
Human Listeria infection is a food-borne disease caused by the consumption of contaminated food products by the bacterial pathogen, Listeria. In this paper, we propose a mathematical model to analyze the impact of media campaigns on the spread and control of Listeriosis. The model exhibited three equilibria namely; disease-free, Listeria-free and endemic equilibria. The food contamination threshold is determined and the local stability analyses of the model is discussed. Sensitivity analysis is done to determine the model parameters that most affect the severity of the disease. Numerical simulations were carried out to assess the role of media campaigns on the Listeriosis spread. The results show that; an increase in the intensity of the media awareness campaigns, the removal rate of contaminated food products, a decrease in the contact rate of Listeria by humans results in fewer humans getting infected, thus leading to the disease eradication. An increase in the depletion of media awareness campaigns results in more humans being infected with Listeriosis. These findings may significantly impact policy and decision-making in the control of Listeriosis disease.
Citation: C. W. Chukwu, F. Nyabadza, Fatmawati. Modelling the potential role of media campaigns on the control of Listeriosis[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7580-7601. doi: 10.3934/mbe.2021375
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Human Listeria infection is a food-borne disease caused by the consumption of contaminated food products by the bacterial pathogen, Listeria. In this paper, we propose a mathematical model to analyze the impact of media campaigns on the spread and control of Listeriosis. The model exhibited three equilibria namely; disease-free, Listeria-free and endemic equilibria. The food contamination threshold is determined and the local stability analyses of the model is discussed. Sensitivity analysis is done to determine the model parameters that most affect the severity of the disease. Numerical simulations were carried out to assess the role of media campaigns on the Listeriosis spread. The results show that; an increase in the intensity of the media awareness campaigns, the removal rate of contaminated food products, a decrease in the contact rate of Listeria by humans results in fewer humans getting infected, thus leading to the disease eradication. An increase in the depletion of media awareness campaigns results in more humans being infected with Listeriosis. These findings may significantly impact policy and decision-making in the control of Listeriosis disease.
Listeriosis is a serious and severe food-borne disease that affects the human population globally. The disease is caused by a bacteria called Listeria monocytogenes which exists in the environment as its primary host (soil, water, ready-to-eat (RTE) foods and contaminated food products)[1,2]. Human beings contract Listeriosis through the ingestion of contaminated RTE food products such as cantaloupes, meat, Ricotta Salata cheese, vegetables, polony, bean sprouts, ham, or directly from the environment [3,4]. The epidemiology of Listeriosis is clearly articulated in [1,4].
Before the 2017 outbreak in South Africa, an average of 60 to 80 confirmed Listeriosis disease cases were recorded annually (i.e., approximately 1 case per week). The recent outbreak in South Africa, which occurred from 1 January 2017 to 17 July 2018, had 1060 confirmed cases, with 216 (26.8%) deaths. This was the world's largest-ever documented Listeriosis outbreak [5]. The source of the disease was traced to be contaminated RTE processed meat products. Also, Listeriosis outbreaks resulting from human consumption of different kinds of contaminated RTE food products occur commonly in the United States of America, Canada and Europe [3].
The media campaign is a series of advertisement messages that share a single idea, beliefs, concepts, and theme, which make up an integrated marketing communication over a particular time frame and target identified audiences [6]. In addition, media campaigns and media-driven awareness programs such as print media, social media, internet, television, radio, and advertisements play an essential role in the disseminating of information about the spread of infectious disease outbreaks [7]. Dissemination of information educates people and helps them take preventive measures such as; practicing better hygiene; factory workers wearing clean gloves to avoid cross-contamination of food products during food production/manufacturing. Further, when a disease breaks out in a human population, changes in behavior in response to the outbreak can alter the progression of the infectious agent. In particular, people aware of a disease in their proximity can take precautionary measures to reduce their susceptibility to infections by isolating a portion of the susceptible population from the infected ones [8].
In recent times, researchers have used mathematical models to model infectious and non-infectious diseases/describe the effect and impact of media campaigns on the dynamics of infectious such as Ebola [9], HIV/AIDS [10,11], Avian Influenza [12], Listeriosis [13], and vector-host disease [14,15]. Misra et al. [16] used a non-linear mathematical model which assumed that due to awareness programs by media, once the population becomes aware of the disease spread, they avoid contact with the infectives and therefore form a new class of individuals called the aware class, who become susceptible again if their awareness wanes over time. The model analysis revealed that the number of infectives decreases with an increase in media campaigns. Kaur [17] extended the work by Misra by assuming that aware susceptibles do not lose awareness but can also interact with infected individuals and get infected, albeit at a lower rate. Their study suggested that with the increase in the rate of implementation of awareness programs via media, there is a subsequent decline in the number of infected in any targeted population under consideration. Authors in [9] used a mathematical model to describe the transmission dynamics of Ebola in the presence of asymptomatic cases and the impact of media campaigns on the disease transmission was represented by a linearly decreasing function. Their results showed that messages sent through media have a more significant effect on reducing Ebola cases if they are more effective and spaced out. The SIRS model was proposed in [18] to investigate the impact of awareness programs by considering private and public awareness, which reduces the contact rate between unaware and aware populations and the effect/impact of public information campaigns on disease prevalence. It was shown that both private and public awareness could reduce the size of epidemic outbreaks. A smoking cessation model with media campaigns was presented [19]. The results showed that the reproduction number was suppressed when media campaigns that focus on smoking cessation were increased. Thus, spreading information to encourage smokers to quit smoking was an effective intervention. According to [20], the SIRS model was used to analyze the role of information and limited optimal treatment on disease prevalence. The model considered the growth rate of information proportional to a saturated function of infected individuals. The results from the mathematical analysis showed that the combined effects of information and treatment is more effective and economical in the control of the infection. Exponential functions have also been used to model the impact of media awareness campaigns on people's behavior, which affects the evolution of infectious diseases. In particular, the effects of Twitter messages on reducing the transmission rate of the influenza virus was studied in [21]. The result revealed that Twitter messages had a substantial influence on the dynamics of influenza disease spread.
To date, there are very few mathematical models on the dynamics of Listeriosis (see, for instance, [22,23,24,25]), let alone those investigating the potential role of media awareness campaigns on the dynamics of Listeriosis. This paper is motivated by the work done in [16]. We formulate a mathematical model to study the impact and effects of media campaigns on the dynamics of Listeriosis disease resulting from the consumption of contaminated RTE in the human population. We describe the model in detail in the following section.
The outline of this paper is as follows; Section 1 introduces the research paper followed by the model described in Section 2. The model basic properties and analyses are presented in Section 3. Numerical simulations were done and presented in Section 4. Section 5 concludes the paper.
The human population is divided into four sub-classes, viz: Susceptibles Sh(t), aware susceptibles Sa(t), the infected Ih(t) and the recovered Rh(t). Individuals are recruited at a rate proportional to the size of the human population N(t) where
N(t)=Sh(t)+Sa(t)+Ih(t)+Rh(t). |
The recruitment rate is given by μhN(t) where μh is natural birth/mortality rate. Upon infection with Listeria from contaminated food, the susceptibles move into the infectious class Ih(t) with a force of infection λh(t), where λh(t)=βf1Fc with βf1 being the rate at which humans gets infected and Fc(t) the contaminated food products. Here, λh(t) describes the force of infection by the consumption of contaminated food products. The susceptible individuals can also move to the awareness class at a rate ρ as a result of the interaction with the media campaigns. We assume that media campaigns wane over time and the aware individuals can revert to being susceptible again at a rate ω1. The infected individuals recover at a rate γ with immunity after treatment. These individuals who recover after some time can also lose their immunity and become susceptible again at a rate δh. We assume a constant human population N(t), which consists of individuals who do not work in the factory over the modelling time. Further, we assume that aware individuals cannot be infected as their awareness protects them from contracting the disease. Given that the bacteria survive even at 40C, it can die or grow in its host or the environment at significantly low temperatures. Let rl and ξ denote the growth and removal rate of the of Listeria, L(t). Our model assumes a logistic growth of Listeria with carrying capacity KL. The non-contaminated food products Fn(t) can be contaminated as a result of interaction with the bacteria from the environment that comes, via the workers, exchange of gloves or utensils during food manufacturing and also through the contact with contaminated food Fc(t) with a force of infection λf(t), where
λf(t)=βLL(t)+βf2Fc(t). |
The parameters βL and βf2 are the contact rate of Listeria and the contamination rate of non-contaminated food by contaminated food products, respectively. The contaminated food products are then responsible for transmitting Listeriosis disease to the human population through ingestion of the contaminated food products. The total amount of food products, F(t), at any given time is given by
F(t)=Fn(t)+Fc(t), |
where μf is the rate of removal of food products through consumption. Let Ma be the cumulative density of media campaigns with maximum intensity, M, at which media awareness campaigns are implemented, π0 the rate of implementation of the media awareness campaigns and μ0 the rate of depletion of media awareness.
The above model descriptions and Figure 1 gives the following systems of non-linear ordinary differential equations:
{dShdt=μhN+δhRh+ω1Sa−λhSh−μhSh−ρMaSh,dSadt=ρMaSh−(ω1+μh)Sa,dIhdt=λhSh−(μh+γ)Ih,dRhdt=γIh−(μh+δh)Rh,dLdt=rlL(1−LKL)−ξL,dMadt=π0Ih−μ0Ma,dFndt=μfF−λfFn−μfFn,dFcdt=λfFn−μfFc. | (2.1) |
All parameters for the model system (2.1) are assumed to be non-negative for all time t>0. By setting
sh=ShN,sa=SaN,ih=IhN,rh=RhN,l=LKL,ma=MaM,fn=FnF,fc=FcF |
and given that rh(t)=1−sh(t)−sa(t)−ih(t) we have the following rescaled system
{dshdt=μh+δh(1−sh−sa−ih)+ω1sa−(˜λh+μh+˜ρma)sh,dsadt=˜ρmash−(ω1+μh)sa,dihdt=˜λhsh−(μh+γ)ih,dldt=rll(1−l)−ξl,dmadt=πih−μ0ma,dfndt=μf−(˜λf+μf)fn,dfcdt=˜λffn−μffc, | (2.2) |
where
˜λh=β1fc,˜λf=β2l+β3fc,α=NM˜ρ=ρM,π=π0α, |
with β1=βf1F,β2=βlKL,β3=βf2F and initial conditions
sh(0)=sh0>0,sa(0)=sa0>0,ih(0)=ih0≥0,l(0)=l0≥0,ma(0)=ma0≥0,fn(0)=fn0≥0,fc(0)=fc0≥0. | (2.3) |
We prove the positivity of the solutions of model system (2.2) with initial conditions (2.3). First, we state the following Lemma as given in [26].
Lemma 1. Suppose Ω⊂R×Cn is open, fi∈C(Ω,R),i=1,2,…,n. If
fi|xi(t)=0,Xt∈Cn0+≥0,Xt=(x1(t),⋯,xn(t))T,i=1,2,⋯,n, |
then Cn+0 is the invariant domain of the following equations
˙xi(t)=fi(t,Xt),t≥0,i=1,2,⋯,n. | (3.1) |
If fi|xi(t)=0,Xt∈Cn−0≤0,Xt=(x1(t),⋯,xn(t))Ti=1,2,⋯,n, then Cn−0 is the invariant domain of Eq (3.1).
We have the following Theorem on the invariance of system (2.2).
Theorem 1. Each solution (sh(t),sa(t),Ih(t),l(t),ma(t),fn(t),fc(t)) of the model system (2.2) with the non-negative initial conditions (2.3) is non-negative for all t>0.
Proof. Let X=(sh,sa,Ih,l,ma,fn,fc)T and
g(X)=(g1(X),g2(X),g3(X),g4(X),g5(X),g6(X),g7(X))T, |
then we can re-write the model system (2.2) as follows:
˙X=g(X) |
where
g(X)=(g1(X)g2(X)g3(X)g4(X)g5(X)g6(X)g7(X))=(μh+δh(1−sh−sa−ih)+ω1sa−(˜λh+μh+˜ρma)sh,˜ρmash−(ω1+μh)sa,˜λhsh−(μh+γ)ih,rll(1−l)−ξl,π0αih−μ0ma,μf−(˜λf+μf)fn,˜λffn−μffc,). | (3.2) |
From (3.2), setting all the classes to zero, we have that
dsh(t)dt|sh=0=(μh+ω1sa)>0,dsa(t)dt|sa=0=ρMmash>0,dih(t)dt|ih=0=˜λhsh>0,dl(t)dt|l=0=0,dma(t)dt|ma=0=π0αih>0,dfn(t)dt|fn=0=μf>0,dfc(t)dt|fc=0=˜λffn>0. |
Thus, it follows that from Lemma 1 that R7+ is an invariant set and positive.
We now show that the solutions of systems (2.2) are bounded. We thus have the following result.
Theorem 2. The solutions of model system (2.2) are contained in the region Ω∈R7+, which is given by Ω={(sh,sa,ih,l,ma,fn,fc)∈R7+:0≤sh+sa+ih≤1,0≤l≤1,0≤ma≤πμ0,0≤fn+fc≤1} for the initial conditions (2.3) in Ω.
Proof. Considering the total change in the human population from the model system (2.2) given by
dndt=μh(1−sh−sa−ih)+δh(1−sh−sa−ih)−γih, | (3.3) |
for n=sh+sa+ih≤1 we obtain
dndt=μh(1−n)+δh(1−n)−γih,≤(μh+δh)(1−n), |
whose solution is
n(t)≤1−n(0)exp[−(μh+δh)t], |
where n(0)=sh(0)+sa(0)+ih(0) is the initial condition. We note that 0≤n≤1−n(0)e−(μh+δh)t, so that n(t) is bounded provided that n(0)≥0.
The equation
dldt=rll(1−l) |
for the Listeria compartment has a standard solution for a logistic equation
l(t)=11+Θ1exp[−rlt], |
which is bounded with Θ1=exp[−c], where c is a constant.
On the other hand, the total change in the amount of food products resulting from summing the last two equations of (2.2) is given by
dfdt=μf−μff,≤μf(1−f), |
whose solution is
f(t)=1−f(0)exp[−μft], |
where f(0)=fn(0)+fc(0) Here, fc(t)≤1 as t→∞ and hence it is bounded above. We thus, conclude that all the solutions of system (2.2) are bounded, biologically feasible and remains in Ω for all t∈[0,∞). This completes the proof.
The steady states of the model system (2.2) are obtained by equating the right side of Eq (2.2) to zero, so that
{μh+δh(1−s∗h−s∗a−i∗h)+ω1s∗a−(β1f∗c+μh+˜ρm∗a)s∗h=0,˜ρm∗as∗h−(ω1+μh)s∗a=0,β1f∗cs∗h−(μh+γ)i∗h=0,rll∗(1−l∗)−ξl∗=0,πi∗h−μ0m∗a=0,μf−(β2l∗+β3f∗c+μf)f∗n=0,(β2l∗+β3f∗c)f∗n−μff∗c=0. | (3.4) |
From the fourth equation of system (3.4), we have l∗=0 or l∗=qrl, where q=(rl−ξ) and rl>ξ. We consider the two cases separately.
CASE A: If l∗=0, (i.e., if there is no Listeria in the environment) then from the second last equation of (3.4) we have that
f∗n=μfβ3f∗c+μf⋅ | (3.5) |
Substituting (3.5) into the last equation of (3.4) we obtain
β3μff∗c−μff∗c(β3f∗c+μf)=0, |
and upon simplification we have
f∗c=0orf∗c=β3−μfβ3⋅ | (3.6) |
Thus, if l∗=0 then f∗c=0,~λh=~λf=0, f∗n=1 and i∗h=0,m∗a=0,s∗a=0. Also, from the first equation of (3.4)
s∗h=1. |
This results in the disease-free steady states (DFS) given by
E∗0=(1,0,0,0,0,1,0). |
On the other hand, from (3.6), we have
f∗c=μfβ3(Rf−1), |
where
Rf=β3μf⋅ |
We thus have the following result on the existence of f∗c.
Lemma 2. The existence of f∗c is subject to Rf>1.
However, we note that β3 is the contamination rate contributed by contaminated food products and 1μf is the duration of food contamination. So, Rf can be defined as the "food contamination threshold'' that measures the growth of contaminated food due to the contamination of uncontaminated food products by contaminated food products. This is equivalent to the basic reproduction number (R0) in disease modelling, see [27].
If f∗c=μfβ3(Rf−1), then expressing the second, third, and fifth equations of (3.4) in terms of i∗h we obtain the following equation
m∗a=ϕ0i∗h,s∗h=ϕ1i∗h,s∗a=ϕ2i∗h2, | (3.7) |
where ϕ0=πμ0,ϕ1=β3(μh+γ)β1μf(Rf−1) and ϕ2=˜ρϕ0ϕ1(μh+ω1)⋅ Substituting all the expressions in Eq (3.7) into the first equation of (3.4) and after some algebraic simplifications we obtain the following quadratic equation
ξ2i∗h2+ξ1i∗h+ξ0=0, | (3.8) |
where
ξ0=−β3(μh+δh)<0,ξ1=γ+μh+δh+β3(μh+γ)(μh+δh)β1μf(Rf−1)>0ifRf>1,ξ2=π˜ρβ3(μh+γ)(μh+δh)β1μ0μf(μh+ω1)(Rf−1)>0ifRf>1. |
However, we note that the solutions of the quadratic equation (3.8) are given by
i∗h=−ξ1±√ξ21−4ξ2ξ02ξ2⋅ |
The solutions to (3.8) has one positive root when Rf>1. Biologically, this implies that the disease will persist and eventually invade the human population. This results in the Listeria steady state (LFS)
E∗1=(ϕ1i∗h,ϕ2i∗h,0,ϕ0i∗h,1Rf,μfβ3(Rf−1)). |
We note that at Listeria disease free steady state, there are contaminated food products which may result in Listeriosis infection in the human population.
Remark 1. We note that, when l∗=0, we have two steady states E∗0 and E∗1. The existence of E∗1 is subject to the contaminated food generation number (Rf) been greater than 1. As long as Rf>1, even without Listeria in the environment, we will have the disease in the human population.
CASE B: If l∗=qrl, then from last equation of (3.4) solving for new f∗n, we have
f∗+n=μff∗cβ3f∗c+β2qrl⋅ | (3.9) |
Substituting (3.9), into second last equation of (3.4) we obtain the following expression in terms of f∗c after some algebraic simplifications
ν2f∗c2+ν1f∗c+ν0=0, | (3.10) |
where
ν0=−β2qμf<0,ν1=β2μfq+μ2frl(1−Rf),ν2=β3μfrl>0. |
The solutions of the quadratic equation (3.10) given by
f∗c=−ν1±√ν21−4ν2ν02ν2, |
has one positive root irrespective of the signs of ν1. The solutions of f∗c say f∗+c exists, but cannot be determined due to its intractability. Hence, as long as l∗=1 we have a positive f∗+c.
Now, we express the second, third and fifth equation of (3.4) in terms of i∗h and obtain the following expressions
m∗a=Ψ0i∗h,s∗h=Ψ1i∗h,s∗a=Ψ2i∗h2, | (3.11) |
respectively, where Ψ0=πμ0,Ψ1=(γ+μh)β1f+c and Ψ2=˜ρΨ0Ψ1μh+ω1⋅ Similarly, substituting all the expressions from Eq (3.11) into the first equation of (3.4) and after some algebraic manipulations we obtain the following quadratic equation in terms of i∗h
ξ5i∗h2+ξ4i∗h+ξ3=0, | (3.12) |
where
ξ3=−(μh+δh)<0,ξ4=(μh+γ)(μh+δh)+β1f∗+c(δh+μh+γ)β1f∗+c>0,ξ5=πρ(γ+μh)(δh+μh)β1μ0f∗+c(μh+ω1)>0. |
The solutions (i∗h) of the quadratic equation (3.12) given by
i∗h=−ξ4±√ξ24−4ξ5ξ32ξ5, |
exists and has one positive root. We thus have the following result on the existence of new i∗h say i∗+h.
Lemma 3. The steady state i∗+h exists whenever f∗+c exists.
This results in the endemic steady states (ESS) given by
E∗2=(s∗+h,s∗+a,l∗1,m∗+a,f∗+n)=(Ψ1i∗+h,Ψ2i∗+h,1,Ψ0i∗+h,μff∗+cβ3f∗+c+β2). |
Hence, at endemic steady state, there are contaminated food products which result in the persistence of the Listeria infections in the human population.
To analyse the local stability of the DFS, we show that the eigenvalues of the Jacobian matrix at DFS have negative real parts. We now state the following theorem for the DFS.
Theorem 3. The disease-free steady state (E∗0) is always stable whenever Rf<1 and rl<ξ.
Proof. The Jacobian of system (2.2) is given by the block matrix
J=(A1A2A3A4), | (3.13) |
where
A1=(−(δh+˜ρm∗a)−δh+ω1−δh0˜ρm∗a−(ω1+μh)00β1f∗c0−(μh+γ)0000F0),A2=(−˜ρs∗h0−β1s∗h˜ρs∗h0000β1s∗h000) |
A3=(00π0000−β2f∗n000β2f∗n),andA4=(−μ0000−(β2l∗+β3f∗c+μf)−β3f∗n0β2l∗+β3f∗cβ3f∗n−μf) |
in which F0=rl−ξ−2rll∗. Evaluating (3.13) at DFS, we have that
J(E∗0)=(J1(E∗0)J2(E∗0)J3(E∗0)J4(E∗0)), |
where
J1(E∗0)=(−δhω1−δh)−δh00−(ω1+μh)0000−(μh+γ)0000rl−ξ),J2(E∗0)=(−˜ρ0−β1˜ρ0000β1000), |
J3(E∗0)=(00π0α0000−β2000β2)andJ4(E∗0)=(−μ0000−μf−β300β3−μf). |
Similar to the approach used in [28], the eigenvalues of J(E∗0) are: λ1=−δh,λ2=−(ω1+μh),λ3=−(μh+γ)
λ4=(rl−ξ)<0, if rl<ξ, λ5=−μ0, λ6=−μf and λ7=μf(Rf−1)<0 when Rf<1. We note that all the eigenvalues are negatives. Hence E∗0 is locally asymptotically stable.
We state the following theorem for the local stability of Listeria-free steady state.
Theorem 4. The Listeria-free steady state (E∗1) is always stable whenever Rf>1 and rl<ξ.
Proof. Evaluating (3.13) at Listeria-free steady state, we have that
J(E∗1)=(J1(E∗1)J2(E∗1)J3(E∗1)J4(E∗1)), |
where
J1(E∗1)=(−(δh+˜ρϕ0i∗h)−δh+ω1−δh0˜ρϕ0i∗h−(ω1+μh)00β1μfβ3(Rf−1)0−(μh+γ)0000rl−ξ),J2(E∗1)=(−˜ρϕ1i∗h0−β1ϕ1i∗h˜ρϕ1i∗h0000β1ϕ1i∗h000), |
J3(E∗1)=(00π0α0000−β2Rf000β2Rf),andJ4(E∗1)=(−μ0000−μfRf−β3Rf0μf(Rf−1)β3Rf−μf). |
The eigenvalues from J1(E∗1) are: λ1=(rl−ξ)<0 if rl<ξ and the solutions to the cubic equation
λ3+a2λ2+a1λ+a0=0, |
where
a0=1β3μ0(β3(γ+μh)(π˜ρμhi∗h+δh(π˜ρi∗h+μ0(μh+ω1)))+(β1μ0μfδh(μh+ω1)(Rf−1)),a1=1β3μ0(β3(π˜ρi∗h(2μh+δh+γ)+μ0((μh+γ)(μh+ω1)+δh(2μh+ω1+γ)))+(β1μ0μfδh)(Rf−1)),a2=2μh+δh+ω1+γ+π˜ρi∗hμ0. |
Given that a0,a1 and a2 are positive provided that Rf>1, we note that
a2a1−a0=1β3μ20(Υ0+β1δhμ0μf(π˜ρi∗h+μ0(γ+δh+μh))(Rf−1)), |
for
Υ0=β3(π˜ρi∗h+μ0(δh+μh+ω1))(π˜ρi∗h(γ+δh+2μh)+μ0(γ+δh+μh)F1) |
and F1=(2μh+ω1+γ). We also, note that a2a1−a0>0 if Rf>1. Hence, by the Routh-Hurwitz criterion the eigenvalues of J(E∗1) have negative real parts. The rest of the eigenvalues are determined from J4(E∗1), which are; λ5=−μ0 and the solutions to the quadratic equation
λ2+a4λ+a3=0, |
where
a3=μfRf+μf(1−1R2f)anda4=μfRf(1−1R3f). |
We thus have that a3>0 and a4>0 if Rf>1 and a4 is always positive. Therefore, the eigenvalues have negative real parts by Routh-Hurwitz criteria. Hence, E∗1 is locally asymptotically stable.
We now state the following theorem on the local stability of endemic steady state.
Theorem 5. The endemic steady state (E∗2), is always locally asymptotically stable if Rf>1 and rl>ξ.
Proof. Evaluating (3.13) at endemic steady state, we have that
J(E∗2)=(J1(E∗2)J2(E∗2)J3(E∗2)J4(E∗2)), |
where
J1(E∗2)=(−(δh+˜ρΨ0i∗+h)−δh+ω1−δh0˜ρΨ0i∗+h−(ω1+μh)00β1f∗+n0−(μh+γ)0000−(rl−ξ)),J2(E∗2)=(−˜ρΨ1i∗+h0−β1Ψ1i∗+h˜ρΨ1i∗+h0000β1Ψ1i∗+h000), |
J3(E∗2)=(00π0α0000−β2f∗+n000β2f∗+n)andJ4(E∗2)=(−μ0000−(β2+β3f∗+c+μf)−β3f∗+n0β2+β3f∗+cβ3f∗+n−μf). |
The eigenvalues from J1(E∗2) are: λ1=−(rl−ξ)<0 if rl>ξ and the solutions to the cubic equation
λ3+b2λ2+b1λ+b0=0, |
where
b0=1μ0(π˜ρi∗+hμh(γ+μh)+δh(π˜ρi∗+h(γ+μh)+μ0D0)),b1=1μ0(π˜ρi∗+h(γ+δh+2μh)+μ0((γ+μh)(μh+ω1)+δhD1)),b2=2μh+δh+ω1+γ+π˜ρi∗+hμ0, |
Note that b0,b1 and b2 are positive and
b2b1−b0=1μ20(Υ1+Υ2)>0, |
for
Υ1=π2˜ρ2i∗+2h(γ+δh+2μh)+μ20(γ+δh+μh)((μh+ω1)(γ+2μh+ω1)+δh(γ+β1f∗+c+2μh+ω1))andΥ2=π˜ρμ0i∗+h(δ2h+(γ+2μh)2+(2γ+3μh)ω1+δh[β1f∗+c+5μh+2(γ+ω1)]). |
Thus, by the Routh-Hurwitz criteria, the eigenvalues have negative real parts. The remaining eigenvalues are determined from J4(E∗2), which are: λ5=−μ0 and the solutions to the quadratic equation
λ2+b4λ+b3=0, |
where
b3=μf((β2+β3f∗+c)2+β2μf)β2+β3f∗+c>0,b4=β2+β3f∗+c+(2β2+β3f∗+c)μfβ2+β3f∗+c>0. |
Hence, the eigenvalues have negative real parts by Routh-Hurwitz criterion. Therefore, all the eigenvalues of J(E∗2) have negative real parts. Thus, E∗2 is locally asymptotically stable and otherwise unstable.
Numerical simulations of the model system (2.2) were done using the set of parameters values given in Table 1. Most of the parameter values were estimated since there are very few mathematical models in literature, that have been done on L. Monocytogenes disease dynamics and hence the parameter values are elusive. We used a fourth order Runge-Kutta numerical scheme to perform the simulations with the initial conditions: sh(0)=0.42,sa(0)=0.53,ih(0)=0.05,l(0)=0.1,ma(0)=0.25,fn=0.2 and fc=0.8. The initial conditions were hypothetically chosen for the numerical simulations presented in section and are thus only for illustrative purpose and do not represent any observed scenario.
Parameter description | Symbol | Value (day−1) | Source |
Mortality rate of humans | μh | 0.02/365 | [24] |
Recovery rate of humans | γ | 0.02 | Assumed |
Rate of loss of immunity for humans | δh | 0.2 | [24] |
Waning rate of aware susceptibles | ω1 | 0.25 | [16] |
Rate of non-aware susceptibles to aware | ˜ρ | 0.9 | Assumed |
Growth rate of Listeria | rl | 0.25 | [24] |
Death rate of Listeria | ξ | 0.056 | Assumed |
Depletion rate of media campaigns | μ0 | 0.03 | [17] |
Implementation rate of media campaigns | π | 0.001 | [17] |
Contact rate between Listeria and humans | β1 | 0.0025 | Assumed |
Food contamination rate by Listeria | β2 | 0.09 | Assumed |
Food contamination rate | β3 | 0.0048 | Assumed |
Removal rate of food products | μf | 0.056 | Assumed |
Sobol sensitivity analysis [29] were used to determine the model parameters that are sensitive to changes in some variable of the model system (2.2). We performed simulations for some chosen parameters π,μ0,˜ρ,ω1,β1,β2,β3 and μf versus some of the model state variables sa,ih and fc to show their respective PRCCs over time. These parameters and state variables were selected because they are the most significant in the Listeria disease transmission and control according to our model formulation relating to the subject under investigation. The simulations were done with 1000 runs over 800 days as depicted in Figures 2 and 3. The scatter plots for parameters with positive and negative PRCCs are also shown in Figure 4. Thus, an increase in the rate at which susceptible individuals move into the aware susceptible class results in a fewer number of humans been infected with the Listeriosis and parameter ω1 with negative correlation signifying that if aware individuals revert to being susceptible, then they are prone to contracting the disease as depicted in Figure 4 respectively. This highlights the importance of media campaigns in disease control.
This subsection is devoted to numerical simulations that show the effects of increasing and decreasing media campaigns over time as depicted in in Figure 5. Figure 5(a), (b) reveal that the increase in the awareness campaigns result in a decrease in the number of infected and increase the aware susceptible humans, respectively. While the reduction of media campaigns results in more humans being infected with Listeriosis and less aware susceptible humans as shown in Figure 5(c), (d), respectively.
In Figure 6, we present a contour plot of the food contamination rate β3 and rate of food product removal, μf, versus the food contamination constant, Rf. An increase in the contamination of non-contaminated food by contaminated food products results in an increase in the value of Rf. Hence, more humans get infected with Listeriosis. Also, an increase in the removal of contaminated food products results in a decrease in the values of Rf which implies that fewer humans contract the disease. Note that, the changes in the value of μf do not significantly impact Rf when compared to β3.
Figure 7 illustrates a mesh plot showing the variation of β3,μf and the contaminated food generation number Rf. It is seen that an increase in the rate of food processing increases Rf, while an increase in the removal of contaminated food results in a decrease in Rf. Note that at the intersection of the plane Rf=1 and the mesh plot, we obtain all values of β3 and μf necessary for the eradication of the disease.
Figure 8 depicts the effects of varying parameters ˜ρ and μf. In Figure 8(a), we observe that increasing ˜ρ decreases the number of infected humans. This implies that media campaigns have the potential to reduce the number of infectives. On the other hand, an increase in ˜ρ does not impact the value of the contaminated food generation number, Rf. On the other hand, increasing μf also reduces the number of infected individuals and Rf (see Figure 8(b)). So the removal of contaminated food products during an outbreak is an important intervention in the control of Listeriosis.
A deterministic model on media campaigns' potential role on Listeriosis disease transmissions was developed and analysed in this manuscript. Stability analyses of the model were done in terms of the food contamination constant Rf. The model exhibited three different steady states, which are: disease-free, Listeria-free, and endemic equilibria. The disease-free and the Listeria-free steady states are locally asymptotically stable if the net growth rate rl<ξ, Rf<1 and rl<ξ, Rf>1 respectively, while the endemic equilibrium state is locally asymptotically stable if rl>ξ and Rf>1. On the other hand from our numerical results, it was established that an increase in the removal of contaminated food products and an increase in the rate at which the susceptible individuals become aware susceptible individuals leads to a decrease in the number of infected humans (see Figure 8). Thus to effectively control Listeria disease spread, policymakers, public health, governments, and global stakeholders are advised to implement media campaigns that do not wane as time progresses. This means that media campaigns need to be effective. Further, these interventions from the campaigns should target mainly susceptible individuals in the danger of contracting Listeriosis. As people adhere to the media campaign effectively, it helps reduce and control the number of infected humans leading to less disease transmission. The model presented in this paper is not without fallibility. The model was not fitted to epidemiological data and we assumed that the infectives do not interact with the aware susceptibles. During Listeriosis spread, the assumption has a negative impact on the human population since fewer un-aware individuals become aware of the disease. Despite these shortcomings, the results obtained in this paper are still implementable to help manage, control or contain Listeriosis disease transmission in the event of an outbreak.
In future work, a non-standard explicit discretization method can be considered for solving the Listeriosis model which developed in [30] and the result can be compared with classical methods such as Euler, Runge–Kutta, and some other established approaches.
The authors would like to thank their respective Universities.
No conflict of interest.
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Parameter description | Symbol | Value (day−1) | Source |
Mortality rate of humans | μh | 0.02/365 | [24] |
Recovery rate of humans | γ | 0.02 | Assumed |
Rate of loss of immunity for humans | δh | 0.2 | [24] |
Waning rate of aware susceptibles | ω1 | 0.25 | [16] |
Rate of non-aware susceptibles to aware | ˜ρ | 0.9 | Assumed |
Growth rate of Listeria | rl | 0.25 | [24] |
Death rate of Listeria | ξ | 0.056 | Assumed |
Depletion rate of media campaigns | μ0 | 0.03 | [17] |
Implementation rate of media campaigns | π | 0.001 | [17] |
Contact rate between Listeria and humans | β1 | 0.0025 | Assumed |
Food contamination rate by Listeria | β2 | 0.09 | Assumed |
Food contamination rate | β3 | 0.0048 | Assumed |
Removal rate of food products | μf | 0.056 | Assumed |
Parameter description | Symbol | Value (day−1) | Source |
Mortality rate of humans | μh | 0.02/365 | [24] |
Recovery rate of humans | γ | 0.02 | Assumed |
Rate of loss of immunity for humans | δh | 0.2 | [24] |
Waning rate of aware susceptibles | ω1 | 0.25 | [16] |
Rate of non-aware susceptibles to aware | ˜ρ | 0.9 | Assumed |
Growth rate of Listeria | rl | 0.25 | [24] |
Death rate of Listeria | ξ | 0.056 | Assumed |
Depletion rate of media campaigns | μ0 | 0.03 | [17] |
Implementation rate of media campaigns | π | 0.001 | [17] |
Contact rate between Listeria and humans | β1 | 0.0025 | Assumed |
Food contamination rate by Listeria | β2 | 0.09 | Assumed |
Food contamination rate | β3 | 0.0048 | Assumed |
Removal rate of food products | μf | 0.056 | Assumed |