A simple vaccination model with multiple endemic states

Abstract

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a disease's reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.

Introduction

Recently there has been interest in the analysis and prediction of consequences of public health strategies designed to control infectious diseases, particularly tuberculosis and AIDS. Attention has been given to vaccination and treatment policies both in terms of the different vaccine classes (all/nothing, leaky, VEI) and efficacy (e.g., [1], [2], [3], [4]) and to application schedules and associated costs (e.g., [5], [6], [7], [8], [9]).

The study of vaccination, treatment and associated behavioral changes related to disease transmission has been the subject of intense theoretical analysis. The literature on these topics is ample and in the references we list but a small subset of it. We want to underline, however, a few of those that directly relate to the approach we take in this work.

The application of a vaccination and treatment programs has the likely effect of inducing behavioral changes in those individuals subjected to it. In particular, in the case of HIV, risk behavior can combine with vaccination or treatment resulting in a possible harmful effect in terms of disease prevalence. Blower and McLean [10] have argued that a mass vaccination campaign can increase the severity of disease if the vaccine being applied resulted in only a 50% coverage and 60% efficacy. Velasco-Hernández and Hsieh [11] and Hsieh and Velasco-Hernández [12] confirmed this result in a theoretical mathematical model of disease transmission. In their case a too-large case treatment rate combined with lengthening of the infectious period could result in the increase of the treatment reproduction number, that is, treatment would contribute to the spread of disease rather than to its elimination.

Of course, these are theoretical investigations on the plausible effects of vaccination and treatment programs. The models are rather simple, but nevertheless they give insight into some of the plausible consequences of public health policies.

A phenomenon of considerable interest recently in theoretical epidemiology is that of the existence of multiple steady-states and the associated population and epidemiological mechanisms that produce them. Mathematical models that give rise to multiple steady-states show bifurcation phenomena. A bifurcation in general is a set of parameter values at which an equilibrium, or fixed point, of the system being considered changes stability and/or appears/disappears. In epidemiology, bifurcation phenomena are associated with threshold parameters, the most common of which is the basic reproduction number, R₀. R₀ is a dimensionless quantity that represents the average number of secondary infections caused by an infective individual introduced into a pool of susceptibles. In the case of the simplest epidemiological models of the SIS and SIR types and a considerable number of generalizations (e.g., [13]), if R₀<1, the pathogen cannot successfully invade the host population, and dies out; if R₀>1, however, the pathogen can invade and successfully colonize hosts, therefore producing an epidemic outbreak that in many cases ends up in the establishment of an endemic disease in steady-state.

The mathematical description of this phenomenon involves a so-called transcritical bifurcation that brings about an exchange in stability between the disease-free equilibrium, which exists for all values of R₀, and an endemic equilibrium which only exists on one side of the bifurcation point. (On the other side, it has a negative value and is therefore outside the biologically feasible state space.) Prototypical R₀ threshold behavior features a `forward' bifurcation, in which the endemic equilibrium exists only for R₀>1, so that there is no possibility of an endemic state when R₀<1. In systems exhibiting a backward bifurcation, however, the endemic equilibrium exists for R₀<1, so that under certain initial conditions it is possible for an invasion to succeed, or for an established endemic state to persist, with R₀<1. See Fig. 1 for an illustration.

The presence of a backward bifurcation has other important consequences for the population dynamics of infectious diseases. In a system with a forward bifurcation, if parameters change and cause R₀ to rise slightly above one, a small endemic state results; that is, the endemic level at equilibrium is a continuous function of R₀ [13]. In a system with a backward bifurcation, the endemic equilibrium that exists for R₀ just above one has a large infective population, so the result of R₀ rising above one would be a sudden and dramatic jump in the number of infectives. Moreover, reducing R₀ back below one would not eradicate the disease, if the infective population size is close to the endemic level at equilibrium: there will be two locally stable equilibrium points, one with no disease and other with a positive endemic level. In this case, in order to eradicate the disease, one must further reduce R₀ so far that it passes a so-called saddle-node bifurcation at R₀^c<1 (see Fig. 2) and enters the region where no endemic equilibria exist, and the disease-free equilibrium is globally asymptotically stable.

This effect is known as hysteresis, and the system is said to have memory: that is, for R₀ between R₀^c and 1, one can tell whether R₀ was most recently less than R₀^c or greater than 1 (barring a sudden significant invasion of infectives).

Models which exhibit backward bifurcations have been studied in an epidemiological context. Castillo-Chávez et al. [14], Dushoff [15], Dushoff et al. [16], Hadeler and Castillo-Chávez [17], Hadeler and van den Driessche [18], Huang et al. [19], and Kribs-Zaleta [20], [21] have all considered models that exhibit backward bifurcations. More recently, Hadeler and van den Driessche [18] reviewed such models and found that multi-group models with asymmetry between groups, or multiple interaction mechanisms, can cause backward bifurcations, and Dushoff et al. [16] derived a criterion for determining the direction of the bifurcation at R₀=1. Finally, Kribs-Zaleta [22] used a more general approach to analyze bifurcations in epidemic models. Here it was also concluded that backward bifurcations tend to arise when the population is compartmentalized in other ways besides infectedness (typically by contact rates), and individuals can move between these compartments. Backward bifurcations have also been found in metapopulation models where the phenomenon has been associated with relatively high impact of migration on local patch dynamics [23].

As mentioned before, behavioral change has been the subject of mathematical analysis in models for the spread of infectious diseases as it relates to isolation [24], mixing patterns [25], treatment [11], [12], [26], and vaccination and education programs [17]. One of the main reasons for modeling these processes is to predict the effect of public health policies, particularly the implementation of treatment and vaccination campaigns. In this paper we concentrate on vaccination policies and vaccine-related parameters: vaccine coverage (rate at which susceptible individuals are immunized per unit time), average duration of immunity acquired by vaccine application (waning period), and the leakiness of the vaccine, that is, the percentage of susceptible individuals left unprotected even though vaccinated.

There are different types of vaccines (see, e.g., [1], [27]): some may give permanent immunity, while others offer only temporary protection; vaccines may not show 100% efficacy (leaky vaccines), and, finally, vaccine coverage may not be 100%. There are other public health strategies that behave in a fashion similar to that of the vaccine, but instead of affecting the immune defenses on individuals, they affect behaviors that may impact (reduce or increase) disease transmission. These we call public education programs (see, e.g., [17]).

Mass vaccination as a control mechanism attempts to lower the degree of susceptibility of a healthy individual against a particular pathogenic agent. Since this decrease of susceptibility occurs in a population, the overall effect of mass vaccination is to decrease the proportion of contacts with infected individuals, giving rise to the concept of herd immunity. At the population level, therefore, one wishes to identify the critical vaccination rate necessary to reduce a certain threshold parameter below one so as to eradicate the disease or prevent infection (epidemic outbreak) [8], [9], [17], [28].

However, as noted above, in models that can exhibit multiple endemic equilibria, ensuring a below-threshold value for the bifurcation parameter is insufficient to ensure that a disease is wiped out [11], [17]. In particular, it has been seen that partially effective disease management programs may actually be worse than none at all.

Because systems with multiple endemic states tend to be relatively complex, it is often difficult to provide a complete mathematical analysis of them. In this paper we present as simple a system of this type as possible – a two-dimensional model that exhibits a backward bifurcation – along with a complete analysis of its behavior.

The system considered in Section 2 models an SIS disease with a vaccinated class in a constant size closed population with homogeneous mixing. The vaccination policy illustrated in the model is one for an all/nothing vaccine which is leaky and confers only temporary immunity. The choice of an SIS disease framework has been made for the sake of generality. Setting the cure rate to zero in our equations leaves all of our results unchanged for an SI disease model. Likewise, the immunity can be made permanent by setting a different parameter to zero, without changing any results.

In Section 3, we extend the system in two different ways. In 3.1 A two-group model, 3.2 Endemic equilibria of the two-group model, the model expands to four dimensions in order to model two coexisting populations with different vaccination rates. This latter model was conceived to model a likely situation that may occur in neighboring border cities such as San Diego and Tijuana, or El Paso and Ciudad Juárez, whose populations interact on a daily basis while being subject to vaccination policies for the same disease which are conceivably very different. In the Mexican–US border example mentioned above, the vaccination policies of BCG and rubella are completely different in each country, and yet disease transmission takes place in either side of the border.

Section 3.3 considers a model which includes as special cases both the model of Section 2 and the model of primary focus in [17], in which some fraction of recovering infecteds pass directly into the vaccinated class, and vaccination is permanent.

Finally, we conclude in Section 4 with a discussion of the mathematical and biological consequences of our findings.