfill \end{array}\end{array}$$

We have just completed the proof of the first assertion of the proposition. The second assertion of the proposition now follows directly from (i). $\square$

We are now in a position to state the stability results of model (2a) and (2b).

Theorem 3.1

• (i)The disease free equilibrium is stable provided that $\beta \le {\beta }_c$. Otherwise, it is unstable.
• (ii)
  Let ${\beta }_u^{\prime }$ be defined as the follow.

  $$\begin{array}{c}\hfill {\beta }_u^{\prime }:=min\{1-\delta +r,1-\delta +\overline{r}\}.\end{array}$$ 16

  Let (I, V), $0<I\le I_0$, be the endemic equilibrium. Then it is stable provided that ${\beta }^{\prime }<{\beta }_u^{\prime }$. Here $\overline{r}$ is defined as in (11e).
• (iii)Let $I>I_0$. Then the corresponding endemic equilibrium, if exists, is unstable whenever $\frac{d\beta }{\mathit{dI}}=\frac{d{h}_2(I)}{\mathit{dI}}<0$. Moreover, its associated endemic equilibrium is stable whenever $\frac{d\beta }{\mathit{dI}}>0$ and ${\beta }^{\prime }\le {\beta }_u^{\prime }$.
• (iv)
  Let ${\beta }_u$ be defined as the follow.

  $$\begin{array}{c}\hfill {\beta }_u:=min\{1+r,1+\overline{r}\}.\end{array}$$ 17

  Let (I, V), $0<I\le I_0$, be the endemic equilibrium. Then it is stable provided that $\beta <{\beta }_u$. Furthermore, if $I>I_0$, then the corresponding endemic equilibrium, if exists, is unstable whenever $\frac{d\beta }{\mathit{dI}}=\frac{d{h}_2(I)}{\mathit{dI}}<0$. Moreover, its associated endemic equilibrium is stable whenever $\frac{d\beta }{\mathit{dI}}>0$ and $\beta \le {\beta }_u$.

Proof

We skip the first assertion of the theorem. To prove (ii), we first note that if I is as assumed, then $\beta >{\beta }_c$, see Figs. 2, 3 and 4. Now $a_I$, as given in (14b), reduces to 0. Let J be the corresponding J(I, V), as defined in (14a). Clearly, $-\text{tr}(J-I_2)$, $\text{tr}(J+I_2)$ and $\text{det}(J-I_2)$ are all greater than zero. Moreover, $\text{det}(J+I_2)=4-2({\beta }^{\prime }I+\alpha +\delta )+\alpha {\beta }^{\prime }I\ge 4-2(1-\alpha )+2\frac{\delta }{\alpha }r+\alpha {\beta }^{\prime }I>0$. We have used (11a) and (16) to justify the last inequality above. The assertion of theorem (ii) now follows from Proposition 3.2 (ii). To prove (iii), we first note, via (11b), that there are possibly two endemic equilibria $(I_{\pm },V_{\pm })$ depending on the range of $I_0$ and ${\beta }^{\prime }$, see Figs. 2, 3 and 4. Specifically, if exists, then $I_{\pm }$ have the following form.

$$\begin{array}{c}\hfill I_{\pm }=\frac{1}{2}-\frac{r_1(\alpha +\delta )}{2\alpha {\beta }^{\prime }}\pm \frac{\alpha +\delta }{2\alpha {\beta }^{\prime }}\sqrt{d},\end{array}$$ 18a

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d& =\frac{{\alpha }^2}{{(\alpha +\delta )}^2}{\beta }^{\prime 2}-\frac{2\alpha (r_1+2r_2{I}_0)}{\alpha +\delta }{\beta }^{\prime }+r_1^2\hfill \\ \hfill & ={\left(\frac{\alpha }{\alpha +\delta }\right)}^2{\left({\beta }^{\prime },-,\frac{(\alpha +\delta )(r_1+2r_2{I}_0)}{\alpha }\right)}^2-{(r_1+2r_2{I}_0)}^2+r_1^2.\hfill \end{array}\end{array}$$ 18b

We next show that $(I_{-},V_{-})$ is unstable. To this end, it suffices to prove that det($J(I_{-},V_{-})-I_2)=:J_{-}<0$. Note, via Proposition 3.1 (iii), that $\sqrt{d}\ge 0$ according as $\beta \ge {\beta }_e$. We have, via (14a) and (14b), that $J_{-}=(\alpha +\delta )r_1-\alpha {\beta }^{\prime }+2\alpha {\beta }^{\prime }{I}_{-}.$ Upon using (18a), we get that $J_{-}=-(\alpha +\delta )\sqrt{d}<0$. Hence, $(I_{-},V_{-})$ is unstable.

To complete the proof of the theorem, it remains to show that $(I_{+},V_{+})$ is stable provided that ${\beta }^{\prime }$ is as assumed. Clearly, tr$(J(I_{+},V_{+})+I_2)>0$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{tr}(J(I_{+},V_{+})-I_2)& =-r_1-\alpha -\delta +\frac{\alpha {\beta }^{\prime }}{\alpha +\delta }-\frac{2\alpha +\delta }{\alpha +\delta }{\beta }^{\prime }{I}_{+}\hfill \\ \hfill & <-\alpha -\delta -\frac{\delta {\beta }^{\prime }}{2(\alpha +\delta )}+\frac{\delta r_1}{2\alpha }<-\alpha -\delta \le 0.\hfill \end{array}\end{array}$$

We have used (18a), (11d), (11e) and (11f) to justify the above inequalities. We also have that

$$\begin{array}{c}\hfill \text{det}(J(I_{+},V_{+})-I_2)=(\alpha +\delta )r_1-\alpha {\beta }^{\prime }+2\alpha {\beta }^{\prime }{I}_{+}=(\alpha +\delta )\sqrt{d}>0.\end{array}$$

Finally,

$$\begin{array}{c}\hfill \begin{array}{cc}& \text{det}(J(I_{+},V_{+})+I_2)\hfill \\ \hfill & =4+2\text{tr}(J(I_{+},V_{+})-I_2)+\text{det}(J(I_{+},V_{+})-I_2)\hfill \\ \hfill & =(-2-\frac{\delta }{\alpha }+\alpha +\delta )\sqrt{d}+4-2\alpha -2\delta +\frac{\delta r_1}{\alpha }-\frac{\delta {\beta }^{\prime }}{\alpha +\delta }\hfill \\ \hfill & >(-2-\frac{\delta }{\alpha }+\alpha +\delta )(\frac{\alpha {\beta }^{\prime }}{\alpha +\delta }-r_1-2r_2{I}_0)+4-2\alpha -2\delta +\frac{\delta r_1}{\alpha }-\frac{\delta {\beta }^{\prime }}{\alpha +\delta }\hfill \\ \hfill & =:\mathrm{\Gamma }(\alpha ,{\beta }^{\prime },\delta ).\hfill \end{array}\end{array}$$

Some direction calculations would yield that

$$\begin{array}{c}\hfill \frac{\partial }{\partial \alpha }\mathrm{\Gamma }(\alpha ,{\beta }^{\prime },\delta )=-\frac{1}{{\alpha }^2}((2-{\beta }^{\prime }+r_1+2r_2{I}_0){\alpha }^2+2\delta (r_1+r_2{I}_0))<0.\end{array}$$ 19

Hence, $\mathrm{\Gamma }(\alpha ,{\beta }^{\prime },\delta )\ge \mathrm{\Gamma }(1,{\beta }^{\prime },\delta )=2-{\beta }^{\prime }+r_1+2r_2{I}_0+(r_1-2)\delta$. We have used the fact that ${\beta }^{\prime }<{\beta }_u^{\prime }\le 2$ to justify the above inequality. It then follows from Proposition 3.1 (v) and the assumption on ${\beta }^{\prime }$ that $\mathrm{\Gamma }(1,{\beta }^{\prime },\delta )\ge \mathrm{\Gamma }(1,1+\overline{r}-\delta ,\delta )$ whenever $I_0\le \frac{r_2}{4r}$ and $\mathrm{\Gamma }(1,{\beta }^{\prime },\delta )\ge \mathrm{\Gamma }(1,1+r-\delta ,\delta )$ whenever $I_0\ge \frac{r_2}{4r}$. We next show that $\frac{1}{1+\overline{r}}$ is an upper bound for $\delta$. To see this, we have, via the assumption on ${\beta }^{\prime }$, that $\frac{(\alpha +\delta )}{\alpha }\overline{r}<1-\delta +\overline{r}$, and so $\delta <\frac{1}{1+\overline{r}}$. Let

$$\begin{array}{c}\hfill \mathrm{\Gamma }(1,1+r-\frac{1}{1+\overline{r}},\frac{1}{1+\overline{r}})=:f_1(r_1,r_2,I_0)\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{\Gamma }(1,1+\overline{r}-\frac{1}{1+\overline{r}},\frac{1}{1+\overline{r}})=:f_2(r_1,r_2,I_0).\end{array}$$

To complete the proof of the theorem, it then suffices to show that $f_1(r_1,r_2,I_0)\ge 0$ (resp., $f_2(r_1,r_2,I_0)\ge 0$) for $I_0\ge \frac{r_2}{4r}$ (resp., $I_0\le \frac{r_2}{4r}$). Now,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_1(r_1,r_2,I_0)& =1+r_1+2r_2{I}_0-\frac{1-r_1}{1+\overline{r}}-r\hfill \\ \hfill & \ge 1+r_1+\frac{r_2^2}{2r}-\frac{1-r_1}{1+r}-r\hfill \\ \hfill & =\frac{-(2r_1+r_2)(r_2^2+(r_1-1)r_2-2r_1)}{2r(1+r)}\hfill \\ \hfill & \ge 0.\hfill \end{array}\end{array}$$

The facts that $0\le r_2\le 1-r_1$ and Proposition 3.1 (v) have been used to justify the last inequalities above. Now,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_2(r_1,r_2,I_0)& =1-\frac{1-r_1}{1+{(\sqrt{r_1+r_2{I}_0}-\sqrt{r_2{I}_0})}^2}-2\sqrt{r_1+r_2{I}_0}\sqrt{r_2{I}_0}\hfill \\ \hfill & =:1-\frac{1-r_1}{1+{(\sqrt{r_1+x}-\sqrt{x})}^2}-2\sqrt{r_1+x}\sqrt{x}\hfill \\ \hfill & =:g_1(x),\hfill \end{array}\end{array}$$

where $x=r_2{I}_0$ and $0\le x\le \frac{r_2^2}{4(r_1+r_2)}:=\widehat{x}$. Furthermore,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_1(x)& =\frac{2\sqrt{r_1+x}}{1+r_1+2x+2\sqrt{x}\sqrt{r_1+x}}(-r_1{x}^{\frac{1}{2}}-2x^{\frac{3}{2}}-2x{(r_1+x)}^{\frac{1}{2}}+{(r_1+x)}^{\frac{1}{2}})\hfill \\ \hfill & =:\frac{2\sqrt{r_1+x}}{1+r_1+2x+2\sqrt{x}\sqrt{r_1+x}}{g}_2(x).\hfill \end{array}\end{array}$$

Then $g_2(x)\ge 0$, if

$$\begin{array}{c}\hfill d(x):=4x^2+(r_1^2+4r_1-1)x-r_1\le 0.\end{array}$$

Indeed, since d(x) is a parabola that intercepts the x-axis at two points $x_{\pm }$ with $(x_{+})(x_{-})=-r_1<0$, we get that $d(x)\le 0$ for $x\in [0,\widehat{x}]$ provided that $d(\frac{{(1-r_1)}^2}{4})\le 0$. A direct calculation yields that

$$\begin{array}{cc}\hfill & d\left(\frac{{(1-r_1)}^2}{4}\right)=\frac{{(1-r_1)}^4}{4}+\frac{(r_1^2+4r_1-1){(1-r_1)}^2}{4}-r_1\hfill \\ \hfill & =\frac{r_1(r_1^3-r_1^2-r_1-1)}{2}\le 0.\hfill \end{array}$$

To show (iv), we first verify that $(1-\delta )(1+r)\le 1-\delta +r$ and $(1-\delta )(1+\overline{r})\le 1-\delta +\overline{r}$. It then follows from (ii) and (iii) that (iv) holds as claimed.

We have completed the proof of the theorem. $\square$

We remark that Theorem 3.1, by replacing $\beta$, ${\beta }_c$, ${\beta }_u$ by $R_0$, 1 and $\frac{\alpha (1-\delta )}{(\alpha +\delta )r}{\beta }_u:=R_u$, respectively, can be easily stated in terms of the basic reproduction number. In Theorem 2.1 (iii), $\beta \le {\overline{\beta }}_u$, or equivalently, $R_0\le \frac{\alpha (1-\delta )}{(\alpha +\delta )r}{\overline{\beta }}_u:={\overline{R}}_u$, is a sufficient condition for $△$ to be invariant.

The model is said to exhibit a type I forward-backward bifurcation provided that it has the following dynamical behavior. The model exhibits a discontinuous outbreak transition from the low prevalence state to the high prevalence state as the basic reproduction number $R_0$ crosses an outbreak threshold ${\overline{R}}_o>1$, an indication of the forward bifurcation occurring at $R_0=1$. Moreover, it exhibits the discontinuous eradication transition from the high prevalence state to the low prevalence state as $R_0$ decreases to an eradication threshold $R_e(>1)$, see Fig. 6. A type II forward-backward bifurcation has a similarly outbreak transition. However, the eradication transition for Type II is from the high prevalence state to the disease free state as $R_0$ decreases to an eradication threshold $R_e(<1)$, see Fig. 7.

Fig. 6.

Let $(\alpha ,\delta ,r_1,r_2)=(0.2,0.2,0.1,0.1)$ and $I_0=0.2$. Then the corresponding $({\beta }_e,{\beta }_o,{\beta }_u,{\overline{\beta }}_u)\approx (0.595,0.625,1.2,1.840)$, or, equivalently, $(R_e,{\overline{R}}_o,R_u,{\overline{R}}_u)\approx (1.190,1.25,2.4,3.679)$ and $\frac{r_2}{4(r_1+r_2)}\le I_0<\frac{r_2}{r_1+2r_2}$. Note that $\beta ={\beta }_c=0.5$ is corresponding to $R_0=1$. The black circles and red stars are the eventual states of I generated by 1,000 iterations of the model (2a) and (2b) with initial values (0.01, 0), (0.99, 0), respectively, for some values of $\beta \in [0,1.15]$. In particular, a red star inside the black circle means that both initial values converge to the same I state (colour figure online)

Fig. 7.

Let $(\alpha ,\delta ,r_1,r_2)=(0.2,0.2,0.1,0.1)$ and $I_0=0.1$. Then the corresponding $({\beta }_e,{\beta }_o,{\beta }_u,{\overline{\beta }}_u)\approx (0.466,0.556,1.186,1.787)$, or, equivalently, $(R_e,{\overline{R}}_o,R_u,{\overline{R}}_u)\approx (0.932,1.111,2.373,3.573)$ and $I_0<\frac{r_2}{4(r_1+r_2)}$. Note that $\beta ={\beta }_c$ is corresponding to $R_0=1$. The black circles and red stars are the eventual states of I generated by 1,000 iterations of the model (2a) and (2b) with initial values (0.01, 0), (0.99, 0), respectively, for some values of $\beta \in [0,1.15]$ In particular, a red star inside the black circle means that both initial values converge to the same I state (colour figure online)

Following Proposition 3.1, we see that the graph of h(I) has three types, see Figs. 2, 3 and 4, depending on the size of $I_0$. Upon using Theorem 3.1, we are able to obtain the following.

Corollary 3.1

• (i)Assume that the medical resources are sufficient, i.e., $I_0\ge \frac{r_2}{r_1+2r_2}$. Then model (2a) and (2b) behaves like a classical epidemic model in which the infected portion of the population depends continuously on parameters.
• (ii)Assume that the medical resources are mildly insufficient, i.e., $\frac{r_2}{4(r_1+r_2)}\le I_0<\frac{r_2}{r_1+2r_2}$. Then model (2a) and (2b) exhibits type I forward-backward bifurcation.
• (iii)Assume that the medical resources are highly insufficient, i.e., $I_0<\frac{r_2}{4(r_1+r_2)}$. Then model (2a) and (2b) exhibits type II forward-backward bifurcation.

Numerical simulations

The purpose of this section is two-fold. First, we provide some numerical simulation results for model (2a) and (2b) to support our main analytical theorem provided in Theorem 3.1. Second, it is numerically demonstrated that the model has period doubling route to chaos.

In Figs. 5, 6 and 7, we run the numerical simulations for model (2a) and (2b) with $(\alpha ,r_1,r_2,\delta )=(0.1,0.1,0.2,0.2)$ and the parameter $I_0$ satisfy the assumptions $I_0\ge \frac{r_2}{r_1+2r_2}$, $\frac{r_2}{4(r_1+r_2)}\le I_0<\frac{r_2}{r_1+2r_2}$, and $I_0<\frac{r_2}{4(r_1+r_2)}$, respectively. We run the numerical simulation of the model with two initial states (0.01, 0), (0.99, 0). The initial state (0.01, 0) (respectively, (0.99, 0)) represents the early stage of the epidemic (respectively, the peak period of the epidemic). The eventual states with two initial states (0.01, 0), (0.99, 0), stopped at $n=1000$, are denoted by the black circle and red star, respectively, which match quite well with our predicted branches, see Figs. 2, 3 and 4. In Figs. 5, 6 and 7, a red star inside the black circle means that both initial states converge to the same I state. As expected, the numerical results in Figs. 5, 6 and 7 are consistent with Corollary 3.1. That is to say that all initial states converge to the increasing branches of the function $\beta =h(I)$ for $\beta <{\beta }_u$. Specifically, in Fig. 5, we see that $I_0=0.45>\frac{r_2}{r_1+2r_2}=\frac{1}{3}$ and the infection portion of the population depends continuously on the parameter $\beta$ as predicated by Corollary 3.1 (i). In Fig. 6, we have that $\frac{r_2}{4(r_1+r_2)}=\frac{1}{8}<I_0=0.2<\frac{1}{3}$. Hence, as $\beta$ races pass ${\beta }_o=0.625$, the infected portion of the population jumps from the lower prevalence state $I_0=0.2$ to the high prevalence state $h_2^{-1}({\beta }_o)=0.4$. Furthermore, at the peak of the epidemic to eradicate the disease we need to drive $\beta$ further down to a number smaller than ${\beta }_e$, which results in the infected portion of the population from the high prevalence state $h_2^{-1}({\beta }_e)\approx 0.290$ to $h_1^{-1}({\beta }_e)\approx 0.160$ as predicted in Corollary 3.1 (ii). It implies the model exhibits type I forward-backward bifurcation whenever $I_0\in [\frac{r_2}{4(r_1+r_2)},\frac{r_2}{r_1+2r_2})$. In Fig. 7, we have that $I_0=0.1<\frac{r_2}{4(r_1+r_2)}=\frac{1}{6}$. The numerical simulation indicates that, in the early stage of the epidemic, the infected portion of the population jumps from $I_0=0.1$ to $h_2^{-1}({\beta }_o)=0.45$ as $\beta$ races pass ${\beta }_o$. To eradicate the disease at the peak of the pandemic, we need to drive $\beta$ down further to a number smaller that ${\beta }_e$. As a result, we observe the infected portion of the population dropping from $h_2^{-1}({\beta }_e)\approx 0.232$ to zero. Hence, a type II forward-backward bifurcation occurs as predicted in Corollary 3.1 (iii). The facts that the numbers in Fig. 7 for three quantities ${\beta }_o-{\beta }_e$, $h_2^{-1}({\beta }_o)-I_0$ and $h_2^{-1}({\beta }_e)-h_1^{-1}({\beta }_e)$ are all greater than those of in Fig. 6 are indications that the phenomena of type II forward-backward bifurcation are even more dire than those of type I. We also note that for the parameters chosen in Figs. 5, 6 and 7 their corresponding $({\beta }_u,{\overline{\beta }}_u)$ are (1.2, 1.969),(1.2, 1.840) and (1.186, 1.787), respectively. All the quantities in Figs. 5, 6 and 7 associated with the transmission rate $\beta$ can be converted, via (13), into the corresponding basic reproduction numbers $R_0$. Specifically, the scale factor $\frac{\alpha (1-\delta )}{(\alpha +\delta )r}=2$, and so $(R_e,{\overline{R}}_o,R_u,{\overline{R}}_u)=2({\beta }_e,{\beta }_o,{\beta }_u,{\overline{\beta }}_u)$.

Fig. 5.

Let $(\alpha ,\delta ,r_1,r_2)=(0.2,0.2,0.1,0.1)$ and $I_0=0.45$. Then the corresponding $({\beta }_o,{\beta }_u,{\overline{\beta }}_u)\approx (0.909,1.2,1.969)$, or, equivalently, $({\overline{R}}_o,R_u,{\overline{R}}_u)\approx (1.818,2.4,3.937)$ and $I_0\ge \frac{r_2}{r_1+2r_2}$. Note that $\beta ={\beta }_c=0.5$ is corresponding to $R_0=1$. The black circles and red stars represent the eventual states of I generated by 1,000 iterations of the model (2a) and (2b) with initial value (0.01, 0), (0.99, 0), respectively, for some values of $\beta \in [0,1.15]$. In particular, a red star inside the black circle means that both initial values converge to the same I state

In Figs. 8, 9, 10, 11 and 12 with the parameters $(\alpha ,\delta ,r_1,r_2)$ set to $(0.5,0.03,0.75,0.15)$, we demonstrate that the period doubling route to chaos can occur regardless of whether medical resources are sufficient. In Figs. 8, 9 and 10, we have that $I_0=0.2>\frac{r_2}{r_1+2r_2}\approx 0.1429$. All three figures demonstrate that the standard period doubling route to chaos can be observed as $\beta$ varies from zero to ${\overline{\beta }}_u\approx 3.6027$ versus eventual states of I, S and V, respectively. Note that ${\overline{\beta }}_u$ is only a sufficient condition on $\beta$ for $△$ being invariant. In Fig. 11, $\frac{r_2}{4(r_1+r_2)}\approx 0.0416<I_0=0.1<0.1429$. The model exhibits both type I forward-backward bifurcation and period doubling route to chaos as $\beta$ varies from zero to ${\overline{\beta }}_u\approx 3.5424$. The zoom in numerical simulation for $\beta \in [1.07,1.1]\supset [{\beta }_e,{\beta }_o]\approx [1.086,1.093]$ is displayed in the upper left corner. We run two initial states for such ranges of $\beta$. Their eventual I states are colored by red circles and blue solid circles. A blue solid circle inside a red circle means that both initial states converge to the same I state. In Fig. 12, $I_0=0.02<\frac{r_2}{4(r_1+r_2)}\approx 0.0416$, we see that model (2a) and (2b) exhibits both type II forward-backward bifurcation and period doubling route to chaos as $\beta$ varies from zero to $\beta ={\overline{\beta }}_u\approx 3.4941$ as predicted in Corollary 3.1 (iii). Figure 13 is to illustrate that if the bifurcation parameters is to be replaced by other variables, say $\alpha$, then similar figures as those provided in Figs. 8, 9, 10, 11 and 12 can still be obtained.

Fig. 8.

Bifurcation diagram of the eventual state of I versus $\beta$. Here $(\alpha ,\delta ,r_1,r_2)=(0.5,0.03,0.75,0.15)$, $I_0=0.2$ and the corresponding $({\beta }_u,{\overline{\beta }}_u)\approx (1.9,3.6027)$ and the corresponding ${\beta }_2\approx 2.93$. Here ${\beta }_2$ is so defined that the smaller eigenvalue of J(I, V), defined in (14a) and (14b), is $-1$ at $\beta ={\beta }_2$. For such choice of the parameters, the model exhibits the period doubling route to chaos as $\beta$ varies from 0 to ${\overline{\beta }}_u$

Fig. 9.

Bifurcation diagram of the eventual state of S versus $\beta$. Here $(\alpha ,\delta ,r_1,r_2)=(0.5,0.03,0.75,0.15)$, $I_0=0.2$ are the same as those in Fig. 8

Fig. 10.

Bifurcation diagram of the eventual state of V versus $\beta$. Here $(\alpha ,\delta ,r_1,r_2)=(0.5,0.03,0.75,0.15)$, $I_0=0.2$ are the same as those in Fig. 8

Fig. 11.

Bifurcation diagram of the eventual state of I versus $\beta$. Here $(\alpha ,\delta ,r_1,r_2)=(0.5,0.03,0.75,0.15)$, $I_0=0.1$ and the corresponding $({\beta }_u,{\overline{\beta }}_u)\approx (1.9,3.5424)$ and the corresponding ${\beta }_2\approx 2.88$, as similarly defined in the caption of Fig. 8. For such choice of the parameters, the model exhibits both Type I forward-backward bifurcation and period doubling route to chaos as $\beta$ varies from 0 to ${\overline{\beta }}_u$

Fig. 12.

Bifurcation diagram of the eventual state of I versus $\beta$. Here $(\alpha ,\delta ,r_1,r_2)=(0.5,0.03,0.75,0.15)$, $I_0=0.02$ and the corresponding $({\beta }_u,{\overline{\beta }}_u)\approx (1.8511,3.4941)$ and the corresponding ${\beta }_2\approx 2.85$, as similarly defined in the caption of Fig. 8. For such choice of the parameters, the model exhibits both Type II forward-backward bifurcation and period doubling route to chaos as $\beta$ varies from 0 to ${\overline{\beta }}_u$

Fig. 13.

Bifurcation diagram of the state of I variable versus $\alpha$. Here $(\beta ,\delta ,r_1,r_2)=(3.5,0.03,0.75,0.15)$ and $I_0=0.1$. The corresponding model exhibits both type I forward-backward bifurcation and period doubling route to chaos as $\alpha$ varies from 0 to 1

The choice of the first two variables $\alpha$ and $\delta$ plays the key roles in producing chaotic dynamic. It is intuitively clear that the model is more prone to chaotic dynamic in case that the vaccine failure rate is high, the immunization rate is low and the medical resource are limited. It is also worth mentioning that the period doubling bifurcation occurs when the smaller eigenvalue of the Jacobian matrix J(I, V), defined in (14a) and (14b), of map F at an endemic point (I, V) is $-1$. Here I is on the increasing branch of $\beta =h_2(I)$. In the cases presented in Figs. 8, 9, 10, 11 and 12 with $(\alpha ,\delta ,r_1,r_2)$ being as given, we denote by ${\lambda }_{\beta }^{-}$ and ${\lambda }_{\beta }^{+}$ the smaller eigenvalue and the larger eigenvalue of J(I, V), respectively. Then there exists a ${\beta }_2$, ${\beta }_u<{\beta }_2<{\overline{\beta }}_u$, such that $|{\lambda }_{\beta }^{\pm }|<1$ for $\beta <{\beta }_2$ and ${\lambda }_{\beta }^{-}<-1$ for $\beta$ slightly larger than ${\beta }_2$. At $\beta ={\beta }_2$, ${\lambda }_{{\beta }_2}^{-}=-1$ and $|{\lambda }_{{\beta }_2}^{+}|<1$. Hence, the corresponding endemic equilibrium becomes unstable for $\beta >{\beta }_2$. The stable period two orbits are then created. This process seems to repeat itself. This sequence of bifurcations is the so called period doubling route to chaos.

Conclusions

In this paper, we consider a discrete epidemic model with vaccination and limited medical resources. We prove, among other things, that our model exhibits classical results, type I forward-backward bifurcation and type II forward-backward bifurcation according as the medical resources are sufficient, mildly insufficient and highly insufficient. Moreover, we numerically demonstrate that period doubling route to chaos occurs provided that the immunization is low, the vaccine failure rate is high and the medical resources are limited.

Declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.