Qualitative Effects of Monovalent Vaccination Against Rotavirus: A Comparison of North America and South America

Abstract

Rotavirus is the most common cause of severe gastroenteritis in young children worldwide. The introduction of vaccination programs has led to a significant reduction in number of hospitalizations due to rotavirus in North and South American countries. Little work has been done, however, to examine the differential impact of vaccination as a function of strain distribution and strain-specific vaccine efficacy. We developed a two-strain epidemiological model of rotavirus transmission, and used it to examine the effects of a monovalent vaccine (Rotarix) on the qualitative behaviors of infection levels in a population. For contrast, we parameterized our model with strain distribution data from North America and from South America. In all cases, the introduction of the vaccine led to significant decreases in the prevalence of primary infection due to both strains for a decade or more, after which the overall prevalence recovers to near pre-vaccination levels. The prevalence of G1P[8] is significantly higher in North America (73 % of all rotavirus infections) compared to that in South America (34 %). Our model predicts that the introduction of Rotarix might result in major strain replacement in regions such as North America where the prevalence of G1P[8] is relatively high, due to higher efficacy of Rotarix against infection caused by G1P[8], while regions with lower prevalence of G1P[8], such as South America, are not susceptible to major strain replacement.

Appendix: Oscillation Analysis

Numerical observations about the behavior of the system drive the study of the oscillations. First, the dynamics of the state variables $Y_1$ and $Y_2$ seen in Fig. 14a, b are slow relative to the dynamics of the infected states shown in Fig. 4a. Second, Fig. 14c shows that the variable H has relatively small changes in amplitude as it oscillates. Indeed, if the dynamics of H are frozen so that H is fixed at some intermediate value, the system still oscillates for a range of $\beta _1$ and $\beta _2$ values.

Fig. 14

a, b $Y_1$ and $Y_2$ are slow variables. c H has relatively small oscillations (Color figure online)

We therefore fix the value of H at an intermediate value and introduce a time-scaling parameter $\tau _y$ to slow down the dynamics of $Y_1$ and $Y_2$ simultaneously. As $\tau _y$ is decreased from 1 to 0, the period of the oscillation increases, and the system stops oscillating when $\tau _y=0$; that is, oscillations vanish when the dynamics of $Y_1$ and $Y_2$ are completely turned off. When $\tau _y$ is small, however, $Y_1$ and $Y_2$ remain (approximately) on a line with respect to one another, as seen in Fig. 15.

Fig. 15

$Y_2$ grows approximately linearly with respect to $Y_1$ for small values of $\tau _y$. Here, $H=0.575$ is fixed, $\tau _y=0.01$, $(\beta _1,\beta _2)=(10,7)$, $\phi =0$, and $Y_2\approx -.02314Y_1+0.0774$ (Color figure online)

We exploit this observation by treating $Y_1$ and $Y_2$ as parameters, with the constraint that they both stay on the line $Y_2=b+aY_2$, where a and b are approximated from Fig. 15.

Treating $Y_1$ and $Y_2$ as parameters in this way eliminates the oscillations in the system. Instead, the system is generically attracted to one of two steady states: one with strain 1 endemic and strain 2 extinct, the other with strain 2 endemic and strain 1 extinct, which we call $E_1^*$ and $E_2^*$, respectively. Figure 16 shows the stability of the two fixed points as the parameter $Y_1$, and consequently $Y_2$, varies. When $Y_1$ is small, the $E_1^*$ is stable and $E_2^*$ is unstable. As $Y_1$ increases past some critical threshold, the two steady states immediately switch stability: $E_1^*$ becomes unstable and $E_2^*$ becomes stable.

Fig. 16

Stability switches from strain 1 being “on” and strain 2 being “off” to strain 1 being “off” and strain 2 being “on” past a critical value of $Y_1$. $(\beta _1,\beta _2)=(10,7)$, $\phi =0$. Bifurcation at $Y_1=0.28828.$ (Color figure online)

This instantaneous switch in the stability of the system is due to a degenerate hyperplane of fixed points at the critical value of $Y_1$. In particular, at $E_i^*$, we must have

$$\begin{aligned} \frac{{\hbox {d}}I_{0i}}{{\hbox {d}}t}= & {} 0\\ \frac{{\hbox {d}}I_{ji}}{{\hbox {d}}t}= & {} 0 \end{aligned}$$

which occurs if and only if

$$\begin{aligned} \beta _i(I_{0i}+\rho I_{ji})S-(\gamma _p+\mu )I_{0i}= & {} 0 \end{aligned}$$

(16)

$$\begin{aligned} \left[ (1-\sigma )Y_j+(1-\theta )H\right] \beta _i(I_{0i}+\rho I_{ji})-(\gamma _s+mu)I_{ji}= & {} 0. \end{aligned}$$

(17)

Equation (16) is satisfied if and only if

$$\begin{aligned} I_{0i}= & {} \frac{\beta _i \rho S}{(\gamma _p+\mu )-\beta _iS}I_{ji}, \end{aligned}$$

(18)

and by plugging (18) into (17), we get

$$\begin{aligned} \big [\left[ (1-\sigma )Y_j+(1-\theta )H\right] \beta _i(\frac{\beta _i \rho S}{(\gamma _p+\mu )-\beta _i S}+\rho )-(\gamma _s+\mu )\big ]I_{ji}= & {} 0. \end{aligned}$$

If $I_{ji}\not =0$, then $\frac{dI_{0i}}{dt}=0$ and $\frac{dI_{ji}}{dt}=0$ if and only if

$$\begin{aligned} \left[ (1-\sigma )Y_j+(1-\theta )H\right] \beta _i\left( \frac{\beta _i \rho S^*}{(\gamma _p+\mu )-\beta _i S^*}+\rho \right) -(\gamma _s+\mu )= & {} 0, \end{aligned}$$

or equivalently,

$$\begin{aligned} S_i^*= & {} \frac{\gamma _p+\mu }{\gamma _s+\mu }\left( \frac{1}{\beta _i}\left( \gamma _s+\mu -\beta _i \rho [(1-\sigma )Y_j+(1-\theta )H]\right) \right) , \end{aligned}$$

(19)

where $S_i^*$ is the value of S at the fixed point $E_i^*$. Thus, $E_1^*$ and $E_2^*$ coincide if and only if

$$\begin{aligned} S_1^*=S_2^*. \end{aligned}$$

(20)

In this case, $I_{01}$, $I_{02}$, $I_{21}$ and $I_{12}$ are all nonzero and from System (1) we have $\frac{dS}{dt}=0$ if and only if

$$\begin{aligned} S_i^*= & {} \frac{\tau _M M}{\beta _1(I_{01}+\rho I_{21})+\beta _2(I_{02}+\rho I_{12})-\mu }\nonumber \\= & {} \frac{\tau _M M}{\beta _1(\alpha _1+\rho )I_{21}+\beta _2(\alpha _2+\rho )I_{12}-\mu }, \end{aligned}$$

(21)

where $\alpha _i=(\beta _i\rho S_i^*)/(\gamma _p+\mu -\beta _i S_i^*)$.

Setting equations (19) and (21) equal defines a line in the $I_{21}$-$I_{12}$ plane, along which the reduced system is at a steady state. Since $I_{0i}=\alpha _i I_{ji}$, the line of fixed points in the $I_{21}$-$I_{12}$ plane defines a hyperplane of fixed points in $I_{01}$-$I_{02}$-$I_{21}$-$I_{12}$ space.

This hyperplane exists only if equation (20) is satisfied. Given our requirement that $Y_2=b+aY_1$, equation (20) is equivalent to

$$\begin{aligned} Y_1=\frac{(\beta _1-\beta _2)(\gamma _s+\mu )}{\beta _1\beta _2\rho (1-\sigma )(1-a)}+\frac{b}{1-a}, \end{aligned}$$

(22)

which, when evaluated at parameter values specified in our simulation, is exactly the bifurcation value of $Y_1$ as in Fig. 16.

To visualize this bifurcation in System (1), we now allow $Y_1$ and $Y_2$ to vary as dynamic variables, but slow down their dynamics by a factor of $\tau _y=0.01$ and continue to treat H as a parameter fixed at $H=0.575$. Figure 17 shows the infected state $I_{02}$ versus the susceptible class $Y_1$. The vertical dashed line coincides with the critical value of $Y_1$ from Fig. 16. Beginning in the lower left corner of the oscillation in Fig. 17, as $Y_1$ increases through the critical value of $Y_1$, the stability switches from strain 1 endemic to strain 2 endemic, and $I_02$ quickly grows to a steady state. After $I_{02}$ increases, $Y_1$ begins to decrease again because all of the individuals who are susceptible to only strain 2 (i.e., individuals in class $Y_1$) quickly begin to become infected. Once $Y_1$ decreases past the critical $Y_1$ value, the incidence of strain 2 drops off sharply and $Y_1$ begins to increase again.

Fig. 17

$Y_1$ plotted against $I_{02}$ in the full system with $Y_1$ and $Y_2$ slowed down by a factor of $\tau _y=0.01$, and H frozen at 0.575. In this case, the transmission rates are $(\beta _1,\beta _2)=(10,7)$, and there is no vaccination ($\phi =0$). After $Y_1$ passes a critical threshold from left to right along the bottom of the blue curve, there is a relative abundance of individuals who are susceptible to strain 2 but not strain 1, and strain 2 gets turned “on.” $I_{02}$ remains on until $Y_1$ falls below the critical value, after which strain 2 is turned “off” and strain 1 is turned “on.” (Color figure online)

In terms of a population, this means that while strain 1 is infecting a large proportion of the population, strain 2 is almost extinct. Consequently, much more individuals become infected with strain 1 than strain 2, and upon recovery, they are only susceptible to strain 2. The switch between strain 1 being the more prevalent strain to strain 2 taking over occurs when the number of individuals susceptible to only strain 2 surpasses a critical threshold. Immediately after the switch, there are much more individuals only susceptible to strain 2, but as these individuals become infected and recover, more will be susceptible to only strain 1, prompting another switch in the dominant strain back to strain 1, and the cycle repeats. The class of individuals who have recovered from strain i and are susceptible only to strain j are therefore the driving force behind the oscillations: Both strains require a sufficient number of susceptible individuals available only to that strain in order to infect a nontrivial proportion of the population. However, since the number of individuals who are susceptible to only strain 1 is inversely proportional to that of those susceptible to only strain 2, the two strains oscillate dominance over time.