# Equilibrium Analysis of a Yellow Fever Dynamical Model with Vaccination

Authors: Silvia Martorano Raimundo; Marcos Amaku; Eduardo Massad
Journal: Computational and Mathematical Methods in Medicine
 (2015)

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## Abstract

We propose an equilibrium analysis of a dynamical model of yellow fever transmission in the presence of a vaccine. The model considers both human and vector populations. We found thresholds parameters that affect the development of the disease and the infectious status of the human population in the presence of a vaccine whose protection may wane over time. In particular, we derived a threshold vaccination rate, above which the disease would be eradicated from the human population. We show that if the mortality rate of the mosquitoes is greater than a given threshold, then the disease is naturally (without intervention) eradicated from the population. In contrast, if the mortality rate of the mosquitoes is less than that threshold, then the disease is eradicated from the populations only when the growing rate of humans is less than another threshold; otherwise, the disease is eradicated only if the reproduction number of the infection after vaccination is less than 1. When this reproduction number is greater than 1, the disease will be eradicated from the human population if the vaccination rate is greater than a given threshold; otherwise, the disease will establish itself among humans, reaching a stable endemic equilibrium. The analysis presented in this paper can be useful, both to the better understanding of the disease dynamics and also for the planning of vaccination strategies.


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## Body

## 1. Introduction

Yellow fever (YF), a hemorrhagic fever caused by aFlavivirus, familyFlaviviridae[1, 2], is characterized by fever, chills, loss of appetite, nausea, muscle pains particularly in the back, and headaches [3]. There are more than 200,000 infections and 30,000 deaths every year [3]. About 90% of YF cases occur in Africa [4], and a billion people live in an area of the world where the disease is common [3]. It also affects tropical areas of South America, but not Asia [3, 5, 6]. The number of cases of yellow fever has been increasing in the last 30 years [3, 7], probably due to fewer people being immune, more people living in cities, people moving frequently, and changing climate [3]. The origin of the disease is Africa, from where it spread in South America through the slave trade in the 17th century [8, 9].The yellow fever virus was the first human virus discovered [10], and its family comprises approximately 70 viruses [2], most of which are transmitted by arthropod insects (hence the name arthropod borne viruses or arboviruses).A safe and effective vaccine against yellow fever exists and some countries require vaccinations for travelers [3]. In rare cases (less than one in 200,000 to 300,000 doses), the vaccination can cause yellow fever vaccine-associated viscerotropic disease (YEL-AVD), which is fatal in 60% of cases, probably due to the genetic morphology of the immune system. Another possible side effect is an infection of the nervous system, which occurs in one in 200,000 to 300,000 cases, causing yellow fever vaccine-associated neurotropic disease (YEL-AND), which can lead to meningoencephalitis, fatal in less than 5% of cases [6]. In some rare circumstances, however, the fatality rate of vaccine induced diseases can reach alarming proportions, as observed recently by Mascheretti et al. [11], who found 1 death per million doses applied in a Southeastern Brazilian region.In this paper, we propose an equilibrium analysis of a dynamical model of yellow fever transmission in the presence of a vaccine. Such a kind of analysis can be useful, both to the better understanding of the disease dynamics and also for the planning of vaccination strategies.


## 2. Model Formulation

The mathematical model described below addresses the transmission dynamics of an infectious agent in a homogeneous population in the presence of an imperfect vaccine. We consider a nonlinear system of ordinary differential equations involving the human and the vector—mosquitoes and their eggs—populations. The term “eggs” also includes the intermediate stages, such as larvae and pupae. It is also worth highlighting that the model proposed here is based on previous papers [12, 13], and we updated the model originally developed by Amaku et al., 2013 [14, 15], for the purpose of investigating the impact of vaccination on population. By including vaccination, in particular vaccines which may have serious adverse effects, the model may help the designing of realistic (from the cost and logistic point of view) vaccination strategies.All variables and parameters in the human system will carry the subscriptH, while those in the vector system will carry one of the subscriptsM(mosquitoes) orE(eggs). In our model the total human population, denoted byNH, is split into four subclasses which are susceptible humans (SH), vaccinated humans (VH), infected humans (IH), and recovered (and immune) humans (RH), so thatNH=SH+IH+RH+VH. The total vector population, which is formed by both total mosquitoes population, denoted byNM, and the total eggs population, denoted byNE, are split into susceptible mosquitoes (SM), infected and latent mosquitoes (LM), infected and infectious mosquitoes (IM), and noninfected eggs (SE), so thatNM=SM+LM+IMandNE=SE.A flow diagram of the model is depicted in Figure1, and the associated variables and parameters are described in Tables 1 and 2, respectively (values from references [16–22]).Table 1
Description of the variables of the yellow fever model (1).
Variable
Biological meaning
S
H
Density of susceptible humans
V
H
Density of vaccinated humans
I
H
Density of infected humans
R
H
Density of recovered  humans
N
H
Density of total human population
S
M
Density of uninfected mosquitoes
L
M
Density of latent mosquitoes
I
M
Density of infected mosquitoes
N
M
Density of total mosquitoes population
S
E
Density of uninfected aquatic forms*
N
E
Density of total aquatic forms*
*Those variables are called “aquatic” because mosquitoes spend the large period of their development history in the water [25].Table 2
Description of the parameters of the yellow fever model (1).
Parameter
Biological meaning
Value (baseline)
a
Average daily biting rate
3.0
b
Fraction of actually infective bites(+)
0.6
μ
H
Humans natural mortality rate
3.5× 10−5 days−1
r
H
Birth rate of humans
9.5× 10−5 days−1
k
H
Humans carrying capacity
5× 106
α
H
Yellow fever mortality in humans
3.5× 10−4 days−1
γ
H
Humans recovery rate
0.143 days−1
p
Susceptible eggs hatching rate
0.15 days−1
ω
H
Rate of waning of immunity induced by vaccination
0.1 days−1
υ
H
Vaccination rate
0.5
f
H
Vaccine efficacy
0.9
γ
M
Mosquitoes latency rate
0.143 days−1
μ
M
Mosquitoes natural mortality rate
0.09 days−1
r
M
Oviposition rate
50 days−1
k
E
Aquatic carrying capacity
9.8× 107
μ
E
Aquatic natural mortality rate
0.1 days−1
c
A. aegypti susceptibility to yellow fever(&)
0.8
c
S
Climatic factor(#)
0.07
(
+
)Probability that an infective bite generates a new infection in humans.
(&)Probability that a new infection in the mosquito is generated when it bites an infective host.
(#)
Ad hoc parameter that modulates seasonality.Figure 1
Schematic diagram of the yellow fever model (1).The model supposes a homogeneous mixing of human and mosquito population based on the idea that the mosquito has a human biting habit, so that each mosquito bite has an equal probability of transmitting the virus to the susceptible human in the population or acquiring infection from an infected human. The equations are derived based on the fact that, in presence of the yellow fever in the population, both mosquitoes and humans can infect each other upon contact. While an infected mosquito remains infected until death, it is assumed that infected humans can recover from the disease (see [23]). We define a logistic recruitment rate of humans, mosquitoes, and eggs, and all new born humans and newly emerged mosquitoes are susceptible (no vertical transmission; see [23]). Susceptible humans become infected through the bite by an infected mosquito and the susceptible mosquitoes become latent infected as result of biting infectious humans. Upon acquiring infection, the susceptible individuals move into the infected compartment. The incidence of new infections is given by the standard incidence (see [24, pp. 602]).Deaths can occur amongst the human population, mosquitoes, and eggs, naturally. In contrast, in the presence of the yellow fever, the human population can either die due to the additional effects of the disease or recover. It is also assumed that recovered human individuals acquire immunity against reinfection, so that they do not acquire yellow fever for a second time.Although there is a vaccine for yellow fever, it is expected that it is imperfect; that is, it does not offer 100% protection against infection in all population. Thus, it is instructive to assess the potential impact of an imperfect yellow fever vaccine.After the duration of protection wanes down, the vaccinated individuals,VH, move to the susceptible class, SH, and they may then acquire a new infection. Hence, the vaccinated population is decreased by the waning of vaccine-induced immunity and by natural death.Combining the above formulation and assumption, it follows that the model for the transmission dynamics of the yellow fever disease in the presence of an imperfect vaccine is given by the following system of nonlinear ordinary differential equations:(1)dSHdt=-abIMSHNH-μHSH+rHNH1-NHκH-fHυHSH+ωHVH,dVHdt=fHυHSH-μH+ωHVH,dIHdt=abIMSHNH-μH+αH+γHIH,dRHdt=γHIH-μHRH,dNHdt=rHNH1-NHκH-μHNH-αHIH,dSMdt=pcSSE-μM+acIHNHSM,dLMdt=acSMIHNH-γM+μMLM,dIMdt=γMLM-μMIM,dNMdt=pcSSE-μMNM,dSEdt=rMNM1-SEκE-μE+pcSSEwith the conditions NH=SH+VH+IH+RH, NM=SM+LM+IM, and NE=SE and the initial conditions SH(0)≥0, VH(0)≥0, IH(0)≥0, RH(0)≥0, SM(0)≥0, LM(0)≥0, IM(0)≥0, NM(0)≥0, and SE(0)≥0. Note that the transmission from mosquitoes to humans is given by ab(IM/NH) (also called “force of infection”) and from humans to mosquitoes is given by ac(IH/NH). This means that the transmission from mosquito to humans depends on the number of infective mosquitoes, but the transmission from human to mosquitoes depends on the density of infective humans.Since system (1) models human, mosquito, and eggs populations, it is assumed that all variables in the system are nonnegative. This assumption yields the epidemiologically feasible domain: (2)Ω=SH,VH,IH,RH,SM,LM,IM,SE∈R8:SH≥0,VH≥0,IH≥0,RH≥0,0≤SH+VH+IH+RH≤NH,SM≥0,LM≥0,IM≥0,0≤SM+LM+IM≤NM,SE=NE≥0.Since the right-hand sides of equations of system (1) and their partial derivatives are continuous in Ω, we will use the techniques described in [21] that there exist a unique solution SH(t)+VH(t)+IH(t)+RH(t)=NH(t), SM+LM(t)+IM(t)=NM(t), and SE(t)=NE(t), for all t≥0, satisfying the initial conditions specified within Ω,SH(0)+VH(0)+IH(0)+RH(0)=NH(0), SM(0)+LM(0)+IM(0)=NM(0), and SE(0)=NE(0), at time t=0. It can also be verified that the given initial conditions make sure that NH≥0. Thus, the total population NH remains positive and bounded for all finite time t>0. Similar arguments can be applied to both mosquito and eggs equations with corresponding expressions. Therefore, all solutions of the model with initial conditions in Ω remain in Ω for all t≥0, the region Ω is positively invariant with respect to model (1), and its solutions are considered epidemiologically and mathematically well posed in Ω.


## 3. The Existence of Equilibria

Our next result concerns the existence of equilibrium points of system (1) that are biologically feasible. Thus, we will find the equilibrium points of system (1) in the region Ω by setting right hand side of all equations in it as equal to zero. First of all, we will seek the conditions for the existence of the equilibria of system (1) which are biologically feasible.From the second and fourth equations of (1) with the right-hand side equal to zero, it can be seen that the equilibrium points must satisfy, respectively, the following relations:(3)VH*=ρvacNH*1-μH+γHμHIH*NH*,(4)RH*=γHμHIH*,where (5)0<ρvac=fHνHfHνH+μH+ωH<1,(6)VH*>0⟺IH*NH*<μHμH+γH<1⟹IH*<NH*.Therefore, VH*>0 is always satisfied.Substituting (3) and (4) into the third equation of system (1), we obtain(7)IM*=μH+αH+γHIH*ab1-ρvac1-1+γH/μH(IH*/NH*),(8)IM*>0⟺IH*NH*<μHμH+γH<1⟹IH*<NH*.Hence, IM*>0 is also always satisfied.From eighth and ninth equations of system (1), we obtain (9)LM*=μMγMIM*,(10)NM*=pcSμMSE*.From tenth equation of system (1), we get either SE*=0 or(11)SE*=kE1-μMμMthres,with μMthres=rMpcS/(μE+pcS). From expression (11) it follows that(12)SE*>0⟺μM<μMthres.Later we will see that for SE*=0 only the trivial equilibrium can exist, while for SE*>0, given by (11), both the trivial and the nontrivial equilibrium may exist for system (1).Substituting (9) into the seventh equation of system (1), we obtain(13)IM*=acNM*(IH*/NH*)ac(IH*/NH*)+μM1+μM/γM.From (7) and (13) we obtain either IH*=0 or (14)IH*=a2bcγM1-ρvacNM*-μH+αH+γHγM+μMμMNH*·a2bcγM1+γHμHγM+μM1-ρvacNM*NH*NM*NH*+acμH+αH+γHγM+μM-1.From expression (14), it should be noted that, for μM<μMthres, IH*>0 whenever(15)a2bcγMNM*μMμH+αH+γHγM+μMNH*>11-ρvac,or(16)0<NH*<a2bcpcSγMkEμM2μH+αH+γHγM+μMiiiiiii·1-μMμMthres1-ρvac=N1.On the other hand, from fifth equation of system (1), we also get(17)IH*=rHαH1-NH*kH-μHαHNH*,and IH*>0 whenever (18)0<NH*<kHrH-μHrH=N2withrH>μH.From expressions (16) and (18), one can note, however, that, N2 is the maximum value of NH*, so it follows that N1<N2 and IH*>0 if and only if condition (18) holds. Furthermore, for IH*>0 the system (1) reaches an endemic equilibrium point.In contrast, from (4), (9), (13), and (18), IH*=0 leads to (19)NH*=N2,IM*=0,LM*=0,VH*=ρvacNH*,SH*=NH*-VH*,RH*=0,where SE*=NE*≠0 and SM*=NM*≠0 are also given by expressions (10) and (11), respectively. Therefore, for IH*=0 only the trivial equilibrium exists for system (1). To be more specific, for SE*=NE*≠0 and SM*=NM*≠0 the trivial equilibrium is given by the densities of humans and vectors, while for SE*=NE*=0 and SM*=NM*=0 the trivial equilibrium is given only by the density of humans.Now, by substituting (14) in (18), it can be shown that the nontrivial equilibria of the model satisfy the following quadratic equation (in terms of NH*):(20)P(NH*)=Q2NH*2+Q1NH*+Q0=0,where (21)Q2=acθrH,Q1=θ1-ρvac1+γHμHkHrH-μHμMR0-Ψ,Q0=τkHμH+γHrHthres-rH,with(22)τ=a2bcNM*1-ρvac,θ=μH+γH+αH1+μMγM,Ψ=αH-acμH(1-ρvac)μH+γHμM,(23)rHthres=μHμH+αH+γHμH+γH,(24)R0=a2bcpcSkErHμM2θkHrH-μH1-μMμMthres,with μMthres satisfying the condition given by (12).The positive endemic equilibrium of model (1) is obtained by solving for NH* from the quadratic (20) and substituting the results (positive values of NH*) into the expressions that give the coordinates of equilibrium point.First, it is straightforward to note thatrHthres given by (23) and θ given by (22) are positive. Moreover, under our assumption Ψ<0 and Q1 is also positive. Clearly, the coefficient Q2 of the quadratic equation (20) is always positive.Also, note that the coefficientQ0 is positive either if τ>0 and rH<rHthres or τ<0 and rH>rHthres. If τ<0 then μM>μMthres and the positive endemic equilibrium does not exist (see (12)) for system (1). Later we will see that, for μM>μMthres, the only equilibrium biologically feasible and stable is the equilibrium given by P0H.Therefore,Q0>0 if and only if τ>0 (i.e., μM<μMthres) and rH<rHthres. In this case, if Q1>0, the quadratic equation (20) does not have a positive solution. It follows then that system (1) does not have positive endemic equilibria whenever rH<rHthres. In this case, note that the only equilibrium biologically feasible and stable for system (1) is the trivial equilibrium given by P0HM (see (19)).Otherwise,τ>0 (i.e., μM<μMthres) but rH>rHthres; then Q0<0, so there is a unique positive real solution of the quadratic equation (20), for any value of Q1. Hence, for rH>μH and μM<μMthres, the system (1) has a unique positive endemic equilibrium when rH>rHthres.IfQ0=0 (i.e., rH=rHthres) and Q1<0 or Q12-4Q1Q0=0, then there is a unique positive real solution for the quadratic equation (20), so the system (1) has a unique positive endemic equilibrium when rH=rHthres.Therefore, forrH>μH, the above analysis indicates the possibility of the existence of the following positive equilibrium points for system (1):(i)
a disease-free equilibrium (DFE) defined only by human population,P0H = SH0,VH0,0,0,NH0,0,0,0,0,0;(ii)
a disease-free equilibrium (DFE) defined by both human and vector populations defined byP0HM = SH0,VH0,0,0,NH0,SM0,0,0,NM0,SE0;(iii)
an endemic equilibrium (EE) point biologically feasible given byP1HM = SH*,IH*,RH*,VH*,NH*,SM*,LM*,IM*,NM*,SE*.Quite apart from this, the existence of the positive equilibrium points for the system (1) can be also summarized as follows.Theorem 1.
Assuming thatrH>μH holds, model (1) has a unique disease-free equilibrium P0H whenever μM>μMthres. Otherwise, if μM<μMthres then model (1) has the trivial equilibrium P0HM and the endemic equilibrium P1HM. For rH<rHthres, the unique equilibrium that exists is given by the trivial equilibrium P0HM. Otherwise, for rH>rHthres, both the trivial equilibrium, P0HM, and the endemic equilibrium, P1HM, exist.Having found the scenarios in which there exist the equilibria for the system (1), it is instructive to analyse whether or not these equilibria are stable under any of these scenarios. Furthermore, together with the threshold vaccination rate, νHC, and the reproduction number, Rvac, we will see that each scenario can be used as a check for the existence, the uniqueness, and the stability of all equilibria. This is explored below forrH>μH.


### 3.1. Disease-Free Equilibria

In the absence of the disease, that is,RH0=IH0=IM0=LM0=0 and for rH>μH, model (1) has two disease-free equilibria given by P0H and P0HM. Thus, if μM<μMthres, then the disease-free equilibrium is given by P0HM:(25)SH0=kHrH-μHrH1-ρvac,VH0=ρvacNH0,NH0=kHrH-μHrH,SM0=NM0=pcSμMSE0,SE0=kE1-μMμMthres,whereρvacis defined by expression (5). In contrast, if μM>μMthres, then the only equilibrium biologically viable is P0H, which is also given by (25), but with SM(0)=NM(0)=SE(0)=NE(0)=0.To establish the stability of both trivial equilibrium, the Jacobian of the system (1) is computed and evaluated at both P0H and P0HM. We will discuss the properties of both trivial equilibrium points making an elementary row-transformation for the Jacobian matrix.Evaluating the system’s Jacobian atP0H, the local stability of P0H is straightforward determined by the six eigenvalues given by τ1=τ2=-μM, τ3=-μH, τ4=-μM+γM, τ5=-μH+αH+γH, and τ6=μH-rH<0 since rH>μH. The other eigenvalues are expressed as the roots of the following submatrix: (26)MP0H=-fHνH+μHωH00fHνH-ωH+μH0000-μMpcS00rM-μE+pcS,where(27)MH0=-fHνH+μHωHfHνH-ωH+μH,MH1=-μMpcSrM-μE+pcS.It is easy to verify that both the traces of the matricestr⁡(MH0) and tr⁡(MH1) are always negative. Moreover, the determinant of the matrix, det⁡(MH0), is always positive, but det⁡(MH1) is positive if and only if μM>μMthres. In other words, it means that the four eigenvalues of matrix MP0H are either negative or have negative real part whenever μM>μMthres.Therefore, all the eigenvalues of the characteristic equation associated with the system (1) at P0H have negative real parts if and only if μM>μMthres and rH>μH. We state then the following result.Lemma 2.
ForrH>μH, the disease-free equilibrium P0H of model (1) is globally asymptotically stable if μM>μMthres. Otherwise, P0H is unstable.The stability of the disease-free equilibriumP0HM is now examined by linearizing the system (1) around P0HM. The characteristic equation of the Jacobian matrix of the system (1) at P0HM is given by(28)Λ1-μM-λ-μH+αH+γH-λ-μM+γM-λ+χ1MP0HM=0,where(29)Λ1=-μH-λ-μM-λμH-rH-λ,(30)χ1=a2bcγMSH0SM0NH02,with SH0,SM0, and NH0 given by (25) and(31)MP0HM=-fHνH+μHωH00fHνH-ωH+μH0000-μMpcS00rM1-SE0kE-μE+pcS-rMkENM0,where MHM0=MH0 (see (27)) and (32)MHM1=-μMpcSrM1-SE0kE-μE-pcS-rMkENM0.From (29) it is easy to verify that λ1=-μH, λ2=-μM, λ3=μH-rH<0, since rH>μH. Moreover, from (27) and (32), both the traces of the matrices tr⁡(MHM0) and tr⁡(MHM1) are always negative; the determinant of the matrix, det⁡(MHM0), is also always positive, but det⁡(MHM1) is positive if and only if μM<μMthres. Therefore, the four eigenvalues of matrix MP0HM are either negative or have negative real parts if and only if μM<μMthres.The other three eigenvalues are associated with the third degree equation in (28) given by(33)-μM-λ-μH+αH+γH-λ-μM+γM-λ+χ1=0.It can be seen after some calculations that the polynomial (33) is equivalent to(34)λ3+g2λ2+g1λ+g0=0,where (35)g2=μM+μH+αH+γH+μM+γM,g1=μMμH+αH+γH+μM+γM+μH+αH+γHμM+γM,g0=μMμH+αH+γHμM+γM1-Rvac,(36)Rvac=R01-ρvac,with ρvac<1 given by (5); R0 is the threshold quantity or the basic reproductive number of the diseases defined by (23) and Rvac<R0. It is worth remembering that P0HM given by (25) exists whenever rH>μH and μM<μMthres, so R0>0 and Rvac>0.By using the Routh-Hurwitz criteria for the polynomial (33), it follows that g2>0, g1>0, and g0>0 if and only if Rvac<1 and g1g2-g0>0. Therefore, the polynomial (33) has negative (or has negative real part) roots if Rvac<1.Hence, forrH>μH, all the eigenvalues of the characteristic equation associated with the system (1) at P0HM are negative or have negative real parts if and only if μM<μMthres and Rvac<1. Hence, the disease-free equilibrium P0HM is locally asymptotically stable when Rvac<1 and μM<μMthres.It is worth remembering that ifμM<μMthres and rH<rHthres there are no positive solutions of the quadratic equation (20) and thus there is no endemic equilibrium of system (1). However, by (19), it follows that the disease-free equilibrium P0HM is the only equilibrium point that exists when rH<rHthres. Otherwise, for rH>rHthres, there is a unique positive real solution of the quadratic equation (20) which indicates the possibility of a unique positive endemic equilibrium given by P1HM. On the other hand, from (25) it can be also noted that the existence of the disease-free equilibrium P0HM does not depend on rHthres, and thus P0HM could also exist in this case. Therefore, the disease-free equilibrium P0HM coexists with an endemic equilibrium P1HM whenever rH>rHthres.It follows from above analyses that the disease-free equilibriumP0HM is a unique equilibrium which is locally asymptotically stable whenever μM<μMthres and rH<rHthres. In such a scenario, disease elimination would depend upon the birth rate of humans, rH, and the mosquitoes natural mortality rateμM. Otherwise, if μM<μMthres and rH>rHthres, then P0HM is locally asymptotically stable when Rvac<1. The epidemiological implication of this is that the requirements of μM<μMthres and rH>rHthres are, although necessary, no longer sufficient for disease elimination. The disease elimination would depend upon the vaccination coverage (υH) and thus will eliminate the disease from the population if the usage of an imperfect vaccine results in making (and keeping) Rvac<1.For the case whenμM<μMthres, for rH<rHthres the quadratic equation has no positive solution. Hence, system (1) has no positive solution (endemic equilibrium) for rH<rHthres. As the only existing equilibrium for μM<μMthres is the trivial one, P0HM, then we can conjecture that, for rH<rHthres, P0HM is the only possible equilibrium for system (1), being, therefore, stable. On the other hand, for rH>rHthres the quadratic equation has a positive solution and hence, system (1) shows an endemic equilibrium, P1HM. As the existence of P0HM, however, does not depend on rHthres (see (25)), we can state that P0HM also exists for rH>rHthres. In addition, if Rvac<1, then P0HM is stable. Note that, in all cases, the mosquito’s mortality rate μM is the critical parameter, in the absence of vaccination, determining the stability of the disease.How about the case whenRvac>1, P0HM becomes unstable? In this case, the critical value of vaccination is given by (38); that is, if υH>υHthres, then P0HM is stable. In contrast, if υH<υHthres, then P0HM is unstable and P1HM becomes stable.We establish then the following stability result for the system (1).Lemma 3.
ForrH>μH and μM<μMthres the disease-free equilibrium of the model (1), P0HM, exists and it is locally asymptotically stable if (a)
Rvac<1 and rH>rHthres,(b)
rH<rHthres,(c)
otherwise,P0HM is unstable.The quantityRvac given by (36) is called the vaccinated reproduction number, since it represents the expected average number of new infections produced by a single infective when introduced into a human community where a fraction of the susceptible population has been vaccinated. For a disease in which the susceptible population is vaccinated it has been demonstrated that Rvac, which is the basic reproduction number R0 [26] modified by vaccination, must be reduced below one in order to ensure that the disease dies out [27]. If there is no vaccination, then Rvac=R0. Therefore, the aim of the vaccination must be to reduce Rvac below one and to provide prolonged protection against the infection.Now, expression (36) for Rvac can be written in terms of SM0, NH0, and SH0 such as(37)Rvac=a2bcγM(SM0/NH0)μH+αH+γHμM+γMμMSH0NH0.SettingRvac=1, and solving (37) for υH, the threshold vaccination rate is found to be(38)υHthres=μH+ωHfHR0-1.It is worth remembering thatR0>0 whenever rH>μH and μM<μMthres (see (24)). More than that, R0<1 implies Rvac<1 (see (36)), according to Lemma 3. P0HM could be stable. Therefore, the disease eradication can be attainable independently of vaccination coverage, that is, even when υHthres=0.In contrast, ifR0>1, υHthres is positive but we also have either Rvac>1 or Rvac<1. In this situation, the disease could be eliminated whenever υH>υHthres and P0HM is globally asymptotically stable. Otherwise, when υH<υHthres, the use of an imperfect vaccine will fail to eliminate the disease from the community, P0HM is unstable, the disease will persist in the community, and P1HM becomes globally asymptotically stable.Finally, concerning the analysis of the polynomial (20), Lemmas 2 and 3, we can establish the following conjecture for the system (1).Conjecture 4.
ForrH>μH the disease-free equilibrium P0H is globally asymptotically stable if μM>μMthres. If μM<μMthres, P0H becomes unstable and for rH<rHthres, P0HM is globally asymptotically stable equilibrium point. If rH>rHthres, then P0HM is globally asymptotically stable if υH>υHthres. Otherwise, if υH<υHthres, then P0HM becomes unstable and the globally asymptotically equilibrium point stable is then given by P1HM.The stability of the equilibrium points can then be summarized in Table3.Table 3
Stability of the equilibrium points of the model (1) for rH>μH.
Conditions
Equilibrium point
μ
M
>
μ
M
thres
P
0
H
M
=
S
H
0
,
V
H
0
,
0
,
0
,
0
,
0
,
0,0 stable
μ
M
<
μ
M
thres, rH<rHthres
P
0
H
M
=
S
H
0
,
V
H
0
,
0,0
,
0,0
,
N
M
*
,
S
E
* stable
μ
M
<
μ
M
thres, rH>rHthres, υH>υHthres
P
0
H
M
=
S
H
0
,
V
H
0
,
0,0
,
0,0
,
N
M
*
,
S
E
* stable
μ
M
<
μ
M
thres, rH>rHthres, υH<υHthres
P
1
H
M
=
V
H
*
,
I
H
*
,
R
H
*
,
L
M
*
,
I
M
*
,
N
M
*
,
S
E
*  stableAs it will be explained in detail below, it is important to point out that by considering our model (1) without vaccine, that is, taking νH=fH=ωH=0, we have ρvac=0; the first term of expression (15) defines then the basic reproduction number used to gauge the severity of an epidemic; it will be denoted by R0. Therefore, according to inequality (15), IH*>0 whenever (39)ρvac<1-1R0.If R0<1, ρvac<0, from (36), Rvac>1. If R0=1, ρvac=0, from (36), Rvac=R0. If R0>1, 0<ρvac<1, from (36), either Rvac>1 or R0<1.From (36) if vaccine has no effect, ωH→∞ (immune protection duration, D=1/ωH→0), then Rvac increases (rH<rHthres or rH>rHthres). If the vaccine induces lifelong immunity, then ωH=0 (D→∞), 0<ρvac≤1, Rvac>1, or R0<1 (see (36)).


## 3.1. Disease-Free Equilibria

In the absence of the disease, that is,RH0=IH0=IM0=LM0=0 and for rH>μH, model (1) has two disease-free equilibria given by P0H and P0HM. Thus, if μM<μMthres, then the disease-free equilibrium is given by P0HM:(25)SH0=kHrH-μHrH1-ρvac,VH0=ρvacNH0,NH0=kHrH-μHrH,SM0=NM0=pcSμMSE0,SE0=kE1-μMμMthres,whereρvacis defined by expression (5). In contrast, if μM>μMthres, then the only equilibrium biologically viable is P0H, which is also given by (25), but with SM(0)=NM(0)=SE(0)=NE(0)=0.To establish the stability of both trivial equilibrium, the Jacobian of the system (1) is computed and evaluated at both P0H and P0HM. We will discuss the properties of both trivial equilibrium points making an elementary row-transformation for the Jacobian matrix.Evaluating the system’s Jacobian atP0H, the local stability of P0H is straightforward determined by the six eigenvalues given by τ1=τ2=-μM, τ3=-μH, τ4=-μM+γM, τ5=-μH+αH+γH, and τ6=μH-rH<0 since rH>μH. The other eigenvalues are expressed as the roots of the following submatrix: (26)MP0H=-fHνH+μHωH00fHνH-ωH+μH0000-μMpcS00rM-μE+pcS,where(27)MH0=-fHνH+μHωHfHνH-ωH+μH,MH1=-μMpcSrM-μE+pcS.It is easy to verify that both the traces of the matricestr⁡(MH0) and tr⁡(MH1) are always negative. Moreover, the determinant of the matrix, det⁡(MH0), is always positive, but det⁡(MH1) is positive if and only if μM>μMthres. In other words, it means that the four eigenvalues of matrix MP0H are either negative or have negative real part whenever μM>μMthres.Therefore, all the eigenvalues of the characteristic equation associated with the system (1) at P0H have negative real parts if and only if μM>μMthres and rH>μH. We state then the following result.Lemma 2.
ForrH>μH, the disease-free equilibrium P0H of model (1) is globally asymptotically stable if μM>μMthres. Otherwise, P0H is unstable.The stability of the disease-free equilibriumP0HM is now examined by linearizing the system (1) around P0HM. The characteristic equation of the Jacobian matrix of the system (1) at P0HM is given by(28)Λ1-μM-λ-μH+αH+γH-λ-μM+γM-λ+χ1MP0HM=0,where(29)Λ1=-μH-λ-μM-λμH-rH-λ,(30)χ1=a2bcγMSH0SM0NH02,with SH0,SM0, and NH0 given by (25) and(31)MP0HM=-fHνH+μHωH00fHνH-ωH+μH0000-μMpcS00rM1-SE0kE-μE+pcS-rMkENM0,where MHM0=MH0 (see (27)) and (32)MHM1=-μMpcSrM1-SE0kE-μE-pcS-rMkENM0.From (29) it is easy to verify that λ1=-μH, λ2=-μM, λ3=μH-rH<0, since rH>μH. Moreover, from (27) and (32), both the traces of the matrices tr⁡(MHM0) and tr⁡(MHM1) are always negative; the determinant of the matrix, det⁡(MHM0), is also always positive, but det⁡(MHM1) is positive if and only if μM<μMthres. Therefore, the four eigenvalues of matrix MP0HM are either negative or have negative real parts if and only if μM<μMthres.The other three eigenvalues are associated with the third degree equation in (28) given by(33)-μM-λ-μH+αH+γH-λ-μM+γM-λ+χ1=0.It can be seen after some calculations that the polynomial (33) is equivalent to(34)λ3+g2λ2+g1λ+g0=0,where (35)g2=μM+μH+αH+γH+μM+γM,g1=μMμH+αH+γH+μM+γM+μH+αH+γHμM+γM,g0=μMμH+αH+γHμM+γM1-Rvac,(36)Rvac=R01-ρvac,with ρvac<1 given by (5); R0 is the threshold quantity or the basic reproductive number of the diseases defined by (23) and Rvac<R0. It is worth remembering that P0HM given by (25) exists whenever rH>μH and μM<μMthres, so R0>0 and Rvac>0.By using the Routh-Hurwitz criteria for the polynomial (33), it follows that g2>0, g1>0, and g0>0 if and only if Rvac<1 and g1g2-g0>0. Therefore, the polynomial (33) has negative (or has negative real part) roots if Rvac<1.Hence, forrH>μH, all the eigenvalues of the characteristic equation associated with the system (1) at P0HM are negative or have negative real parts if and only if μM<μMthres and Rvac<1. Hence, the disease-free equilibrium P0HM is locally asymptotically stable when Rvac<1 and μM<μMthres.It is worth remembering that ifμM<μMthres and rH<rHthres there are no positive solutions of the quadratic equation (20) and thus there is no endemic equilibrium of system (1). However, by (19), it follows that the disease-free equilibrium P0HM is the only equilibrium point that exists when rH<rHthres. Otherwise, for rH>rHthres, there is a unique positive real solution of the quadratic equation (20) which indicates the possibility of a unique positive endemic equilibrium given by P1HM. On the other hand, from (25) it can be also noted that the existence of the disease-free equilibrium P0HM does not depend on rHthres, and thus P0HM could also exist in this case. Therefore, the disease-free equilibrium P0HM coexists with an endemic equilibrium P1HM whenever rH>rHthres.It follows from above analyses that the disease-free equilibriumP0HM is a unique equilibrium which is locally asymptotically stable whenever μM<μMthres and rH<rHthres. In such a scenario, disease elimination would depend upon the birth rate of humans, rH, and the mosquitoes natural mortality rateμM. Otherwise, if μM<μMthres and rH>rHthres, then P0HM is locally asymptotically stable when Rvac<1. The epidemiological implication of this is that the requirements of μM<μMthres and rH>rHthres are, although necessary, no longer sufficient for disease elimination. The disease elimination would depend upon the vaccination coverage (υH) and thus will eliminate the disease from the population if the usage of an imperfect vaccine results in making (and keeping) Rvac<1.For the case whenμM<μMthres, for rH<rHthres the quadratic equation has no positive solution. Hence, system (1) has no positive solution (endemic equilibrium) for rH<rHthres. As the only existing equilibrium for μM<μMthres is the trivial one, P0HM, then we can conjecture that, for rH<rHthres, P0HM is the only possible equilibrium for system (1), being, therefore, stable. On the other hand, for rH>rHthres the quadratic equation has a positive solution and hence, system (1) shows an endemic equilibrium, P1HM. As the existence of P0HM, however, does not depend on rHthres (see (25)), we can state that P0HM also exists for rH>rHthres. In addition, if Rvac<1, then P0HM is stable. Note that, in all cases, the mosquito’s mortality rate μM is the critical parameter, in the absence of vaccination, determining the stability of the disease.How about the case whenRvac>1, P0HM becomes unstable? In this case, the critical value of vaccination is given by (38); that is, if υH>υHthres, then P0HM is stable. In contrast, if υH<υHthres, then P0HM is unstable and P1HM becomes stable.We establish then the following stability result for the system (1).Lemma 3.
ForrH>μH and μM<μMthres the disease-free equilibrium of the model (1), P0HM, exists and it is locally asymptotically stable if (a)
Rvac<1 and rH>rHthres,(b)
rH<rHthres,(c)
otherwise,P0HM is unstable.The quantityRvac given by (36) is called the vaccinated reproduction number, since it represents the expected average number of new infections produced by a single infective when introduced into a human community where a fraction of the susceptible population has been vaccinated. For a disease in which the susceptible population is vaccinated it has been demonstrated that Rvac, which is the basic reproduction number R0 [26] modified by vaccination, must be reduced below one in order to ensure that the disease dies out [27]. If there is no vaccination, then Rvac=R0. Therefore, the aim of the vaccination must be to reduce Rvac below one and to provide prolonged protection against the infection.Now, expression (36) for Rvac can be written in terms of SM0, NH0, and SH0 such as(37)Rvac=a2bcγM(SM0/NH0)μH+αH+γHμM+γMμMSH0NH0.SettingRvac=1, and solving (37) for υH, the threshold vaccination rate is found to be(38)υHthres=μH+ωHfHR0-1.It is worth remembering thatR0>0 whenever rH>μH and μM<μMthres (see (24)). More than that, R0<1 implies Rvac<1 (see (36)), according to Lemma 3. P0HM could be stable. Therefore, the disease eradication can be attainable independently of vaccination coverage, that is, even when υHthres=0.In contrast, ifR0>1, υHthres is positive but we also have either Rvac>1 or Rvac<1. In this situation, the disease could be eliminated whenever υH>υHthres and P0HM is globally asymptotically stable. Otherwise, when υH<υHthres, the use of an imperfect vaccine will fail to eliminate the disease from the community, P0HM is unstable, the disease will persist in the community, and P1HM becomes globally asymptotically stable.Finally, concerning the analysis of the polynomial (20), Lemmas 2 and 3, we can establish the following conjecture for the system (1).Conjecture 4.
ForrH>μH the disease-free equilibrium P0H is globally asymptotically stable if μM>μMthres. If μM<μMthres, P0H becomes unstable and for rH<rHthres, P0HM is globally asymptotically stable equilibrium point. If rH>rHthres, then P0HM is globally asymptotically stable if υH>υHthres. Otherwise, if υH<υHthres, then P0HM becomes unstable and the globally asymptotically equilibrium point stable is then given by P1HM.The stability of the equilibrium points can then be summarized in Table3.Table 3
Stability of the equilibrium points of the model (1) for rH>μH.
Conditions
Equilibrium point
μ
M
>
μ
M
thres
P
0
H
M
=
S
H
0
,
V
H
0
,
0
,
0
,
0
,
0
,
0,0 stable
μ
M
<
μ
M
thres, rH<rHthres
P
0
H
M
=
S
H
0
,
V
H
0
,
0,0
,
0,0
,
N
M
*
,
S
E
* stable
μ
M
<
μ
M
thres, rH>rHthres, υH>υHthres
P
0
H
M
=
S
H
0
,
V
H
0
,
0,0
,
0,0
,
N
M
*
,
S
E
* stable
μ
M
<
μ
M
thres, rH>rHthres, υH<υHthres
P
1
H
M
=
V
H
*
,
I
H
*
,
R
H
*
,
L
M
*
,
I
M
*
,
N
M
*
,
S
E
*  stableAs it will be explained in detail below, it is important to point out that by considering our model (1) without vaccine, that is, taking νH=fH=ωH=0, we have ρvac=0; the first term of expression (15) defines then the basic reproduction number used to gauge the severity of an epidemic; it will be denoted by R0. Therefore, according to inequality (15), IH*>0 whenever (39)ρvac<1-1R0.If R0<1, ρvac<0, from (36), Rvac>1. If R0=1, ρvac=0, from (36), Rvac=R0. If R0>1, 0<ρvac<1, from (36), either Rvac>1 or R0<1.From (36) if vaccine has no effect, ωH→∞ (immune protection duration, D=1/ωH→0), then Rvac increases (rH<rHthres or rH>rHthres). If the vaccine induces lifelong immunity, then ωH=0 (D→∞), 0<ρvac≤1, Rvac>1, or R0<1 (see (36)).


## 4. Numerical Analysis

The numerical analysis of the stability of the equilibria was done with the parameters of the model fixed at the baseline values indicated in Table2. We explore the implications of variable vaccination coverage υH, birth rate of humans rH, and mosquitoes natural mortality rate μM which are chosen for simulations purposes only, so we can illustrate our theoretical results. For the baseline parameters values in Table 2, we have rHthres=0.000035085, μMthres=4.751131, and υHthres=8.0117.Figures2 and 3 show the graph of (20), with P(NH*) plotted versus υH. Figure 2 shows the graph for μM<μMthres and rH<rHthres, with increasing values of υH. One can see two real negative roots of (20), which are not biologically viable for the system (1). Therefore, the unique equilibrium point that is biologically viable and locally asymptotically stable is given by P0HM (see (19)). In this case, the disease can be eradicated from the population. Figure 3 shows the graph for μM<μMthres and rH>rHthres, with increasing values of υH. One can see one positive and one negative real root of (20). Hence, the system (1) has a unique endemic equilibrium P1HM, and the disease persists at an endemic level. Parameter values used are as given in Table 2 (baseline values), except for υH.Figure 2
ForμM<μMthres and rH<rHthres, the quadratic equation P(NH*) is plotted versus υH, corresponding to two negative (which is not biologically relevant) real roots.Figure 3
ForμM<μMthres and rH>rHthres, the quadratic equation P(NH*) is plotted versus υH, corresponding to one negative (which is not biologically relevant) and one positive real roots.Figure4 shows the graph of (37) for μM<μMthres and rH>rHthres, with increasing values of υH. Parameter values used are as given in Table 2 (baseline values), except for υH. For υH>υHthres, Rvac<1; the disease is then eradicated from population (P0HM is locally asymptotically stable). For υH<υHthres, then Rvac>1; the disease persists into the population (P1HM is the locally asymptotically stable).Figure 4
ForμM<μMthres and rH>rHthres. If υH>υHthres, then Rvac<1; if υH<υHthres, then Rvac>1.Figure5 shows the prevalence of infectious individuals as a function of υH. All parameters values used are given in Table 2 (baseline values), except for υH. For υH>υHthres, IH*=0, and IM*=0; thus, P0HM is globally asymptotically stable. For υH<υHthres, then IH*≠0 and IM*≠0 and P1HM is globally asymptotically stable.Figure 5
Prevalence of (a) infectious individuals (IH*) and (b) infectious mosquitoes (IM*) as a function of υH. P0HMis a unique equilibrium globally asymptotically stable for υH>υHthres and P1HM is globally asymptotically stable for υH<υHthres.
(a)
(b)Figures6 and 7 show the profiles of both infectious populations IH* and IM*. In Figure 6, for μM<μMthres and rH<rHthres there is no positive real solution of P(NH*) and the DFE P0HM is globally asymptotically stable. In Figure 7, for μM<μMthres and rH>rHthres the quadratic equation P(NH*) has one positive real solution. For υH<υHthres, the endemic equilibrium P1HM is, therefore, a unique equilibrium globally asymptotically stable. Parameter values used are as given in Table 2 (baseline values), except for μM, rH, and υH.Figure 6
ForμM<μMthres and rH<rHthres, P0HM is a unique equilibrium globally asymptotically stable for any vaccination coverage, υH. (a) Profile of population of both infectious humans (IH*) and mosquitoes (IM*). (b) IH* is plotted versus IM*. The system approaches IH*=0 and IM*=0, that is, the system approaches DFE, P0HM.
(a)
(b)Figure 7
ForμM<μMthres and rH>rHthres, P1HM is a unique equilibrium locally asymptotically stable for υH<υHthres. (a) Profile of populations of both infectious humans (IH*) and mosquitoes (IM*). (b) IH* is plotted versus IM*. The system approaches IH*≠0 and IM*≠0, that is, the system approaches endemic equilibrium, P1HM.
(a)
(b)


## 5. Sensitivity Analysis

In this section, we present the sensitivity analysis of the model to find out the degree to which the parameters influence the outputs of the model. Using the equation described in [15, 28, 29], we investigate only two of the most significant epidemiological concepts that affect the disease dynamics: the force of infection and the prevalence of infection.The force of infection is defined as [28, 29](40)λ=abIM*NH*,where a and b are given in Table 2. Thus, substituting (7) into expression (40) gives(41)λ=μH+αH+γH(IH*/NH*)1-ρvac1-1+γH/μH(IH*/NH*).After some simple algebraic manipulations, (14) reduces to(42)IH*NH*=1-ρvacR*-1k11-ρvacR*+k2,where k1=1+γH/μHμM+γM, k2=ac/μM, and the basic reproduction number of the diseases, R0, is being approximated by (43)R*=a2bcγM(SM*/NH*)μH+αH+γHμM+γMμM,since NH0≅NH* and SM0≅SM* (see (37)).In fact,SM* and NH* are smaller than SM0 and NH0, respectively. This is due to the fact that, in the endemic equilibrium, the disease “consumes” the susceptible and total population. As the prevalence of yellow fever is typically low, the approximation holds.Note that since0<ρvac<1, (42) is satisfied whenever ρvac<1-1/R*, that is, whenever R*>1. In this case, the endemic equilibrium P1HM is a unique equilibrium which is locally asymptotically stable. Moreover, we define IH*/NH*, given by (42), as a prevalence for model (1), and denote it by prev.Now, applying the expression given by [15, 28, 29] to estimate the sensitivity of a variables Vi to the parameters θj,(44)ΔViVi=θjVi∂Vi∂θjΔθjθj,we can then calculate the sensitivity of both the force of infection λ (see (40)) and the prevalence (see (42)), for the parameter ρvac, which are given by(45)Δλλ=ρvacλμH+αH+γH1+γH/μHρvac+prev-1-1Δρvacρvac,Δprevprev=ρvacprevΔρvacρvac,with Δρvac/ρvac=0.01.Finally, applying (45) to estimate the sensitivity of force of infection and the prevalence to the vaccination effort ρvac we can show that for every 1% of variation in ρvac results in a variation of approximated 0.6% and 844% in λ and prev, respectively. Therefore, the prevalence is 1400 more sensitive to vaccination than the force of infection. All parameters values used are given in Table 2 (baseline values).


## 6. Summary and Conclusions

In this paper, we have formulated a model for yellow fever disease with both human and vector populations as variables. We found four threshold parameters that control the development of the disease and the infectious status of the human population in the presence of a preventive vaccine whose protection may wane over time.Our analysis is based on the assumption that the growth rate of the human population is positive, that is,rH-μH>0, which is the case of the yellow fever affected populations. We, therefore, can conclude the following:(a)
if the mortality rate of the mosquitoes is greater than the threshold,μM>μMthres, then the disease is naturally (without intervention) eradicated from the population;(b)
if, in contrast, the mortality rate of the mosquitoes is less than the threshold,μM<μMthres, then the disease is eradicated from the populations only when the growing rate of humans is less than a threshold, rH<rHthres. Otherwise, rH>rHthres; then the disease is eradicated only if Rvac<1;(c)
in caseRvac>1, then the disease will be eradicated from the human population if the vaccination rate is greater than a threshold, υH>υHthres. Otherwise, υH<υHthres; then the disease will establish itself among humans, reaching a stable endemic equilibrium. This conclusion derives from a rearrangement of (37); that is,(46)Rvac=R01-ρvac>1⟹1-ρvac>1R0=S*N;or in words, the vaccination effort should be such that 1 minus it should be greater than the proportion of susceptible at equilibrium;(d)
the prevalence is 1400 more sensitive to vaccination than the force of infection.The model and analyses presented in this paper are intended to serve as a framework for testing alternative vaccination schedules taking into account the vaccine and disease induced mortality rates. This will be the subject of a future publication



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*Source: 482091-2015-03-05.xml*