Considerations for Basic Reproduction Number Variation

aSchool of Mathematical and Geospatial Sciences, RMIT University, Australia

bBiomathematics Unit, Department of Zoology, Faculty of Life Sciences, Tel Aviv University, Israel

a r t i c l e i n f o

Article history:

Received 27 January 2015 Accepted 25 July 2015 Available online 23 October 2015

Keywords:

Epidemic models Basic reproduction number Backward bifurcation

Backward bifurcation thresholdRc

a b s t r a c t

This study addresses problems that have arisen in the literature when calculating backward bifurcations, especially in the context of epidemic modeling. Backward bifurcations are gen- erally studied by varying a bifurcation parameter which in epidemiological models is usually the so-called basic reproduction numberR0. However, it is often overlooked thatR0is an ag- gregate of parameters in the model. One cannot simply vary the aggregateR₀while leaving all model parameters constant as has happened many times in the literature. We investigate two scenarios. For the incorrect approach we fix all parameters in the aggregateR0to constant values, butR₀is nevertheless varied as a bifurcation parameter. In the correct approach, a key parameter inR₀is allowed to vary, and henceR₀itself varies and acts as a natural bifurcation parameter. We explore how the outcomes of these two approaches are substantially different.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

Epidemiological models have become important tools for helping understand the qualitative dynamics controlling the spread of infectious diseases. Many of these models have strong nonlinearities and therefore exhibit complex population dynamics and possess subtle bifurcation properties. Recently there has been interest in so called “backward bifurcations” because of the unusual thresholds they introduce. In this study we examine an overlooked problem in calculating backward bifurcation diagrams which has led to repeated errors in the literature.

As is well known, the qualitative dynamics of most epidemiological models depends on the threshold quantity known as the basic reproduction numberR₀[1,2]. This quantity represents the average number of secondary infections generated by a typical infected individual during the infectious period, when introduced into an entirely susceptible population. Generally, ifR₀<1 an infected individual will on average be unable to replace himself/herself, and the disease will die out in time. IfR₀>1, an infected individual will on average be able to infect more than one other and thus transmit through the population; as such an equilibrium number of infectivesI^∗could plausibly be maintained (I^∗>0) . ThusR₀<1 corresponds to an infection-free equilibrium (I^∗=0), whileR₀>1 corresponds to an endemic equilibriumI^∗>0, and this is the usual forward bifurcation scenario shown inFig. (1).

While the above characterization familiar to most epidemiological modelers is quite general, it has recently been observed that it gives the wrong picture in the presence of a backward bifurcation. Backward bifurcations, which are characterized by multiple coexisting equilibria, allow a disease to persist even thoughR₀<1. More specifically, one can find a stable disease free equilibrium coexisting with two endemic equilibria (I^∗>0), one being unstable and the other stable even thoughR₀<1. A range

∗ Corresponding author. Tel.: +61 452176626.

E-mail addresses:mwangiisaac@aims.ac.za,s3470462@student.rmit.edu.au(I.M. Wangari),stephen.davis@rmit.edu.au(S. Davis),lewistone2@gmail.com (L. Stone).

http://dx.doi.org/10.1016/j.apm.2015.07.022

S0307-904X(15)00480-1/© 2015 Elsevier Inc. All rights reserved.

Fig. 1. The equilibriumI^∗=0 is stable whenR0<1 and unstable whenR0>1. ForI^∗>0 andR0>1 the endemic equilibria is stable. There is a bifurcation and exchange of stability atR0=1.

of different epidemiological models have been found to exhibit backward bifurcation, including models that incorporate behav- ioral responses to perceived risks[3], vaccination[4,5], multiple groups[6], vector-borne diseases[7]and exogenous reinfection [8–10]. The presence of backward bifurcation is important in a practical sense because control programs must reduceR₀further than below unity to eliminate a disease.

The problem we address here stems from studies of backward bifurcation in the literature where there have been instances where authors illustrate the phenomena by varyingR₀without properly considering the fact thatR₀is an aggregate of parameters in the model. One cannot simply vary the aggregateR₀while leaving all model parameters constant as has been the practice in a number of important studies[11–15]. To illustrate this problem we examine a (tuberculosis) TB model that incorporates reinfectionpas the parameter to induce backward bifurcation, although any other example can be used to exhibit the difference.

2. The model

The model consists of four subpopulations; susceptible (S), exposed (E), infectious (I) and recovered (R) individuals, and may be described by the following equations:

dS

dt =

π

⁻

β

^SI−

μ

^S,

dE

dt =

β

^SI−p

β

^EI−

(μ

^+k

)

^E,

dI

dt =p

β

^EI+kE−

(μ

^+r+

μ

d

)

^I,

dR

dt =rI−

μ

^{R. (1)}

The numbers of susceptible individuals increase by recruitment through births and immigration at a rate

π

. Susceptibles who come into contact with infected individuals move straight to the exposedEclass but they are not themselves yet infective.

The susceptible population is thus diminished due to contact with infected individuals at a rate

β

^{SI, where}

β

represents the per-capita effective contact rate of acquiring TB bacteria. Concomitantly, the numbers in the exposed class increase at a rate

β

SI. Progression to the infectious state occurs when an exposed individual harbors a dormant infection that becomes active due to immune system destabilization. This is the usual “slow TB” which can take years or decades before progression. Exposed individuals move to the infected classIat rate (kE). In addition, exposed individuals can encounter infectious individuals (I) and be reinfected leading to an acceleration into the infectious class at ratep

β

EI. The infected subpopulation is diminished when individuals recover from TB due to treatment at raterand disease induced death rate

μ

d. Finally, the recovered sub-population (R) is generated by recovery of infected individuals (at rater). The natural death rate decrease all classes at the same rate via the background mortality parameter

μ

. Though not necessary for our purposes, a more detailed description of the model and parameters can be found in[15].

3. Backward bifurcation

Without going into detailed computation (see[16]forR₀computation), the basic reproduction number of the model is given as

R0= k

βπ

μ(

^k+

μ)(μ

^+r+

μ

d

)

^.

Fig. 2. (a) Illustration of backward bifurcation when all the parameters inR0are fixed at constant values but the aggregatedR0is nonetheless varied. Parame- ters used areπ^=10,μ^=0.016,r=2,μd=0.4,k=0.0005,p=0.2,β^=0.036.(b) Illustration of backward bifurcation when one parameterβ^inR0is varied.

Parameters used areπ^=10,μ^=0.016,r=2,μd=0.4,k=0.0005,p=0.2,β^∈{^0.025,0.175}^.

One can find the endemic equilibrium or equilibria ofEq. (1)by setting the right-hand expressions to zero and solving to find

(

^S∗,E^∗,R^∗

)

=

π

β

^I∗+

μ

^,

(μ

^+r+

μ

d

)

^I∗

p

β

^I∗+k ,rI^∗

μ

,

in terms of the number of infectivesI^∗, whereI^∗can be obtained by solving the quadratic expression:

f

(

^I∗

)

^{=c2I∗2+c1I∗+c0=0, (2)}

where

c₂ =p

β

²

(μ

^+r+

μ

d

)

^,

c₁ =

(μ

+k+

μ

^p

)(μ

+r+

μ

d

)β

−

β

^2p

π

, c0 =

μ(μ

+k

)(μ

+r+

μ

d

)(

¹−R0

)

.

We can summarize the endemic equilibria of modelEq. (1)through the following theorem:

Theorem 1.The model Eq.(1)has:

(i) A unique endemic equilibrium if c₁<0and c₀=0or the discriminant =c²₁−4c₂c₀=0, (ii) A unique endemic equilibrium if c₀<0,

(iii) Two endemic equilibria if c₁<0,c₀>0and c²₁−4c₂c₀>0, (iv) No endemic equilibrium if c₁>0and c₀>0.

Proof.It is easy to note that in polynomial(2)c₂is always positive andc₀>0 ifR₀<1.For Case (i) wherec₁<0 andR₀=1 (i.e.c₀=0) the quadratic equationf(I^∗) reduces toc₂I^∗+c₁=0 and in this case the modelEq. (1)will have a unique positive endemic equilibrium ifc₁<0 and no positive non-trivial equilibrium ifc₁≥0. From Case (ii) wherec₀<0 (that isR₀>1) a unique endemic equilibrium exist since there is only one change of signs according to Descartes rule of signs. However, for Case (iii) wherec₁<0 andR₀<1 we have exactly two changes of signs indicating existence of two non-trivial equilibria. For Case (iv) wherec₁>0 andR₀<1 we do not have any change of signs, thus no endemic equilibrium in such a case. Hence, from Case (iii) we conclude that we have a maximum of two endemic equilibria (I₁^∗_,2) whenR₀<1,c₁<0 andc²₁−4c₂c₀>0.The above equilibrium analysis suggests a possibility of backward bifurcation since two endemic equilibria exist as exhibited by Case (iii) whenR₀<1, which is actually a necessary criterion for the occurrence of backward bifurcation phenomena. For a rigorous proof of stability using Rourth–Hurwitz criterion we refer the reader to work therein[15].

The problem arises when retrieving these two non-trivial solutions

(

^I1^∗,2

)

as a function of the aggregate parameterR₀. A number of studies incorrectly plot the solutions

(

^I∗_1,2

)

simply by varyingR₀inEq. (2)that is, without varying any parameter within R₀. Such a practice leads to an incorrect backward bifurcation diagram yielding a simple (though incorrect) parabolic shape such as shown inFig. 2(a). The correct approach for drawing the bifurcation diagram requires first choosing a proper specific model bifurcation parameter to vary, say the transmission rate

β

. The bifurcation diagram can then be determined through varying this model bifurcation parameter. Once obtained, the bifurcation diagram can then be rescaled so that thex-axis is given in terms of the aggregated parameterR₀. For the particular model above, by varying the transmission rate

β

in the interval

β

∈{0.025, 0.175}

we obtainFig. 2(b). Using this approach the figure no longer appears parabolic in shape and has shifted to the left compared to Fig. 2(a). Another outstanding difference betweenFig. 2(a) and (b) is the gap between the bifurcation branches. From Fig.2(b)

Fig. 3. The red dashed figure is a plot ofRc1as a function ofpwhile the blue solid figure illustrate a plot of the correct backward bifurcation thresholdRc2as a function ofp. Parameters used areπ^=10,μ^=0.016,r=2,μd=0.4,k=0.0005,β^=0.0195.The plot ofRcas a function of reinfection is due to the fact thatRc

decreases as reinfectionpincreases.pminis the minimum value of exogenous reinfection that triggers backward bifurcation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the gap between the bifurcation curves is wider than forFig. 2(a). This implies that the endemic equilibrium predicted when

β

is varied is higher than for the case where all parameters are kept constant i.e., the incorrect approach. Also, one can observe that the critical value of the basic reproduction number denoted byRcwhere the backward bifurcation initiates (defined in more detail below) will be incorrect if estimated using the aggregated parameter method.

4. The backward bifurcation thresholdR_c

As seen in the bifurcation diagram ofFig. 2, there is a threshold quantityRcwhich is the value ofR₀where the two non-trivial endemic equilibria collide and annihilate each other, leaving only the disease free equilibria as the only stationary solution. For instance inFig. 2(b), the thresholdRc=0.38. IfR₀<Rcthen the only model equilibrium is the stable disease free equilibrium.

We now explore the two approaches for computingRc[10–14].

(i) Incorrect aggregated parameter approach.Recall that this approach wrongly assumes that all parameters inR₀ are kept constant whileR₀ may be varied to obtain the backward bifurcation diagram, as inFig. 2(a). If the discriminant ( = c²₁−4c₂c₀) ofEq. (2)is set to zero it is possible to obtain the critical pointR_c1. This is just the value ofR₀where stable and unstable endemic curves coincide, namely

Rc1=1− c²₁ 4c2

^,

where=

μ(μ

+k

)(μ

+r+

μ

d

)

. Using the above equation, it is possible to investigate how the parameters that induce backward bifurcation affectR_c1. For example, a plot ofR_c1as a function ofpis shown inFig. 3.

(ii) Correct approach.The correct value forRcis obtained by first selecting a specific bifurcation parameter of the model. For illustration we choose

β

. By setting the discriminant =0 and evaluating for the critical transmission rate

β

cyields

β

c=−

φ

^1±

φ

1²−4

φ

²

φ

⁰

2

φ

2 ,

where

φ

^2=p2

π

^2,

φ

^1=4p

π

^k

(μ

^+r+

μ

d

)

⁻²

(μ

^+k+

μ

^p

)(μ

^+r+

μ

d

)

^,

φ

0=

(μ

^+k+

μ

^p

)

²

(μ

^+r+

μ

d

)

²⁻⁴

μ

^p

(μ

^+r+

μ

d

)

²

(μ

^+k

)

^.

Replacing

β

^inR0with

β

^cyields Rc2=

k

μ(μ

+k

)(μ

+r+

μ

d

)

φ

1²−4

φ

²

φ

¹⁻

φ

¹

2

π

^p2

. A plot ofR_c2as a function ofpis shown onFig. 3.

In this study we have shown that there is a profound difference in the backward bifurcation characteristics when the parame- ters inR₀are fixed andR₀is wrongly varied, as compared to validly varying a true model bifurcation parameter. Using the wrong approach, for constant values ofR₀a parabolic shape is observed. However, when a true model parameter inR₀is varied this parabolic shape disappears. Moreover, if parameters inR₀are incorrectly fixed researchers may overestimate or underestimate the backward bifurcation thresholdRc, below whichR₀need to be reduced to eradicate the disease from the community. Since the ultimate goal of modeling is to give insight into disease dynamics knowing the correct value ofRcis important to public health. Thus, one has to be careful in plotting and calculating backward bifurcation, since if not done correctly errors can be in- troduced as in the studies[11–15]. We hope our correction on obtaining backward bifurcation in epidemic models will be useful for others investigating this interesting phenomena in the future.

Appendix A. Endemic equilibria in terms of force of infection

λ

^∗

We can also reformulate the modelEq. (1)with a frequency dependent transmission rate as in[11–14]where the endemic equilibrium is given in terms of the force of infection

λ

^∗=^β_N^I∗^∗.Thus, expressing the endemic equilibria in terms of force of infection we have

(

^{S∗,E∗,I∗,R∗}

)

⁼

π

(λ

^∗+

μ)

^,

πλ

^∗

(

^p

λ

^∗+

μ

^+k

)(λ

^∗+

μ)

^,

λ

^∗

π(

^p

λ

^∗+k

)

(

^p

λ

^∗+

μ

^+k

)(λ

^∗+

μ)(μ

^+r+

μ

d

)

^,

rI^∗

μ

,

where

λ

^∗can be obtained by solving the following equation

p

(λ

^∗

)

=A

λ

^∗2+B

λ

^∗+C=0, ^(A.1)

where A=

(μ

+r

)

^p, B=

μ

^p

(μ

+r+

μ

d

)

+

μ(μ

+r+

μ

d

)

+k

(μ

+r

)

−

μ

^p

β

, and C=

μ(μ

+k

)(μ

+r+

μ

d

)(

¹−R₀

)

. Note that

R0= k

β

(μ

^+k

)(μ

^+r+

μ

d

)

^.

Two scenario of a plot of solution ofEq. (A.1)is shown inFig. A.4.

There is a distinctly difference betweenFig. A.4(a) and (b) as exhibited by the variation in their shape.Fig. A.4(a) where all parameters inR₀are fixed to constant values represent incorrect approach of obtaining backward bifurcation and therefore depict a parabolic shape. In factFig. A.4(a) resembles backward bifurcation produced in[10,12–14]where all parameters inR₀ were kept constant. However,Fig. A.4(b) where

β

∈{4, 6} is varied is the correct approach of obtaining backward bifurcation and in fact the parabolic shape is lost. Moreover, there is a huge difference in the gap between bifurcation curves withFig. A.4(b) having wider gap thanFig. A.4(a). In general varying atleast one parameter inR₀, one allows the bifurcation curves to choose the colliding point but keeping parameters constant it is as if you have already determined the meeting point of the two bifurcation curves. Thus, keeping parameters constant inR0when obtaining backward bifurcation may result to either underestimating or overestimating backward bifurcation thresholdRc.

Appendix B. Computation of bifurcation coefficients

Making use of center manifold approach as described in[17]we prove thatp_min=

(

_μ^k

)(

^μ_μ^+k

)

indicated by the arrow on Fig. 3(a) act as a threshold that determine positivity of bifurcation coefficienta.

For simplification and understanding of the center manifold theorem it is convenient to transform the model variables of system(1)as follows:x₁=S,x₂=E,x₃=I,x₄=RandN=4

j=1x_j. Now lettingX=

(

^x1,x₂,x₃,x₄

)

^T (T denote transpose) the model system (1) can be written as ^dX_dt =F

(

^X

)

^whereF=

(

^f1,f2,f3,f4

)

^T.Hence we have

Fig. A.4. Backward bifurcation when endemic equilibria expressed in terms of force of infection. (a) Represent backward bifurcation when all the parameters in R0are fixed at constant values. Parameters used areμ=0.01,r=0.85,μd=0.1,k=0.002316,p=0.5,β=4.5.(b) Represent backward bifurcation when one parameter inR0is varied. Parameters used areμ=0.01,r=0.85,μd=0.1,k=0.002316,p=0.5,β∈{4,6}.

dx1

dt =

π

−

β

^x1x3−

μ

^x1= f1, dx2

dt =

β

^x1x3−p

β

^x2x3−

(μ

+k

)

^x2=f2, dx3

dt =p

β

^x2x3+kx2−

(μ

+r+

μ

d

)

^x3= f3, dx4

dt =rx3−

μ

^{x4=f4. (B.1)}

The Jacobian matrix of the system(B.1)evaluated at the disease free equilibriumP₀=

_π

μ,0,0,0 is obtained as

J=

⎛

⎜ ⎝

−

μ

^{0 −βπ}_μ ⁰

0 −

(μ

^+k

)

^βπ_μ ⁰

0 k −

(μ

^+r+

μ

d

)

⁰

0 0 r −

μ

⎞

⎟ ⎠

^.

AtR₀=1 suppose

β

is the bifurcation parameter, hence solving for

β

^fromR0=1 yields

β

^∗= ^μ(μ+k)(μ_kπ^+r+μd).With

β

=

β

^∗the

transformed system(B.1)has a simple eigenvalue with zero real part and all other eigenvalues are negative (i.e. has a hyperbolic equilibrium point). Thus, we can use the center manifold theory[17,18]to investigate dynamics of transformed system(B.1)near

β

=

β

^∗.It is possible to obtain the right eigenvectors ofJ

(

^P0

)|

β=β^∗which are denoted byw=

(

^w1,w2,w3,w4

)

^Twhere w1=−

βπ

^w3

μ

^{2 , w2=}

βπ

^w3

μ(μ

+k

)

^{, w3=w3>0, w4=}

rw₃

μ

^.

Similarly we can obtain the left eigenvectors denoted as

v

=

( v

1,

v

2,

v

3,

v

4

)

^Twhere

v

1=0,

v

2= k

v

3

μ

^+k,

v

3=

v

3>0,

v

4=0.

Now, we proceed in obtaining the associated bifurcation coefficients respectively denoted byaandbas described in Theorem 4.1 of[17]. As indicated in Theorem 4.1 of[17]if bifurcation coefficientsaandbare both non-negative then the system exhibits backward bifurcation where an unstable and stable non-trivial equilibria coexist with a stable disease free equilibrium.

Hence, a=

4

k,i,j=1

v

kw_iw_j

∂

^2fk

(

^0,0

)

∂

^xi

∂

^xj

=

v

1w2w3

∂

^2f1

(

^0,0

)

∂

^x1

∂

^x3 +

v

2w1w3

∂

^2f2

(

^0,0

)

∂

^x1

∂

^x3 +

v

2w2w3

∂

^2f2

(

^0,0

)

∂

^x2

∂

^x3 +

v

3w2w3

∂

^2f3

(

^0,0

)

∂

^x2

∂

^x3

=

v

3w3

πβ

^∗2

(μ

+k

)

²

p−

k

μ

μ

+k

μ

w₃,

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