Abstract
We describe and analyse by elementary means some simple models for disease transmission with vaccination and treatment. In particular, we give conditions for the existence of multiple endemic equilibria and backward bifurcations. The results are applicable to epidemic models in which demographic effects are ignored, but there is temporary immunity following recovery from infection.
1. Introduction
The central concept in the analysis of compartmental models for the transmission of communicable diseases is the basic reproductive number ℛ0, representing the mean number of secondary infections caused by a single infective introduced into a susceptible population. As ℛ0 increases through 1, there is an exchange of stability between the disease-free equilibrium, which is asymptotically stable for and unstable for , and the endemic equilibrium which exists if . The usual transition is a forward, or transcritical, bifurcation at , with an asymptotically stable endemic equilibrium and an equilibrium infective population size depending continuously on
The behaviour at a bifurcation may be described graphically by the bifurcation curve, which is the graph of the equilibrium infective population size I as a function of the basic reproductive number ℛ0. It has been noted Citation2 Citation3 Citation4 Citation6 that in epidemic models with multiple groups and asymmetry between groups or multiple interaction mechanisms, it is possible to have a very different bifurcation behaviour at . There may be multiple positive endemic equilibria for values of and a backward bifurcation at . The qualitative behaviour of a system with a backward bifurcation differs from that of a system with a forward bifurcation and the nature of these changes has been described in Citation1. Since these behaviour differences are important in planning how to control a disease, it is important to determine whether a system can have a backward bifurcation.
In Citation1, we have analysed a simple SIS disease transmission model which includes vaccination of susceptibles and have obtained by elementary methods conditions for the existence of a backward bifurcation. We were able to give a complete qualitative analysis of the resulting two-dimensional model which had previously been examined in Citation6 using the centre manifold theorem and examination of normal forms Citation5. Here, we consider models that include temporary immunity after recovery and treatment of infectives. The same approach yields conditions for the existence of a backward bifurcation, but the qualitative analysis of these higher-dimensional models remains incomplete. Our treatment is very elementary, depending largely on the analysis of the number of positive solutions of a quadratic equation. We are able to analyse the bifurcation behaviour by elementary methods, without making use of the bifurcation theory.
2. A vaccination model
We begin by adding vaccination to a simple SIRS model with a constant birth rate Λ in the susceptible class, a proportional natural death rate μ in each class and no disease deaths,
To model Equation(1), we add the assumption that in the unit time a fraction ϕ of the susceptible class is vaccinated. The vaccination may reduce but not completely eliminate susceptibility to infection. We model this by including a factor σ, , in the infection rate of vaccinated members with σ=0 meaning that the vaccine is perfectly effective and σ=1 meaning that the vaccine has no effect. We assume also that the vaccination loses effect at a proportional rate θ. We describe the new model by including a vaccinated class V,
In model Equation(2), it is easy to see by adding the equations in EquationEquation (2) that as t→∞. If R=N−S−V−I is substituted into the first equation in EquationEquation (2), then the first three equations uncouple from the fourth and are asymptotically autonomous with the limiting system
Equilibria of model Equation(3) are solutions of
If I=0 then the first equation of EquationEquation (4) is satisfied, and solution of the remaining equilibrium conditions leads to
Since we have a population which is not completely susceptible, we speak of a control reproduction number rather than a basic reproduction number. We will use to denote the control reproductive number of model Equation(3), and the control reproduction number is
If σ=1, meaning that the vaccine has no effect, model Equation(3) is equivalent to the SIRS model Equation(1) and if there is a unique endemic equilibrium which is globally asymptotically stable. If endemic equilibria are solutions of the system of EquationEquations (4).
We use the first two equations of EquationEquation (4) to express S and V in terms of I,
The coefficients A, B, C are functions of β, and we will indicate this dependence explicitly when necessary. When , and there is exactly one positive root of EquationEquation (6), corresponding to an endemic equilibrium.
The condition C=0, or , corresponds to a critical value β c of β, given by
All endemic equilibria lie on the bifurcation curve Equation(6). In drawing the bifurcation curve (the graph of I as a function of ), we think of β as a variable with the other parameters being constant. Since is a constant multiple of β, we may think of β as the independent variable in the bifurcation curve.
Implicit differentiation of the equilibrium condition Equation(6) with respect to β gives
These two possibilities are illustrated by and .
If C>0 and either B≥0 or B 2<4AC, there are no positive solutions of EquationEquation (6) and thus there are no endemic equilibria. EquationEquation (6) has two positive solutions, corresponding to two endemic equilibria, if and only if C>0, or , and B<0, B 2>4AC, or . If B=−2√AC there is one positive solution I=−B/2A of EquationEquation (6).
If , so that there is a backward bifurcation at , there are two endemic equilibria for an interval of values of β from β c corresponding to to a value β0 defined by B=−2√AC. Since both B and AC are linear functions of β, the equation B 2=4AC is quadratic and has two roots. One of these roots, namely the root corresponding to B=−2√AC, satisfies B<0 and determines the value of β0. While it is possible to give an analytic expression for β0, in practice it would be more efficient to compute its value numerically.
The condition , necessary and sufficient for the existence of a positive endemic equilibrium and a backward bifurcation at , is
Algebraic simplification reduces this condition to
If ψ=0, the case of an SIR model, the condition Equation(9) reduces to
The SVIS model is the special case . To analyse this case, we divide both sides of the inequality Equation(9) by μ+ψ and let , so that
The necessary and sufficient condition Equation(9) for a backward bifurcation can be written as a cubic inequality,
For a backward bifurcation, it is necessary that R<0. Normally, the disease time scale is much faster than the demographic time scale, and we have μ small compared with the other rates. There is a backward bifurcation for sufficiently small μ if and only if R<0.
Similarly, examination of the condition Equation(10) shows that a necessary condition for the SVIS model to have a backward bifurcation is
3. A vaccination/treatment model
To the vaccination model Equation(2), we add treatment of infectives at rate γ. It is assumed that treatment eliminates infectivity and changes the recovery rate to η. This leads to the model
In model Equation(12), just as for model Equation(2), it is easy to see that as t→∞. If R=N−S−V−I−T is substituted into the first equation in EquationEquation (2), then the first four equations uncouple from the fifth and are asymptotically autonomous with the limiting system
We note that, just as for the vaccination model Equation(3), C>0 if and C<0 if , and there are two equilibria at if and only if .
The condition C=0 corresponds to
Algebraic simplification reduces this condition to
Much as for the model Equation(3) without treatment, the necessary and sufficient condition Equation(17) for a backward bifurcation can be written as a polynomial inequality of the form g(μ)<0, where g is now a fourth-degree polynomial whose coefficients except for the constant term are positive. For a backward bifurcation, it is necessary that the constant term be negative and there is a backward bifurcation for sufficiently small μ if and only if the constant term is negative. Thus, for the model Equation(13), a necessary condition for a backward bifurcation is
We may summarize our main result as follows.
Theorem 3.1
For both the vaccination model Equation(3) and the vaccination/treatment model Equation(3), there is a necessary and sufficient condition for the existence of an endemic equilibrium and a backward bifurcation at . Also, there is a necessary condition not involving the natural death rate μ that is also sufficient if μ is small.
Strains of communicable diseases such as influenza often mutate rapidly enough that immunity following vaccination or recovery from infection by a specific strain is only temporary. A short-term epidemic of such a disease may be modelled by EquationEquation (3) or Equation(13) with μ=0, and our results indicate the possibility of a backward bifurcation in such models. The condition Equation(18) is satisfied, for example, if θ=0 (no loss of vaccine efficacy), if η=α (treatment does not change the infective period) and
4. The bifurcation curve
In Section 2, we have examined the slope of the bifurcation curve Equation(6) at the equilibrium I=0 when . In general, the slope of the bifurcation curve is given implicitly by
We conjecture that has the same sign as 2AI+B for all β, not just for values corresponding to , but we have not been able to establish this except in the special case studied in Citation1. In this special case, the condition 2AI+B>0 is also significant in the local stability analysis of endemic equilibria. Because the model in this case is two-dimensional, it is not difficult to show that an endemic equilibrium is (locally) asymptotically stable if and only if it corresponds to a point on the bifurcation curve at which the curve is increasing. It is also possible to show that there are no periodic orbits and thus to obtain a complete picture of the global behaviour. We conjecture that these facts extend to the model Equation(13).
5. Conclusions
We have examined simple vaccination models and have shown by elementary algebraic means how to analyse the existence of multiple endemic equilibria when the basic reproduction number is less than 1. In the simplest case, with permanent immunity following recovery and no treatment an equilibrium corresponding to a point on the bifurcation curve with negative slope is unstable, and an equilibrium corresponding to a point on the bifurcation curve with the positive slope is asymptotically stable; we conjecture that this is true also for more complicated vaccination models. Our model appears not to admit the possibility of oscillations about an unstable endemic equilibrium.
The models we have examined could also be treated by using the centre manifold theory and normal forms. However, the required preliminary transformations would be at least as complicated technically as our approach. For models with more compartments, the analysis by elementary means becomes hopelessly complicated and a centre manifold approach would be necessary. It would be of considerable interest to determine if the properties we have found in simple examples carry over to more complicated models such as models involving the age of infection.
Although the introduction of a vaccination policy may lead to backward bifurcations, we emphasize that it always decreases infective population size. The danger of a vaccination policy is that an unforeseen backward bifurcation may require a larger than expected vaccination fraction to control a disease. If a vaccine can be developed which is completely effective, this possibility does not arise, and a programme which decreases the contact rate can also control a disease without leading to backward bifurcations. Nevertheless, a vaccination programme, even one which is not fully effective, may be a useful approach in controlling infections.
Acknowledgements
The exposition of backward bifurcations and multiple endemic equilibria has been improved substantially by the suggestion of an anonymous reviewer. This research was supported by NSERC Research Grant OGPIN #203901-99.
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