Backward bifurcations, turning points and rich dynamics in simple disease models

Abstract

In this paper, dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcations in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcations with varied positions of turning points facilitate the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.

Appendices

Appendix A

The equilibrium solutions of system (7) are obtained by solving the following algebraic equations: $f_1(S,\,I)=0$ and $f_2(S,\,I)=0$, from which the disease-free equilibrium can be easily obtained as $\bar{\mathrm {E}}_0 = (\varLambda /d,\,0)$. For the infected equilibrium $\bar{\mathrm {E}}=(\bar{S},\,\bar{I})$, $\bar{S}$ is solved from $f_1=0$ as $\textstyle \bar{S}(I)=\frac{\varLambda (1+kI)}{(d k+\beta ) I+d}$. Then, substituting $S=\bar{S}(I)$ into $f_2=0$ yields a quadratic equation of the form

$$\begin{aligned} {\mathscr {F}}(I) = {\mathscr {A}}I^2 + {\mathscr {B}} I + {\mathscr {C}} = 0, \end{aligned}$$

(21)

which in turn gives two roots: $\bar{I}_{1,\,2} = \frac{-{\mathscr {B}}\pm \sqrt{{\mathscr {B}}^2-4{\mathscr {A}}\,{\mathscr {C}}}}{2{\mathscr {A}}}$, where, ${\mathscr {A}} = (d+\gamma +\epsilon )(dk+\beta )$, ${\mathscr {B}} = [(dk +\beta )\omega +d](d+\gamma +\epsilon ) +(d k+\beta )\alpha -\beta \varLambda$, ${\mathscr {C}} = [(d+\gamma +\epsilon )\omega +\alpha ]d - \beta \varLambda \omega$ for system (7). Since all parameters take positive values, we have ${\mathscr {A}}>0$. To get the two positive roots essential for backward bifurcation, it is required that ${\mathscr {B}}<0$ and ${\mathscr {C}}>0$. Noticing that $\beta , \, \varLambda ,\, \omega >0$, we can see that the infection force, $\beta$, the constant influx of the susceptibles, $\varLambda$, and the effect of medical treatment $\frac{\alpha I}{\omega + I}$ are indispensable terms for backward bifurcation. The number of positive infected equilibrium solutions changes from two to one when the value of ${\mathscr {C}}$ passes from negative to positive, which gives a critical point at ${\mathscr {C}}=0$, that is, $[(d+\gamma +\epsilon )\omega +\alpha ]d = \beta \varLambda \omega$, which is equivalent to $R_0^{(7)} = 1$.

On the other hand, we may infer the emergence of backward bifurcation without solving the equilibrium conditions. When we introduce the loss of the infectives due to medical treatment, the dynamics of system (7) differ greatly from system (6). In particular, backward bifurcation emerges and complex dynamical behaviors may occur. To clarify this effect, we obtain the function $f_4(S,\,I)$ from the equation $\frac{dI}{dt}=0$ of (7). Note that $f_4(S,\,I)$ is not an incidence rate. But, if we fix $S=\tilde{S}>0$, there exist $0<I_1<I_2<+\infty$, such that $\frac{\partial f_4(\tilde{S},\,I)}{\partial I} =\frac{1}{(1+kI)^2(\omega +I)^2} [\beta \tilde{S}(\omega +I)^2-\alpha \omega (1+kI)^2]>0$, $\forall \; I \in (0,\,I_2)$; and $\frac{\partial ^2 f_4(\tilde{S},\,I)}{\partial I^2} = -2k\beta \tilde{S} (1+kI)^{-3} + 2\alpha \omega (\omega +I)^{-3}>0$, $\forall \; I \in (0,\,I_1)$, $\frac{\partial ^2 f_4(\tilde{S},\,I)}{\partial I^2}=0$, for $I=I_1$, $\frac{\partial ^2 f_4(\tilde{S},\,I)}{\partial I^2}<0$, $\forall \; I \in (I_1,\,I_2)$. Thus, $f_4(\tilde{S},\,I)$ actually has a convex-concave ‘S’ shape, and may have two positive intersection points with the ray line, $g_1(I)$, in the first quadrant.

The infected equilibrium of (7) is denoted as $\bar{\mathrm {E}}_1 = (\bar{S},\,\bar{I})$, where $\bar{I}$ is solved from the equation ${\mathscr {F}}(I) =0$ in (21). The turning point is determined by both the quadratic equation (21) and the relation $\frac{\mathrm {d}\varLambda }{\mathrm {d}I}=-\frac{\partial {\mathscr {F}}}{\partial I}/ \frac{\partial {\mathscr {F}}}{\partial \varLambda }=0$, which is equivalent to $\frac{\partial {\mathscr {F}}}{\partial I}=0$. Solving $\frac{\partial {\mathscr {F}}}{\partial I}=0$ yields the expression of the turning point of I, given in (14). To find the stability of the infected equilibrium $\bar{\mathrm {E}}_1$, evaluating the Jacobian matrix at $\bar{\mathrm {E}}_1$, and further denoting it as $J_2=J|_{(7)}(\bar{\mathrm {E}}_1)$, we obtain the characteristic polynomial in the form of (13), with $\mathrm {Tr}(J_2)=a_{11}/[(\omega +I)^2(kI+1)(dkI+\beta I+d)]$ and $\mathrm {Det}(J_2)=a_{21}/[(\omega +I)^2(kI+1)(dkI+\beta I+d)]$, where $a_{11}=a_{1a}-a_{1b}$ and $a_{21}=a_{2a}-a_{2b}$, with $a_{1b}= \beta \varLambda (\omega +I)^2$ and $a_{2b}=d a_{1b}$, and $a_{1a}$ and $a_{2a}$ only contain positive terms (their expressions are omitted here for brevity). Determining whether a Hopf bifurcation can occur from $\bar{\mathrm {E}}$ is equivalent to finding whether $\mathrm {Det}(J_2)$ remains positive when $\mathrm {Tr}(J_2)=0$. Consider the following expression:

$$\begin{aligned} h_1(I)= & {} \Big [\mathrm {Tr}(J_2)- \frac{1}{d} \mathrm {Det}(J_2) \Big ] (\omega +I)^2(kI+1)(dkI+\beta I+d) \nonumber \\= & {} a_{11}-\frac{1}{d}a_{21}=a_{1a}-\frac{1}{d}a_{2a} \nonumber \\= & {} \frac{1}{d}(dkI+\beta I+d)[(kI+1)d^2(\omega +I)^2- \beta \epsilon I (\omega +I)^2-\alpha \beta \omega I],\nonumber \\ \end{aligned}$$

(22)

where the expressions of $a_{1a}$ and $a_{2a}$ have been used. This indicates that $h_1(I)$ may take negative values, for which $\mathrm {Det}(J_2)>0$.

Appendix B

It is easy to find the uninfected equilibrium of model (4), $\bar{\mathrm {E}}_0=(\bar{X}_0,\,\bar{Y}_0)=(\frac{1}{D},\,0)$, whose characteristic polynomial has two roots: $\lambda _1=-D<0$, and $\lambda _2 = \frac{B}{D}-1$, which gives $R_0=\frac{B}{D}$. Consequently, $\bar{\mathrm {E}}_0$ is stable (unstable) for $R_0<1\,({>}1)$. To find the infected equilibrium solution, setting $f_6(X,\,Y) =0$ yields $\bar{X}_1(Y)=\frac{Y+C}{(A+B)Y+BC}$, which is then substituted into $f_5(X,\,Y)=0$ to give the following quadratic equation:

$$\begin{aligned} {\mathscr {F}}_5(Y)=(A+B)Y^2+(BC+D-A-B)Y+C(D-B)=0. \end{aligned}$$

(23)

In order to have two real, positive roots, two conditions must be satisfied, that is, $BC+D-A-B<0$ and $D-B>0$, or in compact form, $0<D-B<A-BC$. The condition $D-B>0$ is equivalent to $0<R_0=\frac{B}{D}<1$, which is a necessary condition for backward bifurcation. Moreover, the positive influx constant, having been scaled to 1, is a necessary term for the positive equilibrium of Y. Therefore, the positive influx rate term and the increasing and saturating infectivity function are necessary for backward bifurcation.

We further examine the incidence function, $f_7(X,\,Y)$ defined in (12), without solving for the equilibrium solutions. The incidence function $f_7$ obviously satisfies the condition (3a), as well as the condition (3b) since $\frac{\partial }{\partial X}f_7(X,\,Y)=[B+AY(Y+C)^{-1}]Y >0$ and $\frac{\partial }{\partial Y}f_7(X,\,Y)= ACXY(Y+C)^{-2}+[B+AY(Y+C)^{-1}]X>0$ for all $X,\,Y>0$. However, the second partial derivative of $f_7(X,\,Y)$ with respect to Y, $\frac{\partial ^2}{\partial Y^2}f_7(X,\,Y)= 2AC^2X(X+C)^{-3}>0$ for all $X,\,Y>0$, showing that $f_7(X,\,Y)$ is a convex function with respect to the variable Y. Consequently, $f_7(X,\,Y)$ can only have one intersection with $g_2(Y)=Y$, implying that only one equilibrium solution would exist if we only consider the second equation in (11), as shown Fig. 2a. However, when considering both conditions given in (11) for equilibrium solutions, we will have two intersection points between $f_7$ and $g_2$. According to the first equation in (11), that is $f_5(X,\,Y)=0$, we can use Y to express X in the equilibrium state as $\bar{X}(Y)=(Y+C)[(A+B)Y^2+(BC+D)Y+DC]^{-1}$. Substituting $\bar{X}(Y)$ into $f_7(X,\,Y)$ in (12), we obtain

$$\begin{aligned} f_7(Y)=Y[(A+B)Y+BC][(A+B)Y^2+(BC+D)Y+CD]^{-1}, \end{aligned}$$

(24)

and $\frac{\partial }{\partial Y}f_7(Y)= D[(A+B)Y^2+2(A+B)CY+BC^2][(A+B)Y^2+(BC+D)Y+CD]^{-2}>0$ for all $X,\,Y>0$. However, the sign of $\frac{\partial ^2}{\partial Y^2}f_7(Y)= -2D[(A+B)^2Y^3+3C(A+B)^2Y^2+3(A+B)BC^2Y+(B^2C-AD)C^2] [(A+B)Y^2+(BC+D)Y+CD]^{-3}$, could alter at the inflection point from positive to negative as Y increases. Therefore, with appropriate parameter values, $f_7(Y)$ can have a convex-concave ‘S’ shape.

If we choose the parameter B as the bifurcation parameter, then $R_0=\frac{B}{D}=1$ defines $B_S=D$ where the ‘S’ in subscript stands for static bifurcation. Further, it can be proved that this is a transcritical bifurcation. Therefore, $\bar{\mathrm {E}}_0$ is stable when $B<D$ (or $R_0<1$), loses its stability and becomes unstable when B increases to pass through $B_S=D$, that is $B>D$ (or $R_0>1$), and no other bifurcations can happen.

Next, we examine the infected equilibrium $\bar{\mathrm {E}}_1=(\bar{X},\,\bar{Y})$. Since $\bar{X}(Y)=\frac{Y+C}{(A+B)Y+BC}$, $\bar{Y}$ is determined by the quadratic equation (23), which gives the turning point $(B_T,\,Y_T)$, as given in (15). We perform a further bifurcation analysis on its corresponding characteristic polynomial (13), which takes the form

$$\begin{aligned} \displaystyle P|_{\bar{\mathrm {E}}_1}(\lambda ,Y) \!= & {} \! \lambda ^2\!+\!\frac{a_{1a}}{[(A\!+\!B)Y\!+\!BC](Y\!+\!C)} \lambda \!+\!\frac{a_{2a}}{[(A\!+\!B)Y\!+\!BC](Y\!+\!C)},\quad \text {where} \nonumber \\ a_{1a}= & {} (A+B)^2Y^3+(2BC+D)(A+B)Y^2+(B^2C^2+ACD+2BCD\nonumber \\&-AC)Y+BC^2D, \nonumber \\ a_{2a}= & {} (A+B)^2Y^3+2(A+B)BCY^2+(B^2C-AD)CY. \end{aligned}$$

(25)

Therefore, the sign of the subtraction between the trace and determinant is determined by $h_2(Y)=a_{1a}-a_{2a}=D(A+B)Y^2+[2CD(A+B)-AC]Y+BC^2D$. Here the equilibrium solution of Y and other parameters satisfy the quadratic equation (23), which leads to an explicit expression, given by $\bar{B}= -\frac{AY^2+(D-A)Y+CD}{Y^2+(C-1)Y-C}$. Substituting $B=\bar{B}$ into $h_2(Y)$, we obtain

$$\begin{aligned} \displaystyle h_2(Y)|_{B\!=\!\bar{B}}\!=\! a_{1a} \!-\! a_{2a} \!=\! \frac{[AC(D-1)-D^2]Y^2-[AC(D-1)+2CD^2]Y-C^2D^2}{Y-1}. \end{aligned}$$

(26)

Hopf bifurcation may occur when the trace is zero, while the determinant is still positive. This implies $h_2(Y)<0$, which is possible with appropriately chosen parameter values. Hence, by solving $a_{1a}=0$ in (25) together with the quadratic equation (23), we get two pairs of points denoted by $(B_{h1},\,Y_{h1})$ and $(B_{h2},\,Y_{h2})$, which are candidates for Hopf bifurcation. Then validating the above two points by substituting them back into the characteristic polynomial (25), respectively, we denote the Hopf bifurcation point as $(B_H,\,Y_H)$ if this validation confirms their existence. According to Yu et al. (2015), Hopf bifurcation can happen only from the upper branch of the infected equilibrium $\bar{\mathrm {E}}_1$.