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Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics

Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics Hindawi International Journal of Differential Equations Volume 2017, Article ID 3758269, 13 pages https://doi.org/10.1155/2017/3758269 Research Article Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics Sylvie Diane Djiomba Njankou and Farai Nyabadza Department of Mathematical Science, Stellenbosch University, Private Bag X1, Matieland 7600, South Africa Correspondence should be addressed to Farai Nyabadza; f.nyaba@gmail.com Received 27 July 2016; Revised 1 November 2016; Accepted 15 November 2016; Published 12 January 2017 Academic Editor:PatriciaJ.Y.Wong Copyright © 2017 S. D. Djiomba Njankou and F. Nyabadza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. eTh presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out. 1. Introduction immune system. So, depending on the state of the infected individual’s immune system, death can directly follow or eTh worldfaced oneofthe most devastatingEbola virus recovery can occur aer ft treatment. According to the World disease (EVD) outbreaks ever in between 2014 and 2015. Health Organisation (WHO), a suspected case of EVD is any EVDiscausedbyaviruscalledEbola,which wasdiscovered person, alive or dead, suffering or having suffered from a in the Democratic Republic of Congo in 1976 near a river sudden onset of high fever and having had contact with a called Ebola[1].Thereare vfi eknown speciesofEbola:Zaire suspected or conrfi med Ebola case, a dead or sick animal, and ebolavirus which has caused the 2014 Ebola disease outbreak at least three of the following symptoms: headaches, anorexia, [2], Sudan ebolavirus, Cote d’Ivoire ebolavirus, Bundibugyo lethargy, aching muscles or joints, breathing difficulties, vom- iting, diarrhoea, stomach pain, inexplicable bleeding, or any ebolavirus (Uganda), and Reston ebolavirus which has not yet caused disease in humans [3]. This virus lives in animals sudden inexplicable death [9]. Confirmed cases of EVD are like bats and primates, mostly found in Western and Central individualswho wouldhavetestedpositivefor thevirus anti- Africa. eTh virus can be transmitted from animals to humans gen either by detection of virus RNA by Reverse Transcriptase when an individual comes into contact with an infectious Polymerase Chain Reaction or by detection of IgM antibodies animal through handling of contaminated meat, for example, directed against Ebola [9]. Ebola seropositive individuals can and contamination is also possible among animals. Contam- be either asymptomatic or symptomatic. Post-Ebola survey ination can occur among humans when they have nonpro- results showed that 71% of seropositive individuals monitored were asymptomatic [7]. Symptomless EVD patients have low tected contact with an infectious individual’s uid fl s like faeces, vomit, saliva, sweat, and blood [4]. It can also happen in hos- infectivity due to their very low viral load whereas the symp- pitals, where healthcare practitioners paid a heavy price [1]. tomatic cases transmit the disease through their u fl ids [8]. Symptoms can appear after 2 to 21 days following contam- Media campaigns have been included in mathematical ination and the infectious period can last from 4 to 10 days [5]. models in recent years. Exponential functions are mostly used Some contaminated individuals become symptomatic aer ft to represent their impact on people’s behaviour which aeff cts 21 days [6], whereas others will never develop symptoms disease evolution [10, 11]. and remain asymptomatic [4, 7, 8]. When the virus gets A model where media coverage influences the trans- into a human body, it rapidly replicates and attacks the missionrateofagivendisease is presentedin[12]. An (1 − p) 2 International Journal of Differential Equations exponentially decreasing function is used to describe the media coverage over time. The results show that media coverage has a short-term beneficial effect on the targeted population. A smoking cessation model with media cam- 1 𝜇 paign was given in [13] and results showed that the repro- 𝛼 𝛼 2 4 ductionnumberissuppressedwhenmedia campaignsthat focus on smoking cessation are increased. u Th s, spreading 𝛿 information to encourage smokers to quit smoking was an Λ S 𝜆 effective intervention. eTh impact of Twitter on influenza 𝜇𝜇 was studied in [14]. An exponential term was associated to model the effect of Twitter messages on reducing the transmission rate of influenza. It was noted that Twitter can have a substantial influence on the dynamics of inu fl enza virus infection and can provide a good real-time assessment of the current disease condition. There is no large-scale treatment for EVD as yet, so stopping the transmission chain remains the only viable Figure 1: Flow diagram for EVD. form of control. Media campaigns publicise the means of contracting the disease and the behaviour to adopt when a suspected or conrfi med Ebola case is detected. The potential the incubation period, a proportion 𝑝 of the exposed do effect of media campaigns on Ebola transmission dynamics not develop symptoms and become infected asymptomatic is thus of great interest. This paper is motivated by the individuals who may recover at a rate 𝛿 .Theasymptomatic work in [14] and was done as an M.S. research work by the rfi st author [15]. We use a mathematical model to individuals may develop symptoms and become symptomatic describe the transmission dynamics of EVD in the presence at a rate 𝜃 . eTh rest of the exposed individuals develop symp- of asymptomatic cases and the impact of media campaigns on toms andbecomesymptomatic.Theinfectedsymptomatic the disease transmission is represented by a linear decreasing class is diminished by EVD related deaths at a rate 𝜎 or function. eTh ecffi acy of media campaigns is a state variable recovery at a rate 𝛿 . Recovered individuals can only leave in this model and a differential equation describing its the compartment through natural death and dead bodies are variation is given. We examine the long-term dynamics of disposed of at a rate 𝜌 . EVD and evaluate the potential impact of media campaigns The general objective of media campaigns against a on reducing thenumberofEbola cases. eTh paperisarranged disease is to increase the population’s awareness of the disease as follows: the model formulation is presented in Section 2, and correct misperceptions about how it is spread and how and the model properties and analysis are given in Section 3. it is and is not acquired [18]. eTh efficacy of messages sent The numerical simulations are presented in Section 4 and we through media is thus their ability to produce the intended give concluding remarks in the last section. results. We consider here that Ebola disease related messages are exchanged by individuals from each of the compartments 2. Model Formulation at any time 𝑡 . Aeft r receiving tweets or messages related to Ebola disease, the population decides on the means of A deterministic model with six independent compartments preventing or even treating the disease. Messages are assumed comprising individuals that are susceptible (𝑆 ), exposed to get outdated at a rate 𝜔 . 𝑀(𝑡) is den fi ed as the fraction of (𝐸 ), infected asymptomatic (𝐼 ), infected symptomatic (𝐼 ), 𝑎 𝑠 effective messages sent by individuals of the respective classes recovered (𝑅 ), and deceased (𝐷 ) is formulated. The total at any time 𝑡 .Thus, 𝑀(𝑡) is the ratio of effective messages population size 𝑁 is given by to thetotal messages sent.Thecontributions to 𝑀 from the 𝑁 (𝑡 )=𝑆 (𝑡 )+𝐸 (𝑡 )+𝐼 (𝑡 )+𝐼 (𝑡 )+𝑅 (𝑡 )+𝐷 (𝑡 ), 𝑎 𝑠 living compartments are, respectively, 𝛼 , 𝛼 , 𝛼 , 𝛼 ,and 𝛼 . 1 2 3 4 5 (1) The use of the campaigns is to reduce EVD transmission. ∀𝑡 ≥ 0. We assume here that media campaigns primarily target the We only consider the Zaire Ebola virus strain which caused transmission process and 0<𝑀()𝑡 ≤1 , ∀𝑡 ≥ 0 . the 2014 Ebola outbreak in West Africa. We assume a constant eTh force of infection will be given by natural death rate 𝜇 forthe wholemodel.Thestudy is made over a relatively large period so that those who recover from (𝐼 𝑡 +𝜂𝐷 𝑡 ) ( ) ( ) (2) 𝜆 (𝑡 )=𝛽𝑐 (1−𝑀 (𝑡 )) , EVD gain permanent immunity against the strain. 𝑁 (𝑡 ) Recruitment into the susceptibles class is done through birth or migration at a constant rate Λ and susceptible where 𝛽 is the probability that a contact will result in an individuals become exposed aer ft unsafe contact with Ebola infection and 𝑐 is the number of contacts between susceptible virus. After contamination, susceptibles move to compart- and infectious individuals. eTh parameter 𝜂>1 describes the ment 𝐸 and, considering 1/𝛾 as the incubation period, high infectivity of dead bodies. The flow diagram is presented individuals leave the exposed compartment at a rate 𝛾 .Aeft r in Figure 1. International Journal of Dieff rential Equations 3 2.1. Model Equations. The system of differential equations We have lim 𝑁(𝑡) < Λ/𝜇 when 𝑁(0) ≤ Λ/𝜇. However, 𝑡→∞ describing the variation of the state variables within the if 𝑁(0) ≥ Λ/𝜇 , 𝑁(𝑡) will decrease to Λ/𝜇 .So, 𝑁(𝑡) is thus a model is as follows: bounded function of time. Together with 𝑀 which is already bounded (see proof (𝑡 ) in Appendix A), we can say that Ω is bounded and at =Λ − (𝜇 + 𝜆 (𝑡 ))𝑆 (𝑡 ), (3) limiting equilibrium lim 𝑁(𝑡) = Λ/𝜇. Besides, any sum 𝑡→∞ or difference of variables in Ω with positive initial values will ( ) (4) =𝜆 (𝑡 )𝑆 (𝑡 )− (𝜇+𝛾 )𝐸 (𝑡 ), remain in Ω or in a neighbourhood of Ω.Thus, Ω is positively invariantandattractingwithrespecttotheflowofsystem(3)– (𝑡 ) 𝑎 (9). (5) =𝑝𝐸𝛾 (𝑡 )−(𝜇+𝜃+𝛿 )𝐼 (𝑡 ), 1 𝑎 (𝑡 ) 3.2. Positivity of Solutions (6) =(1 − )𝑝 𝐸𝛾 (𝑡 )+𝜃𝐼 (𝑡 )−(𝜇+𝛿 +𝜎)𝐼 (𝑡 ), 𝑎 2 𝑠 Theorem 2. The existing solutions of system (3)–(9) are all (𝑡 ) positive. (7) =𝛿 𝐼 (𝑡 )+𝛿 𝐼 (𝑡 )−𝜇𝑅 (𝑡 ), 1 𝑎 2 𝑠 Proof. From (3), we can have (𝑡 ) =𝜎𝐼 (𝑡 )−𝜌𝐷 (𝑡 ), (8) (𝑡 ) (13) ≥− (𝜆 (𝑡 )+𝜇 )𝑆 (𝑡 ),∀𝑡≥0. ( ) =𝛼 𝑆 (𝑡 )+𝛼 𝐸 (𝑡 )+𝛼 𝐼 (𝑡 )+𝛼 𝐼 (𝑡 ) 1 2 3 𝑎 4 𝑠 (9) Solving for (13) yields +𝛼 𝑅 (𝑡 )−𝜔𝑀 (𝑡 ). 𝑆 (𝑡 )=𝑆 (0)exp [− ∫ 𝜆 (𝜏 ) − 𝑡]𝜇 , (14) We set 𝑆(0) > 0 , 𝐸(0) ≥ 0 , 𝐼 (0) ≥ 0, 𝐼 (0) ≥ 0, 𝑅(0) ≥ 𝑎 𝑠 0, 𝐷(0) ≥ 0 ,and 𝑀(0) ≥ 0 as the initial values of each of the state variables 𝑆 , 𝐸 , 𝐼 , 𝐼 , 𝑅 , 𝐷 ,and 𝑀 , all assumed to be 𝑎 𝑠 which is positive given that 𝑆(0) is also positive. positive. Similarly, from (4), we have ( ) 3. Model Properties and Analysis (15) ≥−(𝛾 + )𝜇 𝐸 (𝑡 ),∀𝑡≥0, 3.1. Existence and Uniqueness of Solutions. The right hand so that side of system (3)–(9) is made of Lipschitz continuous func- tions since they describe the size of a population. According 𝐸 (𝑡 )=𝐸 (0)exp [− (𝛾 + )𝜇 ]𝑡 , (16) to Picard’s Existence eTh orem, with given initial conditions, thesolutions of oursystemexist andtheyare unique. which thus shows that 𝐸(𝑡) is positive since 𝐸(0) is also Theorem 1. eTh system makes biological sense in the region positive. Similarly, from (5), we can write Ω={(𝑆 (𝑡 ),𝐸 (𝑡 ),𝐼 (𝑡 ),𝐼 (𝑡 ),𝑅 (𝑡 ),𝐷 (𝑡 ),𝑀 (𝑡 )) 𝑎 𝑠 (𝑡 ) ≥ −(𝜇 +𝜃+𝛿 )𝐼 (𝑡 ),∀𝑡≥0, (17) 1 𝑎 (10) ∈𝑅 :𝑁 𝑡 ≤ ,0 < 𝑀 𝑡 ≤1} ( ) ( ) from which we obtain which is attracting and positively invariant with respect to the 𝐼 𝑡 ≥𝐼 0 exp[−(𝜇 +𝜃+𝛿 )𝑡]. ( ) ( ) (18) 𝑎 𝑎 1 flow of system (3)–(9). Thus, 𝐼 is positive since 𝐼 (0)is positive. 𝑎 𝑎 Proof. We rfi st assume that 𝜌>𝜇 during the modelling time. The remaining equations yield This assumption makes sense since EVD death rate is higher than the natural death rate in the course of an EVD epidemic. 𝐼 (𝑡 )≥𝐼 (0)exp [− (𝜇 + 𝜎 + 𝛿 )𝑡], 𝑠 𝑠 2 By adding (3)–(8), we have 𝑅 (𝑡 )≥𝑅 (0)exp [−𝜇𝑡] , ( ) (19) (11) ≤Λ − 𝑁𝜇 (𝑡 ). 𝐷 𝑡 ≥𝐷 0 exp [−𝜌𝑡] , ( ) ( ) Integrating (11) gives the following solution: 𝑀 (𝑡 )≥𝑀 (0)exp [−𝜔𝑡 ] . Λ Λ So, 𝐼 (𝑡) , 𝑅(𝑡) ,and 𝑀(𝑡) are all positive for positive initial 0≤𝑁 (𝑡 ) ≤ ( 𝑁 (0)− ) exp [−𝜇𝑡 ] + ,∀𝑡≥0. (12) 𝑠 𝜇 𝜇 conditions. u Th s, all the state variables are positive. 𝑑𝑡 𝑑𝑁 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐸 𝑑𝜏 𝑑𝑡 𝑑𝑡 𝑑𝑀 𝑑𝑆 𝑑𝑡 𝑑𝐷 𝑑𝑡 𝑑𝑅 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐸 𝑑𝑡 𝑑𝑆 4 International Journal of Differential Equations 3.3. Steady States Analysis. Thismodel hastwo steady states: Proof. Let us define 𝑉(𝑡) = 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐷(𝑡) as the 𝑎 𝑠 thedisease-freeequilibrium (DFE)which describes thetotal Lyapunov function. absence of EVD in the studied population and the endemic 𝑉(𝑡) > 0 since 𝐸(𝑡) > 0 , 𝐼 (𝑡) > 0 , 𝐼 (𝑡) > 0,and 𝑎 𝑠 equilibrium (EE) which exists at any positive prevalence of 𝐷()𝑡 >0 ∀𝑡 >0. EVD in the population. This section is dedicated to the study of local and global stability of these steady states. 𝑉(𝑡) = 0 if 𝐸(𝑡) = 𝐼 (𝑡) = 𝐼 (𝑡) = 𝐷(𝑡) = 0 (at DFE). 𝑎 𝑠 Thus, 𝑉 is a positive definite function at the DFE. 3.4. eTh Disease-Free Equilibrium and 𝑅 . The disease- ∗ ∗ ∗ ∗ ∗ ∗ ∗ The derivative of 𝑉 is given by free equilibrium is given by (𝑆 ,𝐸 ,𝐼 ,𝐼 ,𝑅 ,𝐷 ,𝑀 )= 𝑎 𝑠 (Λ/,0 𝜇 ,0,0,0,0,Λ𝛼 /𝜔𝜇) .Tocompute themedia campaigns ̇ ̇ ̇ ̇ 𝑉= 𝐸+ 𝐼 + 𝐼 𝑎 𝑠 reproduction number 𝑅 ,weusethenextgenerationmethod comprehensively discussed in [19]. eTh renewal matrix 𝐹 and (24) =(𝛽𝑐 (1−𝑀 ) −𝑄 +𝜎)𝐼 +(𝛾−𝑄 )𝐸 transfer matrix 𝑉 at DFE are 3 𝑠 1 ∗ ∗ 00𝑐𝛽(1−𝑀 )𝑐𝜂𝛽(1−𝑀 ) +(𝜃−𝑄 )𝐼 −𝜌𝐷. 2 𝑎 [ ] [ 00 0 0 ] [ ] 𝐹= , Also, 𝑆/𝑁 ≤ 1 and at equilibrium [ ] 00 0 0 [ ] 00 0 0 [ ] 𝐸= 𝐼 , (20) 𝑄 00 0 [ ] [𝑝𝜃 + (1 − )𝑝 𝑄 ] −𝛾𝑝 𝑄 00 [ ] (25) 𝐼 = 𝐼 , 𝑠 𝑎 [ ] 𝑉= , 𝑝𝑄 [ ] 3 (𝑝 − 1) 𝛾 −𝜃 𝑄 0 [ ] 𝐷= 𝐼 . 00 𝜎 −𝜌 [ ] where Plugging (25) into (24) yields 𝑄 =𝛾 + ,𝜇 𝑄 𝑄 𝑄 1 2 3 𝑄 = 𝜇 +𝜃+𝛿 , (21) 𝑉≤ (𝑅 (𝑀 (𝑡 ))−1)𝐼 (26) 2 1 𝑠 𝛾 [ + (1 − 𝑝) 𝑄 ] 𝑄 =𝛿 +𝜎+𝜇. 3 2 with The media campaigns reproduction number 𝑅 is the −1 𝑅 (𝑀 (𝑡 )) spectral radius of the matrix 𝐹𝑉 and is given by (27) (1−𝑀 (𝑡 )) (1 − 𝑀 ) = (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + ). 𝜂𝜎 𝑅 = (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + ). 𝜂𝜎 (22) 2 𝑀 2 𝜌𝑄 𝑄 𝑄 𝜌𝑄 𝑄 𝑄 1 2 3 1 2 3 We can rewrite 𝑅 =𝑅 +𝑅 for elucidation purposes where ̇ ̇ 𝑀 1 2 Thus, 𝑉≤0 when 𝑅(𝑀(𝑡)) ≤ 1 and, particularly, 𝑉=0 only if 𝐸=𝐼 =𝐼 =𝐷=0 .Since 𝑀(𝑡) ≥ 𝑀 for all 𝑎 𝑠 (1 − 𝑀 ) 𝑡>0 (see proof in Appendix A), we have 𝑅 < 𝑅(𝑀(𝑡)) . 𝑅 = (1 − 𝑝 ]), 𝜌𝑄 𝑄 1 3 Because the largest invariant set for which 𝑉=0 in Ω is the (23) ∗ DFE and 𝑉≤0 if 𝑅(𝑀(𝑡)) ≤ 1 ,byusing theinvariance (1 − 𝑀 ) principle of LaSalle [20], we can conclude that the DFE is 𝑅 = (1 − 𝑝 ])𝜂,𝜎 𝜌𝑄 𝑄 1 3 globally asymptotically stable for 𝑅 < 𝑅(𝑀(𝑡)) < 1 . Together with the existence of a backward bifurcation later and ] =(𝜇 + 𝛿 )/𝑄 . 1 2 proven, we na fi lly obtain the global stability of the DFE for Note here that 𝛾/𝑄 is the probability that an individual 𝑅 <𝑅 <1. in 𝐸 moves either to 𝐼 or to 𝐼 . 𝜎/𝑄 is the proportion 𝑀 𝑀 𝑎 𝑠 3 of symptomatic individuals who die from EVD. u Th s, the Analysis of the Reproduction Number 𝑅 . 𝑅 is considered as media campaigns reproduction number is a sum of secondary 𝑀 𝑀 a reproduction number whose values depend on the fraction infections due to infectious individuals in 𝐼 and the deceased of eeff ctive messages on EVD at a given time. Assuming 𝑀 to in 𝐷 .Noticeherethereductionfactor 1−𝑀 which represents be constant, Figure 2 graphically describes the relationship the attenuating effect of media campaigns on the future between two concepts: reproduction number and media number of EVD cases. campaigns ecffi acy. It shows the reducing eect ff of media Theorem 3. The DFE is globally asymptotically stable when- campaigns on the number of EVD infected individuals and 𝑐 𝑐 ever 𝑅 <𝑅 <1,where 𝑅 = min(𝑅(𝑀(𝑡)), 𝑅(𝑀, ]))and indicates as well how we can test the efficacy of Ebola related 𝑀 𝑀 𝑀 𝑅(𝑀, ]) will be den fi ed later. When 𝑅 <𝑅 <1,the DFEis messages through the pace of the disease transmission. locally stable. Otherwise, the DFE is unstable. In fact, for each value of 𝑀 ,the correspondingvalue of 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑝𝜃 𝑝𝛾 International Journal of Dieff rential Equations 5 3.5 Table 1: Roots signs. ] >0 2.5 ] >0 ] <0 1 1 ] >0 ] <0 ] >0 ] <0 0 0 0 0 1.5 (𝑅 <1) (𝑅 >1) (𝑅 <1) (𝑅 >1) 𝑀 𝑀 𝑀 𝑀 ∗∗ 𝜆 −−+− ∗∗ 0.5 𝜆 −+++ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 𝜙 =𝛾(𝛼𝜇 +𝛼 𝛿 )(𝜃𝑝 + 𝑄 (1 − )) 𝑝 Media campaigns efficacy ( M) 2 4 5 2 2 Figure 2: Time dependent reproduction number. The parameters +𝑄 (𝜇 (𝑄 𝛼 +𝑝𝛼𝛾 )+𝑝𝛼𝛾 𝛿 ). 3 2 2 3 5 1 values used for this plot are 𝜇 = 0.008 , 𝛽 = 0.2 , 𝜎 = 0.58 , 𝛾 = 0.845 , −4 (29) 𝑝 = 0.85 , 𝜃 = 0.1 , 𝛿 = 0.15, 𝛿 = 0.6, 𝑐=12 , 𝜔=4 × 10 , 1 2 −7 −7 −6 −5 −6 𝛼 =9×10 , 𝛼 =2×10 , 𝛼 =5×10 , 𝛼 =8×10 , 𝛼 =10 , 1 2 3 4 5 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Set 𝑃(𝜆 )=𝜆 −𝛽(𝑐 1−𝑀 )((𝐼 +𝜂𝐷 )/𝑁 ).By 𝜌 = 0.97 ,and 𝜂 = 3.5. ∗∗ ∗∗ ∗∗ ∗∗ replacing 𝑀 , 𝐼 , 𝐷 ,and 𝑁 by their values expressed ∗∗ as functions of 𝜆 and by setting the reproduction number can be found and then used to ∗∗ 𝑃(𝜆 )=0, (30) analysethe diseaseevolution.For instance,when 𝑅 =1, the critical value of media campaigns efficacy 𝑀 can be 𝑐 we obtain the following equation: determined. Since the behaviour of the system changes when ∗∗ ∗∗ ∗∗ the reproduction number crosses the value one, 𝑀 can also 𝑐 𝜆 [(] (𝜆 ) + ] 𝜆 + ] )] =0, (31) 2 1 0 be used as a threshold parameter that indicates a behavioural change of thesystemand thus canhelpinthe diseasecontrol where for any given set of parameter values. 2 2 2 2 ] =𝜇 𝑄 𝑄 𝑄 (1 − 𝑅 ), 0 1 2 3 𝑀 3.5. Existence and Stability of the Endemic Equilibrium. ] =𝑄 𝑄 𝑄 (𝜉 +𝜉 )𝜇𝜔 + 𝜉 , 1 1 2 3 1 2 3 In this section, we show the existence of the endemic (32) equilibrium (EE). We denote the endemic equilibrium by ] =𝑄 𝑄 𝑄 [𝛾 (𝜇 (𝜌 + )𝜎 + 𝜌𝛿 ) ( + (1 − 𝑝) 𝑄 ) 2 1 2 3 2 2 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ (𝑆 ,𝐸 ,𝐼 ,𝐼 ,𝑅 ,𝐷 ,𝑀 ).Atequilibrium, (3)–(9) 𝑎 𝑠 give +𝜌𝑄 (𝜇𝑄 +𝑝𝛾(𝜇+𝛿 ))] 𝜔, 3 2 1 ∗∗ with 𝑆 = 𝑄 𝑄 𝑄 , 1 2 3 ∗∗ 𝜆 +𝜇 𝜉 =𝜌(1 + 𝑄 𝑄 𝑄 +𝑄 𝑄 𝜇+𝑝𝛾𝑄 𝜇) , 1 1 2 3 2 3 3 ∗∗ ∗∗ 𝐸 = 𝑄 𝑄 , 2 3 ∗∗ 𝜉 = ( + (1 − 𝑝) 𝑄 )(𝜌(−𝛾𝑐𝛽 +)𝜇𝛾 (𝜆 +𝜇) 2 2 ∗∗ +𝛾𝜎(−𝜂𝑐𝛽 +𝜇)), ∗∗ 𝐼 = 𝑄 , ∗∗ (𝜆 +𝜇) 𝜉 = 𝑐𝛾Λ𝜇 𝛽 (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + )𝜂𝜎 (33) 3 2 ∗∗ [𝑝𝜃 + 𝑄 (1 − )] 𝑝 ∗∗ 2 𝐼 = , (28) ∗∗ ⋅(𝑄 (𝑄 𝛼 +𝑝𝛼𝛾 ) + (𝑝𝜃 + (1 − )𝑝 𝑄 )𝛼 3 2 2 3 2 4 (𝜆 +𝜇) ∗∗ [𝑝 (𝑄 𝛿 +𝜃𝛿 )+𝑄 𝛿 (1 − )] 𝑝 ∗∗ 3 1 2 2 2 𝑅 = , −(𝑝𝑄 𝛿 +(𝜃𝑝 +(1−𝑝)𝑄 ) )𝛼 ). 3 1 2 5 ∗∗ 𝜇(𝜆 +𝜇) ∗∗ ∗∗ 𝜎[𝜃𝑝 + 𝑄 (1 − )] 𝑝 2 From (31), 𝜆 =0 corresponds to the DFE discussed in the ∗∗ 𝐷 = , ∗∗ previous section. The signs of the solutions of the quadratic 𝜌(𝜆 +𝜇) equation ∗∗ ∗∗ 𝑀 = (𝜙 +𝜙 𝜆 ), 1 2 2 ∗∗ ∗∗ ∗∗ 𝜇𝜔 (𝜆 +𝜇) ] (𝜆 ) + ] 𝜆 + ] =0 (34) 2 1 0 where are given in Table 1. ∗∗ ∗∗ From Table 1, we notice that, for the existence and (𝐼 +𝜂𝐷 ) ∗∗ ∗∗ 𝑠 𝜆 =𝛽𝑐(1 − 𝑀 ) , uniqueness of the endemic equilibrium, ] must be negative. ∗∗ 0 Thisisonlypossibleif 𝑅 >1.Thus,wehavethe following theorem on the existence of the endemic equilibrium. 𝜙 =𝜇𝑄 𝑄 𝑄 𝛼 , 1 1 2 3 1 Reproduction number R 𝛾𝜆 𝛾𝜆 𝛾𝜆 𝑝𝛾 𝑝𝜃 𝑝𝜃 𝜔𝜌 6 International Journal of Differential Equations Theorem 4. (i) If 𝑅 >1,(34)has auniquepositivesolution described in [21]. us, Th there exists a critical value of 𝑅 , 𝑀 𝑀 and system (3)–(9) has a unique endemic equilibrium. denoted by 𝑅 ,for whichthere is achangeinthe qualitative 𝑐 ∗∗ (ii) If 𝑅 <𝑅 <1 and ] <0,the roots 𝜆 and behaviour of our model. 𝑀 𝑀 1 1 ∗∗ 𝜆 are both positive, and system (3)–(9) admits two endemic At the bifurcation point, there is an intersection between 𝑐 ∗∗ equilibria. the line 𝑅 =𝑅 and the graph of 𝑃(𝜆 ). eTh discriminant (iii) If 𝑅 =𝑅 ,then(34)has arepeatedpositiverootand Δ is equal to zero at 𝑅 = 𝑅(𝑀, ]), which is solution of 𝑀 𝑀 𝑀 a unique endemic equilibrium exists for system (3)–(9). ] −4𝜔𝑄 𝑄 𝑄 𝜇 (1−𝑅 (𝑀, ]))] =0. (35) (iv) If 0<𝑅 <𝑅 , then system (3)–(9) does not admit 1 2 3 2 𝑀 1 any endemic equilibrium and only the DFE exists. Equation (35) implies Provided ] <0, the existence of two endemic equilibria for 𝑅 <1 suggests the existence of a backward bifurcation 𝑅 (𝑀, ])=1 − . (36) since the DFE also exists in that particular domain. The 4𝜓 ] coexistence of DFE and endemic equilibrium when 𝑅 < Considering as well the threshold value of the reproduc- 1 is a well known characteristic of a backward bifurcation tion number from eTh orem 3, we can conclude that 𝑅 = min(𝑅(𝑀(𝑡)), 𝑅(𝑀, ])).So, 0<𝑅 <𝑅 , the DFE is globally stable, 𝑀 𝑀 𝑅 <𝑅 <1, (37) the DFE is locally stable and two endemic equilibria exist with one which is stable and the other one unstable. 𝐼 =𝑥 , The DFE and EE both describe different qualitative behav- 𝑠 4 iours of our epidemic. Let us set 𝜙=𝑐(𝛽 1 − 𝑀 ) as our 𝑅=𝑥 , bifurcation parameter, so that 𝐷=𝑥 , 𝜌𝑄 𝑄 𝑄 ∗ 1 2 3 𝜙=𝜙 = , 𝛾 ( + (1 − 𝑝) 𝑄 )(𝜌 + )𝜂𝜎 𝑀=𝑥 , (38) for 𝑅 =1. 𝑆=𝑓 , In order to describe the stability of the endemic equilibrium, 𝐸=𝑓 , we use the theorem, remark, and corollary in [22] which arebased on theCentreManifoldTheory,and formulated in ̇ 𝐼 =𝑓 , 𝑎 3 Appendix B. 𝐼 =𝑓 , 𝑠 4 Theorem 5. Auniqueendemic equilibriumexistswhen 𝑅 > 𝑅=𝑓 , 1 and is locally asymptotically stable. 𝐷=𝑓 , Proof. For model (3)–(9), the DFE (𝐸 )isnot equalto 6 zero. According to Remark 1 in [22], we notice that if the 𝑀=𝑓 . equilibrium of interest in eTh orem B.1 is a nonnegative (40) equilibrium 𝑥 , then the requirement that 𝑤 is nonnegative in Theorem B.1 is not necessary. When some components in 𝑤 The equilibrium of interest here is the DFE denoted by 𝐸 = ∗ ∗ ∗ are negative, one can still apply eTh orem B.1 on condition that (𝑆 ,0,0,0,0,0,𝑀 ) and the bifurcation parameter is 𝜙 . The linearisation matrix 𝐴 of our model at (𝐸 ,𝜙 ) is 𝑤 (𝑗) > 0, if 𝑥 (𝑗) = 0, 0 (39) ∗ ∗ −𝜇 0 0 −𝜙 0−𝜂𝜙 0 if 𝑥 (𝑗) > 0, 𝑤 (𝑗) does not need to be positive, [ ] ∗ ∗ [ ] 0−𝑄 0𝜙 0𝜂𝜙 0 [ ] where 𝑤(𝑗) and 𝑥 (𝑗) denote the 𝑗 th component of 𝑤 and 𝑥 , 0 0 [ ] [ 0𝑝𝛾 −𝑄 00 0 0 ] respectively. [ ] Firstly, let us rewrite system (3)–(9) introducing [ ] 𝐴= [ 0(1−)𝑝 𝛾 𝜃 −𝑄 00 0 ] . (41) [ ] 𝑆=𝑥 , [ ] 00 𝛿 𝛿 −𝜇 0 0 [ 1 2 ] [ ] [ ] 00 0 𝜎 0 −𝜌 0 𝐸=𝑥 , [ ] 𝛼 𝛼 𝛼 𝛼 𝛼 0−𝜔 1 2 3 4 5 [ ] 𝐼 =𝑥 , 𝑎 3 𝑝𝜃 International Journal of Dieff rential Equations 7 The eigenvalues of 𝐴 are −𝜇 (twice), −𝜔 , 0,and theroots of we have polynomial (42) below: −𝑄 𝑄 𝑄 𝜌 1 2 3 𝑤 = , 3 2 (42) 𝑄 (𝜍 )=𝜍 +𝑑 𝜍 +𝑑 𝜍+𝑑 , 0 1 2 V =0, where 𝜌𝑄 𝑄 2 3 𝑤 = , 𝑑 =𝜌 + 𝑄 +𝑄 +𝑄 , 0 1 2 3 −𝜓 (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + )𝜂𝜎 1 2 V = , 𝑑 =𝑄 (𝑄 +𝑄 )+𝑄 𝑄 +𝜌(𝑄 +𝑄 +𝑄 ) 1 1 2 3 2 3 1 2 3 𝜓 −𝜙 (1 − )𝑝 𝛾, (43) 𝑝𝑄𝜌𝛾 𝑤 = , 𝑑 =𝑄 𝑄 𝑄 +𝑄 𝑄 𝜌+𝑄 (𝑄 +𝑄 )𝜌 2 1 2 3 2 3 1 2 3 −𝜃𝜓 (𝜌 + 𝜂𝜎) 𝑄 ∗ 1 1 −𝛾(𝜃+( 𝑝 1 −𝑝)(𝑄 +𝜌+𝜂)𝜎 )𝜙 . V = , 2 3 𝛾𝜓 Our linearisation matrix 𝐴 will thus have zero as simple 𝑤 = , (46) eigenvalue. Statement (A1) is verified. We now show that (A2) is satisfied. −𝑄 𝑄 (𝜌 + 𝜂𝜎) 1 2 V = , The right eigenvector 𝑊=[𝑤 ,𝑤 ,𝑤 ,𝑤 ,𝑤 ] and the 1 2 3 4 5 𝛾𝜓 left eigenvector 𝑉=[ V , V , V , V , V , V ] associated with the 1 2 3 4 5 6 eigenvalue 0 such that 𝑉𝑊 = 1 are solutions of the system: 1 𝑄 𝜌𝛿 𝜌𝛿 3 1 2 𝑤 = ( − ), 𝜇 𝜓 𝜎 = [0, 0, 0, 0, 0, 0] , V =0, (44) = [0, 0, 0, 0, 0, 0] , 𝑤 =1, 𝑉𝑊 = 1. −𝜂𝑄 𝑄 𝑄 𝜓 1 2 3 1 V = , 𝛾𝜓 Setting 𝑤 = , 𝜓 = ( + (1 − 𝑝) 𝑄 )𝛾,𝜎 1 2 V =0. 𝑄 𝑄 𝜓 2 3 1 2 𝜓 =−𝜌(𝜌 + )𝜂𝜎 ( +𝑄 𝑝𝑄 𝜃) + (𝜌 2 1 3 𝛾𝜎 We notice that 𝐸 (𝑥 )=0, 𝑤 >0, +𝜂𝜎(𝑄 +𝜌))(𝑄 (𝑝 − 1) 𝑄 −𝑝𝑄 𝜃) , 0 2 2 3 1 2 2 𝐸 (𝑥 )=0, 𝑤 >0, 𝜓 =2𝑄 𝑄 𝑄 𝜌 (𝜌 + 𝜂𝜎) 𝜔 (𝜇 (𝑄 𝑄 +(𝜌+𝜎)(𝜃𝑝 0 3 3 3 1 2 3 1 2 𝐸 (𝑥 )=0, 𝑤 >0, (47) +(1−𝑝)𝑄 )+𝑝𝛾𝑄 𝜌) 𝜔 + 𝜌 (𝜇𝑄 𝑄 𝛼 0 4 4 2 3 2 3 2 𝐸 (𝑥 )=0, 𝑤 >0, +𝛾(𝑝𝑄 +(𝜃𝑝 +(1−𝑝)𝑄 )𝜇𝛼 0 5 5 3 3 2 4 (45) 𝐸 (𝑥 )=0, 𝑤 >0. +(𝜔+𝛼 )(𝑝𝑄 𝛿 + ( + (1 − 𝑝) 𝑄 )𝛿 )))) , 0 6 6 5 3 1 2 2 2 2 Besides, since 𝐸 (𝑥 ) and 𝐸 (𝑥 ) are positive, 𝑤 and 𝑤 𝜓 =𝛾Λ𝜎(−𝑝𝑄 𝜌 𝜃−𝑄 (𝑝𝜃 + (1 − )𝑝 𝑄 )𝜌 0 1 0 7 1 7 4 3 2 2 do not need to be positive according to Remark 1 in [22]. So, +𝑄 𝜂(−𝑝𝑄 − 𝑄 (𝑄 + 𝜌) ( + (1 − 𝑝) 𝑄 )) statement (A2) is verified. 1 3 2 3 2 eTh formulas of the constants 𝑎 and 𝑏 are ⋅𝜎−𝑄 𝑄 (𝑝𝜃 + (1 − )𝑝 𝑄 )𝜌(𝜌 + )𝜂𝜎 )(𝜔 2 3 2 𝑛 2 𝜕 𝑓 𝑘 ∗ 𝑎= ∑ V 𝑤 𝑤 (𝐸 ,𝜙 ), 𝑘 𝑖 𝑗 0 +𝛼 ) , 1 𝑖 𝑗 𝑘,𝑖,𝑗=1 (48) 𝜓 =𝑄 𝑄 𝜌(−𝑄 𝛼 +𝜇𝛼 )+𝛾𝜌(𝑝𝑄 (𝜇𝛼 +𝛿 𝛼 ) 𝑛 2 5 2 3 1 1 2 3 3 1 5 𝜕 𝑓 𝑏= ∑ V 𝑤 (𝐸 ,𝜙 ). 𝑘 𝑖 0 𝜕𝜙 + ( + (1 − 𝑝) 𝑄 )𝛾𝜌(𝛼𝜇 +𝛿 𝛼 )) , 𝑖 𝑘,𝑖=1 2 4 2 5 𝑝𝜃 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝑝𝜃 𝜃𝜌 𝑝𝜃 𝜇𝛼 𝑝𝜃 𝜔𝜓 𝑉𝐴 𝐴𝑊 𝑝𝛾 𝜇𝜓 8 International Journal of Differential Equations 0.5 1 0.45 0.9 0.4 0.8 0.35 0.7 𝜆 𝜆 0.3 0.6 Stable EE 0.25 0.5 Stable EE 0.2 0.4 0.15 0.3 Unstable EE 0.1 0.2 Unstable DFE Unstable DFE Stable DFE 0.1 Stable DFE 0.05 0 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 R 1 1.5 Reproduction number R Reproduction number R M M (a) (b) Figure 3: Forward bifurcation for 𝑅 = 3.11 in (a) and backward bifurcation for 𝑅 = 0.95 in (b) with 𝑅 = 0.68. 𝑀 𝑀 𝑀 0.014 After multiple derivations, we have 𝜓 0.012 𝑎= <0, 0.010 (49) 𝛾(𝑝 − 𝜃 + (1 − )𝑄𝑝 ) (𝜌 + 𝜂𝜎) 𝜔 𝑏= >0. 0.008 −𝜓 0.006 Since 𝑎<0 and 𝑏>0 ,byusing thefourthitemof eTh orem B.1, we can conclude that when 𝜙 changes from 0.004 negative to positive, 𝐸 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium 0.002 becomes positive and locally asymptotically stable and a forward bifurcation appears [21]. 0.000 0 200 400 600 800 1000 Time (days) 3.6. Bifurcation Analysis. The study of the DFE and EE led to the proof of the existence of a backward and a forward R = 0.60 bifurcation for our model. Graphically, they are, respectively, R = 0.96 R = 1.21 represented in Figures 3(a) and 3(b) where 𝑅 is chosen as the bifurcation parameter. We have shown that a forward Figure 4: Time series variation of the force of infection for different bifurcation exists for values of 𝑅 greater than one. This values of the reproduction number. meansthatEVD will persistaslongassecondary infections will occur and reducing 𝑅 to values less than one is enough to eradicate EVD. However, the existence of a backward zero. When the DFE is locally stable and the EE is unstable bifurcation makes it difficult to control the epidemic. In fact, (𝑅 <𝑅 <1), the force of infection reflects a persistent the coexistence of the DFE and the EE for 𝑅 in [𝑅 ,1] 𝑀 𝑀 𝑀 𝑀 infection. When the EE is unstable (𝑅 >1), the force of showsthatreducingthe number of secondaryinfections infection is maximal. This confirms the results obtained at the to less than one is not enough to eradicate EVD. Other bifurcation analysis and describes the unstable nature of EVD control measures like quarantine and contact tracing should which can easily become an explosive epidemic aer ft a small be implemented together with media campaigns to reach a increase in its force of infection as 𝑅 passes through 1. globally stable DFE and wipe out EVD. Figure 4 shows time series plots for the force of infection 𝜆 for varying initial conditions. The trajectories converge 4. Numerical Simulations to steady states depending on the initial conditions and the values of 𝑅 .Wecan observethatwhenthe DFEisasymp- In this section, we use Matlab to carry out simulations for totically stable (𝑅 <𝑅 ), the force of infection reaches our model. We rfi st verify our theoretic conclusions related ∗∗ Force of infection ∗∗ Force of infection Force of infection 𝜆(t) International Journal of Dieff rential Equations 9 to stability analysis of system (3)–(9) and then we vary our Table 2: Parameter values and their description. parameters values to better understand how media cam- Parameters Description (per day) Range Source paigns influence the prevalence and transmission of EVD. Λ Recruitment rate 20000 Estimated It is important to note that the figures chosen are for illustrative purposes only, as we endeavour to verify the Probability for a contact 𝛽 [0.02, 1] [7] to be infectious analytic results. 𝑐 Number of contacts [1, 500] Estimated Rate of exposed 4.1. Parameters’ Estimation. The parameters used in the sim- 𝛾 individuals becoming [0.04, 0.5] [5] ulations are either obtained from the literature or estimated. infectious Since the mean infectious period is set to be from 4 to 𝜇 Natural death rate 0.2 Estimated 10 days, the highest recovery rate 𝛿 is set to 1/4.The Proportion of recovery rate of asymptomatic individuals is assumed to be 𝑝 asymptomatic infected [0.15, 0.7] [7] greaterthanthe oneofthe symptomaticindividuals since individuals theformerhavestrongerresistancetoEVD.Without any Rate of asymptomatic reliable source for EVD media related data, we assume that 𝜃 individuals becoming 0.12 [6] individuals can send EVD related messages through media symptomatic independently of their disease status. At the beginning of the 𝜎 Disease related death rate [0.2, 0.9] [1] epidemic, there is neither a recovered nor an asymptomatic Disposal rate of dead infected individual since only symptomatic persons transmit 𝜌 0.497 [16] bodies thedisease.Wealsoassume that messages aretransmitted Recovery rate of through media at time 𝑡=0 ,atleast forpreventivepurpose. 𝛿 [0, 0.6] Estimated asymptomatic individuals The setting of the initial conditions is driven by the fact that the population of Nzer ´ ek ´ ore, ´ the region where this 2014 Recovery rate of 𝛿 [0, 0.25] [5] symptomatic individuals Ebola disease outbreak started in Guinea, is estimated to be 1,663,582 individuals [23]. We consider the introduction Rate of messaging by −1 𝛼 [0, 10 ] Estimated susceptible individuals of infectives in the population and high infectivity of dead bodies (𝜂 = 1.5 ). eTh initial conditions are then Rate of messaging by −1 𝛼 [0, 10 ] Estimated exposed individuals 𝑆 = 990000, Rate of messaging by −1 𝛼 infected asymptomatic [0, 10 ] Estimated 𝐸 = 8000, individuals Rate of messaging by 𝐼 =0, −1 𝑎0 𝛼 infected symptomatic [0, 10 ] Estimated individuals 𝐼 = 2000, (50) 𝑠0 Rate of messaging by −1 𝛼 [0, 10 ] Estimated recovered individuals 𝑅 =0, Outdating rate of media 𝜔 [0.2, 0.5] [17] 𝐷 =0, campaigns 𝑀 = 0.4. Den fi ition 6. The normalized forward sensitivity index of a Table 2 gives the description of parameters and their values. variable, 𝑢 , that depends differentiably on a parameter, 𝑝 ,is defined as 4.2. Sensitivity Analysis. In mathematical modelling, param- 𝜕𝑢 𝑝 eters whose values are not precisely known are oeft n used Υ fl × . (51) and may vary within some ranges. Numerical methods used to solve equations derived from models may introduce Media campaigns in this paper contribute to the limita- numerical errors in the results. eTh eect ff s of such errors tion of the disease transmission. eTh reproduction number or uncertainties in the model’s parameters are quantiefi d 𝑅 is an important concept when it comes to the disease through sensitivity analysis. eTh aim of sensitivity analysis transmission, because it helps to determine EVD incidence. is to quantify the inu fl ence of parameters variation on The normalized forward sensitivity indices of 𝑅 with calculated results [24]. respect to each parameter 𝑢 in expression (22) are given by Sensitivity indices allow us to measure the relative change in a state variable when a parameter changes. The normalized Υ fl × . (52) forward sensitivity index of a variable to a parameter is the 𝜕𝑢 𝑅 ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable Table 3 represents the numerical values of the sensitivity function of the parameter, the sensitivity index may be indices of the reproduction number 𝑅 for the parameters alternatively defined using partial derivatives (see [25]). used in the model. eTh most important parameters are 𝜕𝑅 𝜕𝑝 10 International Journal of Dieff rential Equations Table 3: Sensitivity indices for EVD reproduction number. 5. Discussion and Conclusion Parameter Sensitivity index To model the potential eeff ct of media campaigns on Ebola 𝜔 +1.5 transmission, we used a deterministic model, with compart- ments comprising individuals with different EVD infection 𝜇 +1.37 status, who send EVD related messages through media. eTh 𝛽+1 eect ff of media campaigns on people’s behaviour is repre- 𝑐+1 sented by a reduction factor which decreases the number of 𝜂 +0.613 new EVD cases. Stability analysis was presented in terms of 𝛾 +0.074 the model reproduction number 𝑅 .Itwas shownthatthe 𝜃 +0.037 disease-free and the endemic equilibria are locally stable if Λ −1.5 𝑅 <1 and 𝑅 >1, respectively. The inclusion of the 𝑀 𝑀 𝛼 −1.5 asymptomatic infected class resulted in the model exhibiting 𝜌 −0.613 a backward bifurcation, emphasizing the necessity of intense 𝜎 −0.347 efforts against EVD as a result of undetected asymptomatic cases. eTh existence of a backward bifurcation has important 𝑝 −0.05 implications in the design of policies and strategies to erad- 𝛿 −0.022 icate or control an epidemic. In the presence of a backward 𝛿 −0.002 bifurcation, classical policies on disease eradication need to be changed as EVD can persist even when the threshold parameter 𝑅 is less than one. thosewiththe highestabsolutevalues. Negative andpositive To be able to control EVD, governments and interna- correlations of the parameters to the reproduction number tional stakeholders should implement feasible campaigns are indicated by negative and positive signs. eTh parameters taking into account the social, economic, and mainly the Λ and 𝛼 have the largest absolute negative numerical values 1 cultural realities of the aeff cted countries. Interventions from with negative sensitivity index values. us, Th increasing their these campaigns should target the aeff cted populations and values will decrease EVD incidence. This result can be help them to understand the disease, comply with control explained by the fact that any increase in the two parameters measures, which sometimes seem severe, and change their leadstoanincreaseintheecffi acyofmediacampaigns.So,the behaviour in order to stop the disease transmission chain [1]. more the media campaigns are ecffi acious, the less the value The best way to contain this outbreak is to jointly implement of the reproduction number is. Another important parameter case isolation, contact tracing with quarantine, and sanitary with a negative index is 𝜌 , which is an expected result since funeral practices as suggested in [26]. burials of EVD dead bodies limit the disease transmission due This model is not without shortcomings. People’s reaction to infected corpses. The outdating rate of media campaigns to media campaigns does not always lead to a reduction in 𝜔 has the most positive influence on EVD reproduction the number of future contamination cases. So, a function rep- number. This means that the more frequently the messages resenting the influence of media campaigns on individuals’ spread by media on EVD are updated, the lower the number behaviour which takes into account the dieff rent cultural set- of new infections is. eTh larger the values of 𝜔 and 𝜇 are, the tings would be an innovative and informative addition to this less the ecffi acy of the messages is and the more the disease model. Aspects of quarantine, contact tracing, and case iden- spreads. The parameters 𝛽 and 𝑐 form the transmission rate ticfi ation initiatives are possible additions that can make this and their increase will directly contribute to an increase in the model more reliable. Despite these shortcomings, this model number of EVD cases. provides a good description of EVD outbreak. The model investigates a very important aspect in disease control in our times, that is, the use of social media in spreading messages. 4.3. Simulations Results and Interpretation. Figures 5(a) and 5(b) confirm the results on stability analysis. It follows that Appendix when 𝑅 <1, the epidemic dies out and, for 𝑅 >1,EVD 𝑀 𝑀 becomes endemic. This is a graphical description of the fact A. Differential Inequalities that the DFE is locally stable for 𝑅 <1 and the EE is locally stable whenever 𝑅 >1. 𝑀 Corollary A.1. Let 𝑥 and 𝑦 be real numbers, 𝐼=[𝑥 ,+∞), 0 0 0 The media campaigns reproduction number 𝑅 is made and 𝑎,𝑏 ∈𝐶(𝐼). Suppose that 𝑦∈𝐶 (𝐼) satisfies the following of parameters which differently influence its values in a inequality: variety of ways. The relationship between those parameters 𝑦 𝑥 ≤𝑎 𝑥 𝑦 𝑥 +𝑏 𝑥 , 𝑥≥𝑥 ,𝑦(𝑥 )=𝑦 . (A.1) ( ) ( ) ( ) ( ) 0 0 0 canbeevaluated throughcontour plots. We chosetwo parameters, 𝛼 and 𝜔 , whose influence on the reproduction Then, number is clearly significant as shown in the expression of 𝑦 (𝑥 )≤𝑦 exp [ ∫ 𝑎 (𝑡 )𝑑𝑡] 𝑅 .Figure6showsthat 𝛼 largely influences 𝑅 when 𝑀 1 𝑀 compared to 𝜔 .Increasingthe values of 𝛼 decreases 𝑅 . (A.2) 1 𝑀 𝑥 𝑥 u Th s, in order to eradicate EVD, the exchange of EVD related + ∫ 𝑏 (𝑠 )exp [ ∫ 𝑎 (𝑡 )]𝑡𝑑 ,𝑠𝑑 𝑥≥𝑥 . 𝑥 𝑠 messages is critical in eradicating the epidemic. 0 International Journal of Dieff rential Equations 11 14000 250000 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Time (days) Time (days) E R E R I D I D a a I I s s (a) (b) Figure 5: Population size at DFE (a) for 𝑅 = 0.61 and EE (b) for 𝑅 = 2.09 with Λ = 20000, 𝜇 = 0.02 , 𝛽 = (0.2, 0.105) , 𝑐 = (8, 12) , 𝜎 = 0.525 , 𝑀 𝑀 −6 −7 −7 −9 𝛾 = (0.2, 0.25) , 𝑝 = 0.17 , 𝜃 = 0.12 , 𝛿 = (0.199, 0.0313), 𝛿 = (0.0001, 0.0013), 𝜔 = 0.2 , 𝛼 = (1.8 × 10 , 1.2 × 10 ), 𝛼 =(2 × 10 ,2 × 10 ), 1 2 1 2 −6 −8 −7 −8 −7 −8 𝛼 =(5 × 10 ,5 × 10 ), 𝛼 =(8 × 10 ,8 × 10 ), 𝛼 = (9.99 × 10 , 9.9 × 10 ), 𝜂 = 1.5 ,and 𝜌 = 0.497 . 3 4 5 0.9 which yields 𝛼 𝑆 𝛼 𝑆 0.8 1 1 𝑀 (𝑡 ) ≥ exp [−𝜔𝑡 ] (𝑀 (0)− )+ . (A.5) 𝜔 𝜔 0.7 Before the disease is spread, we assume that 𝑀 is at the steady state level. So, 𝑀(0) = 𝛼 𝑆/𝜔 which is equivalent to 𝑀(0) = 0.6 𝑀 and (A.5) will give 𝜔 0.5 𝛼 𝑆 (A.6) 𝑀 (𝑡 ) ≥ . 0.4 Together with the assumption 0<𝑀≤1 ,wethushave 0.3 𝑀 ≤𝑀 (𝑡 )≤1. (A.7) 0.2 0.1 0.005 0.01 0.015 0.02 0.025 0.03 B. An Approach to Determine the Direction of Figure 6: Reproduction number contour plot. the Bifurcation eTh orem B.1. Consider a general system of ordinary dieff ren- tial equations with a parameter 𝜙 : If the converse inequality holds in (A.1), then the converse inequality holds in (A.2) too. =𝑓(𝑥,)𝜙 , Let us prove that 𝑀(𝑡) is bounded. (B.1) 𝑛 𝑛 2 𝑛 Proof. From (9), we have 𝑓: 𝑅 ×𝑅→ 󳨀 𝑅 ,𝑓 ∈𝐶 (𝑅 ×𝑅). (𝑡 ) Withoutlossofgenerality, it is assumedthat 0 is an (A.3) ≥𝛼 𝑆 (𝑡 )−𝜔𝑀 (𝑡 ). equilibrium for system (B.1) for all values of the parameter 𝜙 ; that is, 𝑓(0, 𝜙) ≡ 0 for all 𝜙. For system (3)–(9), 𝑀 (𝑡) ≥ 𝛼 𝑆−𝜔.𝑀 By applying Corollary A.1, we have Assume that 𝑀 (𝑡 )≥𝑀 (0)exp [ ∫ (−𝜔 )𝑑𝑢] (A1) 𝐴=𝐷 𝑓(0, 0) = ((𝜕𝑓 /𝜕𝑥 )(0, 0))is the linearisation 𝑥 𝑖 𝑗 (A.4) matrix of system (B.1) around equilibrium 0 with 𝜙 𝑡 𝑡 evaluated at 0. Zero is a simple eigenvalue of 𝐴 and all + ∫ 𝛼 𝑆 exp [ ∫ (−𝜔 )𝑑 V]𝑑,𝑧 ∀𝑡 ≥ 0, 0 𝑧 other eigenvalues of 𝐴 have negative real parts; 0.8 1.2 1.2 1.6 1.6 1.6 2 2 2 Population Population 𝑑𝑡 𝑑𝑀 𝑑𝑡 𝑑𝑥 12 International Journal of Dieff rential Equations (A2) matrix 𝐴 has a nonnegative right eigenvector 𝑤 References and a left eigenvector V corresponding to the zero [1] World Health Organisation, Ebola and Marburg virus dis- eigenvalue. Let 𝑓 be the 𝑘 th component of 𝑓 and ease epidemics: preparedness, alert, control, and evaluation, WHO/HSE/PED/CED/2014.05, 2014. 𝑛 2 𝜕 𝑓 𝑘 [2] Centers for Disease Control and Prevention, Morbidity and 𝑎= ∑ V 𝑤 𝑤 (0, 0), 𝑘 𝑖 𝑗 Mortality Weekly Report, https://www.cdc.gov/mmwr/pre- 𝑖 𝑗 𝑘,𝑖,𝑗=1 view/mmwrhtml/mm6325a4.htm. (B.2) 𝑛 2 [3] J.J.Muyembe-Tamfum, S. Mulangu, J. Masumu,J.M.Kayembe, 𝜕 𝑓 𝑏= ∑ V 𝑤 (0, 0). 𝑘 𝑖 A. Kemp, and J. T. Paweska, “Ebola virus outbreaks in Africa: 𝜕𝜙 𝑘,𝑖=1 past and present,” Onderstepoort Journal of Veterinary Research, vol. 79,no. 2, 8pages,2012. [4] E.M.Leroy,S.Baize,V.E.Volchkovetal.,“Humanasymp- The local dynamics of (B.1) around 0 are totally determined tomatic Ebola infection and strong inflammatory response,” The by 𝑎 and 𝑏 . Lancet, vol. 355, no. 9222, pp. 2210–2215, 2000. [5] J. Astacio, D. Briere, M. Guillen, J. Martinez, F. Rodriguez, and N. Valuenzuela-Campos, “Mathematical Models to study the (1)𝑎>0 , 𝑏>0 .When 𝜙<0 with |𝜙| ≪ 1 , outbreaks of Ebola,” http://ecommons.cornell.edu/bitstream/ 0 is locally asymptotically stable, and there exists a 1813/31962/1/BU-1365-M.pdf. positive unstable equilibrium; when 0<𝜙≪1 , [6] C. N. Haas, “On the quarantine period for ebola virus,” PLoS 0 is unstable and there exists a negative and locally Currents Outbreaks,2014. asymptotically stable equilibrium. [7] S. E. Bellan, J. R. C. Pulliam, J. Dusho,ff and L. A. Meyers, (2) 𝑎<0 , 𝑏<0 .When 𝜙<0 with |𝜙| ≪ 1 , 0 is unstable; “Ebola control: effect of asymptomatic infection and acquired when0<𝜙=1 , 0 is locally asymptotically stable, and immunity,” The Lancet ,vol.384,no. 9953,pp. 1499–1500, 2014. there exists a positive unstable equilibrium. [8] A. G. Baxter, “Symptomeless infection with Ebola virus,” Com- mentary,vol.355,article 2210,2014. (3)𝑎>0 , 𝑏<0 .When 𝜙<0 with |𝜙| ≪ 1 , 0 [9] World Health Organisation, Case Den fi ition Recommendations is unstable, and there exists a locally asymptotically for Ebola or Marburg Virus Diseases, WHO, Geneva, Switzer- stable negative equilibrium; when 0<𝜙≪1 , 0 is land, 2014. stable, and a positive unstable equilibrium appears. [10] Y. Xiao, T. Zhao, and S. Tang, “Dynamics of an infectious dis- eases with media/psychology induced non-smooth incidence,” (4) 𝑎<0 , 𝑏>0 .When 𝜙 changes from negative to Mathematical Biosciences and Engineering,vol.10, no.2,pp. positive, 0 changes its stability from stable to unsta- 445–461, 2013. ble. Correspondingly, a negative unstable equilibrium [11] Y. Xiao, S. Tang, and J. Wu, “Media impact switching surface becomes positive and locally asymptotically stable. during an infectious disease outbreak,” Scienticfi Reports ,vol.5, article 7838, 2015. Corollary B.2. When 𝑎>0 and 𝑏>0 ,the bifurcationat [12] J. M. Tchuenche and C. T. Bauch, “Dynamics of an infectious 𝜙=0 is subcritical or backward. disease where media coverage influences transmission,” ISRN Biomathematics,vol.2012, ArticleID581274, 10 pages, 2012. Competing Interests [13] A. Sharma and A. K. Misra, “Backward bifurcation in a smoking cessation model with media campaigns,” Applied Mathematical The authors declare that there are no competing interests Modelling,vol.39, no.3-4,pp. 1087–1098, 2015. regarding the publication of this paper. [14] K. A. Pawelek, A. Oeldorf-Hirsch, and L. Rong, “Modeling the impact of Twitter on influenza epidemics,” Mathematical Biosciences and Engineering. MBE, vol. 11, no. 6, pp. 1337–1356, Authors’ Contributions [15] S.D.DjiombaNjankou andF.Nyabadza, Modelling the potential Farai Nyabadza developed the model and supervised Sylvie role of control strategies on Ebola virus disease dynamics (Master Diane Djiomba Njankou during her M.S. which resulted in thesis in Mathematical Biology) [M.S. thesis], Stellenbosch Uni- this paper. Sylvie Diane Djiomba Njankou carried out the versity, Stellenbosch, South Africa, 2015. analysis and model simulations. [16] C. M. Rivers,E.T.Lofgren,M.Marathe,S.Eubank, andB.L. Lewis, “Modelling the impact of interventions on an epidemic of EbolainSierraLeone andLiberia,” PLOS Current Outbreaks, Acknowledgments vol. 6, 2014. This publication has benefited from the intellectual and [17] Centers for Disease Control and Prevention: 2014 Ebola out- material contribution of the Organization for Women in break in West Africa, http://www.cdc.gov/vhf/ebola/outbreaks/ Sciencefor theDevelopingWorld (OWSD) andthe Swedish 2014-west-africa/case-counts.html. International Development Cooperation Agency (SIDA). The [18] S. L. Coyle, R. F. Boruch, and C. F. Turner, Evaluating AIDS second author acknowledges the support of Stellenbosch Prevention Programs: Expanded Edition, National Academies University in the production of this manuscript. Press, Washington, DC, USA, 1991. 𝜕𝑥 𝜕𝑥 𝜕𝑥 International Journal of Dieff rential Equations 13 [19] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental mod- els of disease transmission,” Mathematical Biosciences,vol.180, no. 1-2, pp. 29–48, 2002. [20] J. P. LaSalle and Z. 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Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics

International Journal of Differential EquationsJan 1, 2017

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Hindawi International Journal of Differential Equations Volume 2017, Article ID 3758269, 13 pages https://doi.org/10.1155/2017/3758269 Research Article Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics Sylvie Diane Djiomba Njankou and Farai Nyabadza Department of Mathematical Science, Stellenbosch University, Private Bag X1, Matieland 7600, South Africa Correspondence should be addressed to Farai Nyabadza; f.nyaba@gmail.com Received 27 July 2016; Revised 1 November 2016; Accepted 15 November 2016; Published 12 January 2017 Academic Editor:PatriciaJ.Y.Wong Copyright © 2017 S. D. Djiomba Njankou and F. Nyabadza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. eTh presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out. 1. Introduction immune system. So, depending on the state of the infected individual’s immune system, death can directly follow or eTh worldfaced oneofthe most devastatingEbola virus recovery can occur aer ft treatment. According to the World disease (EVD) outbreaks ever in between 2014 and 2015. Health Organisation (WHO), a suspected case of EVD is any EVDiscausedbyaviruscalledEbola,which wasdiscovered person, alive or dead, suffering or having suffered from a in the Democratic Republic of Congo in 1976 near a river sudden onset of high fever and having had contact with a called Ebola[1].Thereare vfi eknown speciesofEbola:Zaire suspected or conrfi med Ebola case, a dead or sick animal, and ebolavirus which has caused the 2014 Ebola disease outbreak at least three of the following symptoms: headaches, anorexia, [2], Sudan ebolavirus, Cote d’Ivoire ebolavirus, Bundibugyo lethargy, aching muscles or joints, breathing difficulties, vom- iting, diarrhoea, stomach pain, inexplicable bleeding, or any ebolavirus (Uganda), and Reston ebolavirus which has not yet caused disease in humans [3]. This virus lives in animals sudden inexplicable death [9]. Confirmed cases of EVD are like bats and primates, mostly found in Western and Central individualswho wouldhavetestedpositivefor thevirus anti- Africa. eTh virus can be transmitted from animals to humans gen either by detection of virus RNA by Reverse Transcriptase when an individual comes into contact with an infectious Polymerase Chain Reaction or by detection of IgM antibodies animal through handling of contaminated meat, for example, directed against Ebola [9]. Ebola seropositive individuals can and contamination is also possible among animals. Contam- be either asymptomatic or symptomatic. Post-Ebola survey ination can occur among humans when they have nonpro- results showed that 71% of seropositive individuals monitored were asymptomatic [7]. Symptomless EVD patients have low tected contact with an infectious individual’s uid fl s like faeces, vomit, saliva, sweat, and blood [4]. It can also happen in hos- infectivity due to their very low viral load whereas the symp- pitals, where healthcare practitioners paid a heavy price [1]. tomatic cases transmit the disease through their u fl ids [8]. Symptoms can appear after 2 to 21 days following contam- Media campaigns have been included in mathematical ination and the infectious period can last from 4 to 10 days [5]. models in recent years. Exponential functions are mostly used Some contaminated individuals become symptomatic aer ft to represent their impact on people’s behaviour which aeff cts 21 days [6], whereas others will never develop symptoms disease evolution [10, 11]. and remain asymptomatic [4, 7, 8]. When the virus gets A model where media coverage influences the trans- into a human body, it rapidly replicates and attacks the missionrateofagivendisease is presentedin[12]. An (1 − p) 2 International Journal of Differential Equations exponentially decreasing function is used to describe the media coverage over time. The results show that media coverage has a short-term beneficial effect on the targeted population. A smoking cessation model with media cam- 1 𝜇 paign was given in [13] and results showed that the repro- 𝛼 𝛼 2 4 ductionnumberissuppressedwhenmedia campaignsthat focus on smoking cessation are increased. u Th s, spreading 𝛿 information to encourage smokers to quit smoking was an Λ S 𝜆 effective intervention. eTh impact of Twitter on influenza 𝜇𝜇 was studied in [14]. An exponential term was associated to model the effect of Twitter messages on reducing the transmission rate of influenza. It was noted that Twitter can have a substantial influence on the dynamics of inu fl enza virus infection and can provide a good real-time assessment of the current disease condition. There is no large-scale treatment for EVD as yet, so stopping the transmission chain remains the only viable Figure 1: Flow diagram for EVD. form of control. Media campaigns publicise the means of contracting the disease and the behaviour to adopt when a suspected or conrfi med Ebola case is detected. The potential the incubation period, a proportion 𝑝 of the exposed do effect of media campaigns on Ebola transmission dynamics not develop symptoms and become infected asymptomatic is thus of great interest. This paper is motivated by the individuals who may recover at a rate 𝛿 .Theasymptomatic work in [14] and was done as an M.S. research work by the rfi st author [15]. We use a mathematical model to individuals may develop symptoms and become symptomatic describe the transmission dynamics of EVD in the presence at a rate 𝜃 . eTh rest of the exposed individuals develop symp- of asymptomatic cases and the impact of media campaigns on toms andbecomesymptomatic.Theinfectedsymptomatic the disease transmission is represented by a linear decreasing class is diminished by EVD related deaths at a rate 𝜎 or function. eTh ecffi acy of media campaigns is a state variable recovery at a rate 𝛿 . Recovered individuals can only leave in this model and a differential equation describing its the compartment through natural death and dead bodies are variation is given. We examine the long-term dynamics of disposed of at a rate 𝜌 . EVD and evaluate the potential impact of media campaigns The general objective of media campaigns against a on reducing thenumberofEbola cases. eTh paperisarranged disease is to increase the population’s awareness of the disease as follows: the model formulation is presented in Section 2, and correct misperceptions about how it is spread and how and the model properties and analysis are given in Section 3. it is and is not acquired [18]. eTh efficacy of messages sent The numerical simulations are presented in Section 4 and we through media is thus their ability to produce the intended give concluding remarks in the last section. results. We consider here that Ebola disease related messages are exchanged by individuals from each of the compartments 2. Model Formulation at any time 𝑡 . Aeft r receiving tweets or messages related to Ebola disease, the population decides on the means of A deterministic model with six independent compartments preventing or even treating the disease. Messages are assumed comprising individuals that are susceptible (𝑆 ), exposed to get outdated at a rate 𝜔 . 𝑀(𝑡) is den fi ed as the fraction of (𝐸 ), infected asymptomatic (𝐼 ), infected symptomatic (𝐼 ), 𝑎 𝑠 effective messages sent by individuals of the respective classes recovered (𝑅 ), and deceased (𝐷 ) is formulated. The total at any time 𝑡 .Thus, 𝑀(𝑡) is the ratio of effective messages population size 𝑁 is given by to thetotal messages sent.Thecontributions to 𝑀 from the 𝑁 (𝑡 )=𝑆 (𝑡 )+𝐸 (𝑡 )+𝐼 (𝑡 )+𝐼 (𝑡 )+𝑅 (𝑡 )+𝐷 (𝑡 ), 𝑎 𝑠 living compartments are, respectively, 𝛼 , 𝛼 , 𝛼 , 𝛼 ,and 𝛼 . 1 2 3 4 5 (1) The use of the campaigns is to reduce EVD transmission. ∀𝑡 ≥ 0. We assume here that media campaigns primarily target the We only consider the Zaire Ebola virus strain which caused transmission process and 0<𝑀()𝑡 ≤1 , ∀𝑡 ≥ 0 . the 2014 Ebola outbreak in West Africa. We assume a constant eTh force of infection will be given by natural death rate 𝜇 forthe wholemodel.Thestudy is made over a relatively large period so that those who recover from (𝐼 𝑡 +𝜂𝐷 𝑡 ) ( ) ( ) (2) 𝜆 (𝑡 )=𝛽𝑐 (1−𝑀 (𝑡 )) , EVD gain permanent immunity against the strain. 𝑁 (𝑡 ) Recruitment into the susceptibles class is done through birth or migration at a constant rate Λ and susceptible where 𝛽 is the probability that a contact will result in an individuals become exposed aer ft unsafe contact with Ebola infection and 𝑐 is the number of contacts between susceptible virus. After contamination, susceptibles move to compart- and infectious individuals. eTh parameter 𝜂>1 describes the ment 𝐸 and, considering 1/𝛾 as the incubation period, high infectivity of dead bodies. The flow diagram is presented individuals leave the exposed compartment at a rate 𝛾 .Aeft r in Figure 1. International Journal of Dieff rential Equations 3 2.1. Model Equations. The system of differential equations We have lim 𝑁(𝑡) < Λ/𝜇 when 𝑁(0) ≤ Λ/𝜇. However, 𝑡→∞ describing the variation of the state variables within the if 𝑁(0) ≥ Λ/𝜇 , 𝑁(𝑡) will decrease to Λ/𝜇 .So, 𝑁(𝑡) is thus a model is as follows: bounded function of time. Together with 𝑀 which is already bounded (see proof (𝑡 ) in Appendix A), we can say that Ω is bounded and at =Λ − (𝜇 + 𝜆 (𝑡 ))𝑆 (𝑡 ), (3) limiting equilibrium lim 𝑁(𝑡) = Λ/𝜇. Besides, any sum 𝑡→∞ or difference of variables in Ω with positive initial values will ( ) (4) =𝜆 (𝑡 )𝑆 (𝑡 )− (𝜇+𝛾 )𝐸 (𝑡 ), remain in Ω or in a neighbourhood of Ω.Thus, Ω is positively invariantandattractingwithrespecttotheflowofsystem(3)– (𝑡 ) 𝑎 (9). (5) =𝑝𝐸𝛾 (𝑡 )−(𝜇+𝜃+𝛿 )𝐼 (𝑡 ), 1 𝑎 (𝑡 ) 3.2. Positivity of Solutions (6) =(1 − )𝑝 𝐸𝛾 (𝑡 )+𝜃𝐼 (𝑡 )−(𝜇+𝛿 +𝜎)𝐼 (𝑡 ), 𝑎 2 𝑠 Theorem 2. The existing solutions of system (3)–(9) are all (𝑡 ) positive. (7) =𝛿 𝐼 (𝑡 )+𝛿 𝐼 (𝑡 )−𝜇𝑅 (𝑡 ), 1 𝑎 2 𝑠 Proof. From (3), we can have (𝑡 ) =𝜎𝐼 (𝑡 )−𝜌𝐷 (𝑡 ), (8) (𝑡 ) (13) ≥− (𝜆 (𝑡 )+𝜇 )𝑆 (𝑡 ),∀𝑡≥0. ( ) =𝛼 𝑆 (𝑡 )+𝛼 𝐸 (𝑡 )+𝛼 𝐼 (𝑡 )+𝛼 𝐼 (𝑡 ) 1 2 3 𝑎 4 𝑠 (9) Solving for (13) yields +𝛼 𝑅 (𝑡 )−𝜔𝑀 (𝑡 ). 𝑆 (𝑡 )=𝑆 (0)exp [− ∫ 𝜆 (𝜏 ) − 𝑡]𝜇 , (14) We set 𝑆(0) > 0 , 𝐸(0) ≥ 0 , 𝐼 (0) ≥ 0, 𝐼 (0) ≥ 0, 𝑅(0) ≥ 𝑎 𝑠 0, 𝐷(0) ≥ 0 ,and 𝑀(0) ≥ 0 as the initial values of each of the state variables 𝑆 , 𝐸 , 𝐼 , 𝐼 , 𝑅 , 𝐷 ,and 𝑀 , all assumed to be 𝑎 𝑠 which is positive given that 𝑆(0) is also positive. positive. Similarly, from (4), we have ( ) 3. Model Properties and Analysis (15) ≥−(𝛾 + )𝜇 𝐸 (𝑡 ),∀𝑡≥0, 3.1. Existence and Uniqueness of Solutions. The right hand so that side of system (3)–(9) is made of Lipschitz continuous func- tions since they describe the size of a population. According 𝐸 (𝑡 )=𝐸 (0)exp [− (𝛾 + )𝜇 ]𝑡 , (16) to Picard’s Existence eTh orem, with given initial conditions, thesolutions of oursystemexist andtheyare unique. which thus shows that 𝐸(𝑡) is positive since 𝐸(0) is also Theorem 1. eTh system makes biological sense in the region positive. Similarly, from (5), we can write Ω={(𝑆 (𝑡 ),𝐸 (𝑡 ),𝐼 (𝑡 ),𝐼 (𝑡 ),𝑅 (𝑡 ),𝐷 (𝑡 ),𝑀 (𝑡 )) 𝑎 𝑠 (𝑡 ) ≥ −(𝜇 +𝜃+𝛿 )𝐼 (𝑡 ),∀𝑡≥0, (17) 1 𝑎 (10) ∈𝑅 :𝑁 𝑡 ≤ ,0 < 𝑀 𝑡 ≤1} ( ) ( ) from which we obtain which is attracting and positively invariant with respect to the 𝐼 𝑡 ≥𝐼 0 exp[−(𝜇 +𝜃+𝛿 )𝑡]. ( ) ( ) (18) 𝑎 𝑎 1 flow of system (3)–(9). Thus, 𝐼 is positive since 𝐼 (0)is positive. 𝑎 𝑎 Proof. We rfi st assume that 𝜌>𝜇 during the modelling time. The remaining equations yield This assumption makes sense since EVD death rate is higher than the natural death rate in the course of an EVD epidemic. 𝐼 (𝑡 )≥𝐼 (0)exp [− (𝜇 + 𝜎 + 𝛿 )𝑡], 𝑠 𝑠 2 By adding (3)–(8), we have 𝑅 (𝑡 )≥𝑅 (0)exp [−𝜇𝑡] , ( ) (19) (11) ≤Λ − 𝑁𝜇 (𝑡 ). 𝐷 𝑡 ≥𝐷 0 exp [−𝜌𝑡] , ( ) ( ) Integrating (11) gives the following solution: 𝑀 (𝑡 )≥𝑀 (0)exp [−𝜔𝑡 ] . Λ Λ So, 𝐼 (𝑡) , 𝑅(𝑡) ,and 𝑀(𝑡) are all positive for positive initial 0≤𝑁 (𝑡 ) ≤ ( 𝑁 (0)− ) exp [−𝜇𝑡 ] + ,∀𝑡≥0. (12) 𝑠 𝜇 𝜇 conditions. u Th s, all the state variables are positive. 𝑑𝑡 𝑑𝑁 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐸 𝑑𝜏 𝑑𝑡 𝑑𝑡 𝑑𝑀 𝑑𝑆 𝑑𝑡 𝑑𝐷 𝑑𝑡 𝑑𝑅 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝐸 𝑑𝑡 𝑑𝑆 4 International Journal of Differential Equations 3.3. Steady States Analysis. Thismodel hastwo steady states: Proof. Let us define 𝑉(𝑡) = 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐷(𝑡) as the 𝑎 𝑠 thedisease-freeequilibrium (DFE)which describes thetotal Lyapunov function. absence of EVD in the studied population and the endemic 𝑉(𝑡) > 0 since 𝐸(𝑡) > 0 , 𝐼 (𝑡) > 0 , 𝐼 (𝑡) > 0,and 𝑎 𝑠 equilibrium (EE) which exists at any positive prevalence of 𝐷()𝑡 >0 ∀𝑡 >0. EVD in the population. This section is dedicated to the study of local and global stability of these steady states. 𝑉(𝑡) = 0 if 𝐸(𝑡) = 𝐼 (𝑡) = 𝐼 (𝑡) = 𝐷(𝑡) = 0 (at DFE). 𝑎 𝑠 Thus, 𝑉 is a positive definite function at the DFE. 3.4. eTh Disease-Free Equilibrium and 𝑅 . The disease- ∗ ∗ ∗ ∗ ∗ ∗ ∗ The derivative of 𝑉 is given by free equilibrium is given by (𝑆 ,𝐸 ,𝐼 ,𝐼 ,𝑅 ,𝐷 ,𝑀 )= 𝑎 𝑠 (Λ/,0 𝜇 ,0,0,0,0,Λ𝛼 /𝜔𝜇) .Tocompute themedia campaigns ̇ ̇ ̇ ̇ 𝑉= 𝐸+ 𝐼 + 𝐼 𝑎 𝑠 reproduction number 𝑅 ,weusethenextgenerationmethod comprehensively discussed in [19]. eTh renewal matrix 𝐹 and (24) =(𝛽𝑐 (1−𝑀 ) −𝑄 +𝜎)𝐼 +(𝛾−𝑄 )𝐸 transfer matrix 𝑉 at DFE are 3 𝑠 1 ∗ ∗ 00𝑐𝛽(1−𝑀 )𝑐𝜂𝛽(1−𝑀 ) +(𝜃−𝑄 )𝐼 −𝜌𝐷. 2 𝑎 [ ] [ 00 0 0 ] [ ] 𝐹= , Also, 𝑆/𝑁 ≤ 1 and at equilibrium [ ] 00 0 0 [ ] 00 0 0 [ ] 𝐸= 𝐼 , (20) 𝑄 00 0 [ ] [𝑝𝜃 + (1 − )𝑝 𝑄 ] −𝛾𝑝 𝑄 00 [ ] (25) 𝐼 = 𝐼 , 𝑠 𝑎 [ ] 𝑉= , 𝑝𝑄 [ ] 3 (𝑝 − 1) 𝛾 −𝜃 𝑄 0 [ ] 𝐷= 𝐼 . 00 𝜎 −𝜌 [ ] where Plugging (25) into (24) yields 𝑄 =𝛾 + ,𝜇 𝑄 𝑄 𝑄 1 2 3 𝑄 = 𝜇 +𝜃+𝛿 , (21) 𝑉≤ (𝑅 (𝑀 (𝑡 ))−1)𝐼 (26) 2 1 𝑠 𝛾 [ + (1 − 𝑝) 𝑄 ] 𝑄 =𝛿 +𝜎+𝜇. 3 2 with The media campaigns reproduction number 𝑅 is the −1 𝑅 (𝑀 (𝑡 )) spectral radius of the matrix 𝐹𝑉 and is given by (27) (1−𝑀 (𝑡 )) (1 − 𝑀 ) = (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + ). 𝜂𝜎 𝑅 = (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + ). 𝜂𝜎 (22) 2 𝑀 2 𝜌𝑄 𝑄 𝑄 𝜌𝑄 𝑄 𝑄 1 2 3 1 2 3 We can rewrite 𝑅 =𝑅 +𝑅 for elucidation purposes where ̇ ̇ 𝑀 1 2 Thus, 𝑉≤0 when 𝑅(𝑀(𝑡)) ≤ 1 and, particularly, 𝑉=0 only if 𝐸=𝐼 =𝐼 =𝐷=0 .Since 𝑀(𝑡) ≥ 𝑀 for all 𝑎 𝑠 (1 − 𝑀 ) 𝑡>0 (see proof in Appendix A), we have 𝑅 < 𝑅(𝑀(𝑡)) . 𝑅 = (1 − 𝑝 ]), 𝜌𝑄 𝑄 1 3 Because the largest invariant set for which 𝑉=0 in Ω is the (23) ∗ DFE and 𝑉≤0 if 𝑅(𝑀(𝑡)) ≤ 1 ,byusing theinvariance (1 − 𝑀 ) principle of LaSalle [20], we can conclude that the DFE is 𝑅 = (1 − 𝑝 ])𝜂,𝜎 𝜌𝑄 𝑄 1 3 globally asymptotically stable for 𝑅 < 𝑅(𝑀(𝑡)) < 1 . Together with the existence of a backward bifurcation later and ] =(𝜇 + 𝛿 )/𝑄 . 1 2 proven, we na fi lly obtain the global stability of the DFE for Note here that 𝛾/𝑄 is the probability that an individual 𝑅 <𝑅 <1. in 𝐸 moves either to 𝐼 or to 𝐼 . 𝜎/𝑄 is the proportion 𝑀 𝑀 𝑎 𝑠 3 of symptomatic individuals who die from EVD. u Th s, the Analysis of the Reproduction Number 𝑅 . 𝑅 is considered as media campaigns reproduction number is a sum of secondary 𝑀 𝑀 a reproduction number whose values depend on the fraction infections due to infectious individuals in 𝐼 and the deceased of eeff ctive messages on EVD at a given time. Assuming 𝑀 to in 𝐷 .Noticeherethereductionfactor 1−𝑀 which represents be constant, Figure 2 graphically describes the relationship the attenuating effect of media campaigns on the future between two concepts: reproduction number and media number of EVD cases. campaigns ecffi acy. It shows the reducing eect ff of media Theorem 3. The DFE is globally asymptotically stable when- campaigns on the number of EVD infected individuals and 𝑐 𝑐 ever 𝑅 <𝑅 <1,where 𝑅 = min(𝑅(𝑀(𝑡)), 𝑅(𝑀, ]))and indicates as well how we can test the efficacy of Ebola related 𝑀 𝑀 𝑀 𝑅(𝑀, ]) will be den fi ed later. When 𝑅 <𝑅 <1,the DFEis messages through the pace of the disease transmission. locally stable. Otherwise, the DFE is unstable. In fact, for each value of 𝑀 ,the correspondingvalue of 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑐𝛽𝛾 𝑝𝜃 𝑝𝛾 International Journal of Dieff rential Equations 5 3.5 Table 1: Roots signs. ] >0 2.5 ] >0 ] <0 1 1 ] >0 ] <0 ] >0 ] <0 0 0 0 0 1.5 (𝑅 <1) (𝑅 >1) (𝑅 <1) (𝑅 >1) 𝑀 𝑀 𝑀 𝑀 ∗∗ 𝜆 −−+− ∗∗ 0.5 𝜆 −+++ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 𝜙 =𝛾(𝛼𝜇 +𝛼 𝛿 )(𝜃𝑝 + 𝑄 (1 − )) 𝑝 Media campaigns efficacy ( M) 2 4 5 2 2 Figure 2: Time dependent reproduction number. The parameters +𝑄 (𝜇 (𝑄 𝛼 +𝑝𝛼𝛾 )+𝑝𝛼𝛾 𝛿 ). 3 2 2 3 5 1 values used for this plot are 𝜇 = 0.008 , 𝛽 = 0.2 , 𝜎 = 0.58 , 𝛾 = 0.845 , −4 (29) 𝑝 = 0.85 , 𝜃 = 0.1 , 𝛿 = 0.15, 𝛿 = 0.6, 𝑐=12 , 𝜔=4 × 10 , 1 2 −7 −7 −6 −5 −6 𝛼 =9×10 , 𝛼 =2×10 , 𝛼 =5×10 , 𝛼 =8×10 , 𝛼 =10 , 1 2 3 4 5 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Set 𝑃(𝜆 )=𝜆 −𝛽(𝑐 1−𝑀 )((𝐼 +𝜂𝐷 )/𝑁 ).By 𝜌 = 0.97 ,and 𝜂 = 3.5. ∗∗ ∗∗ ∗∗ ∗∗ replacing 𝑀 , 𝐼 , 𝐷 ,and 𝑁 by their values expressed ∗∗ as functions of 𝜆 and by setting the reproduction number can be found and then used to ∗∗ 𝑃(𝜆 )=0, (30) analysethe diseaseevolution.For instance,when 𝑅 =1, the critical value of media campaigns efficacy 𝑀 can be 𝑐 we obtain the following equation: determined. Since the behaviour of the system changes when ∗∗ ∗∗ ∗∗ the reproduction number crosses the value one, 𝑀 can also 𝑐 𝜆 [(] (𝜆 ) + ] 𝜆 + ] )] =0, (31) 2 1 0 be used as a threshold parameter that indicates a behavioural change of thesystemand thus canhelpinthe diseasecontrol where for any given set of parameter values. 2 2 2 2 ] =𝜇 𝑄 𝑄 𝑄 (1 − 𝑅 ), 0 1 2 3 𝑀 3.5. Existence and Stability of the Endemic Equilibrium. ] =𝑄 𝑄 𝑄 (𝜉 +𝜉 )𝜇𝜔 + 𝜉 , 1 1 2 3 1 2 3 In this section, we show the existence of the endemic (32) equilibrium (EE). We denote the endemic equilibrium by ] =𝑄 𝑄 𝑄 [𝛾 (𝜇 (𝜌 + )𝜎 + 𝜌𝛿 ) ( + (1 − 𝑝) 𝑄 ) 2 1 2 3 2 2 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ (𝑆 ,𝐸 ,𝐼 ,𝐼 ,𝑅 ,𝐷 ,𝑀 ).Atequilibrium, (3)–(9) 𝑎 𝑠 give +𝜌𝑄 (𝜇𝑄 +𝑝𝛾(𝜇+𝛿 ))] 𝜔, 3 2 1 ∗∗ with 𝑆 = 𝑄 𝑄 𝑄 , 1 2 3 ∗∗ 𝜆 +𝜇 𝜉 =𝜌(1 + 𝑄 𝑄 𝑄 +𝑄 𝑄 𝜇+𝑝𝛾𝑄 𝜇) , 1 1 2 3 2 3 3 ∗∗ ∗∗ 𝐸 = 𝑄 𝑄 , 2 3 ∗∗ 𝜉 = ( + (1 − 𝑝) 𝑄 )(𝜌(−𝛾𝑐𝛽 +)𝜇𝛾 (𝜆 +𝜇) 2 2 ∗∗ +𝛾𝜎(−𝜂𝑐𝛽 +𝜇)), ∗∗ 𝐼 = 𝑄 , ∗∗ (𝜆 +𝜇) 𝜉 = 𝑐𝛾Λ𝜇 𝛽 (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + )𝜂𝜎 (33) 3 2 ∗∗ [𝑝𝜃 + 𝑄 (1 − )] 𝑝 ∗∗ 2 𝐼 = , (28) ∗∗ ⋅(𝑄 (𝑄 𝛼 +𝑝𝛼𝛾 ) + (𝑝𝜃 + (1 − )𝑝 𝑄 )𝛼 3 2 2 3 2 4 (𝜆 +𝜇) ∗∗ [𝑝 (𝑄 𝛿 +𝜃𝛿 )+𝑄 𝛿 (1 − )] 𝑝 ∗∗ 3 1 2 2 2 𝑅 = , −(𝑝𝑄 𝛿 +(𝜃𝑝 +(1−𝑝)𝑄 ) )𝛼 ). 3 1 2 5 ∗∗ 𝜇(𝜆 +𝜇) ∗∗ ∗∗ 𝜎[𝜃𝑝 + 𝑄 (1 − )] 𝑝 2 From (31), 𝜆 =0 corresponds to the DFE discussed in the ∗∗ 𝐷 = , ∗∗ previous section. The signs of the solutions of the quadratic 𝜌(𝜆 +𝜇) equation ∗∗ ∗∗ 𝑀 = (𝜙 +𝜙 𝜆 ), 1 2 2 ∗∗ ∗∗ ∗∗ 𝜇𝜔 (𝜆 +𝜇) ] (𝜆 ) + ] 𝜆 + ] =0 (34) 2 1 0 where are given in Table 1. ∗∗ ∗∗ From Table 1, we notice that, for the existence and (𝐼 +𝜂𝐷 ) ∗∗ ∗∗ 𝑠 𝜆 =𝛽𝑐(1 − 𝑀 ) , uniqueness of the endemic equilibrium, ] must be negative. ∗∗ 0 Thisisonlypossibleif 𝑅 >1.Thus,wehavethe following theorem on the existence of the endemic equilibrium. 𝜙 =𝜇𝑄 𝑄 𝑄 𝛼 , 1 1 2 3 1 Reproduction number R 𝛾𝜆 𝛾𝜆 𝛾𝜆 𝑝𝛾 𝑝𝜃 𝑝𝜃 𝜔𝜌 6 International Journal of Differential Equations Theorem 4. (i) If 𝑅 >1,(34)has auniquepositivesolution described in [21]. us, Th there exists a critical value of 𝑅 , 𝑀 𝑀 and system (3)–(9) has a unique endemic equilibrium. denoted by 𝑅 ,for whichthere is achangeinthe qualitative 𝑐 ∗∗ (ii) If 𝑅 <𝑅 <1 and ] <0,the roots 𝜆 and behaviour of our model. 𝑀 𝑀 1 1 ∗∗ 𝜆 are both positive, and system (3)–(9) admits two endemic At the bifurcation point, there is an intersection between 𝑐 ∗∗ equilibria. the line 𝑅 =𝑅 and the graph of 𝑃(𝜆 ). eTh discriminant (iii) If 𝑅 =𝑅 ,then(34)has arepeatedpositiverootand Δ is equal to zero at 𝑅 = 𝑅(𝑀, ]), which is solution of 𝑀 𝑀 𝑀 a unique endemic equilibrium exists for system (3)–(9). ] −4𝜔𝑄 𝑄 𝑄 𝜇 (1−𝑅 (𝑀, ]))] =0. (35) (iv) If 0<𝑅 <𝑅 , then system (3)–(9) does not admit 1 2 3 2 𝑀 1 any endemic equilibrium and only the DFE exists. Equation (35) implies Provided ] <0, the existence of two endemic equilibria for 𝑅 <1 suggests the existence of a backward bifurcation 𝑅 (𝑀, ])=1 − . (36) since the DFE also exists in that particular domain. The 4𝜓 ] coexistence of DFE and endemic equilibrium when 𝑅 < Considering as well the threshold value of the reproduc- 1 is a well known characteristic of a backward bifurcation tion number from eTh orem 3, we can conclude that 𝑅 = min(𝑅(𝑀(𝑡)), 𝑅(𝑀, ])).So, 0<𝑅 <𝑅 , the DFE is globally stable, 𝑀 𝑀 𝑅 <𝑅 <1, (37) the DFE is locally stable and two endemic equilibria exist with one which is stable and the other one unstable. 𝐼 =𝑥 , The DFE and EE both describe different qualitative behav- 𝑠 4 iours of our epidemic. Let us set 𝜙=𝑐(𝛽 1 − 𝑀 ) as our 𝑅=𝑥 , bifurcation parameter, so that 𝐷=𝑥 , 𝜌𝑄 𝑄 𝑄 ∗ 1 2 3 𝜙=𝜙 = , 𝛾 ( + (1 − 𝑝) 𝑄 )(𝜌 + )𝜂𝜎 𝑀=𝑥 , (38) for 𝑅 =1. 𝑆=𝑓 , In order to describe the stability of the endemic equilibrium, 𝐸=𝑓 , we use the theorem, remark, and corollary in [22] which arebased on theCentreManifoldTheory,and formulated in ̇ 𝐼 =𝑓 , 𝑎 3 Appendix B. 𝐼 =𝑓 , 𝑠 4 Theorem 5. Auniqueendemic equilibriumexistswhen 𝑅 > 𝑅=𝑓 , 1 and is locally asymptotically stable. 𝐷=𝑓 , Proof. For model (3)–(9), the DFE (𝐸 )isnot equalto 6 zero. According to Remark 1 in [22], we notice that if the 𝑀=𝑓 . equilibrium of interest in eTh orem B.1 is a nonnegative (40) equilibrium 𝑥 , then the requirement that 𝑤 is nonnegative in Theorem B.1 is not necessary. When some components in 𝑤 The equilibrium of interest here is the DFE denoted by 𝐸 = ∗ ∗ ∗ are negative, one can still apply eTh orem B.1 on condition that (𝑆 ,0,0,0,0,0,𝑀 ) and the bifurcation parameter is 𝜙 . The linearisation matrix 𝐴 of our model at (𝐸 ,𝜙 ) is 𝑤 (𝑗) > 0, if 𝑥 (𝑗) = 0, 0 (39) ∗ ∗ −𝜇 0 0 −𝜙 0−𝜂𝜙 0 if 𝑥 (𝑗) > 0, 𝑤 (𝑗) does not need to be positive, [ ] ∗ ∗ [ ] 0−𝑄 0𝜙 0𝜂𝜙 0 [ ] where 𝑤(𝑗) and 𝑥 (𝑗) denote the 𝑗 th component of 𝑤 and 𝑥 , 0 0 [ ] [ 0𝑝𝛾 −𝑄 00 0 0 ] respectively. [ ] Firstly, let us rewrite system (3)–(9) introducing [ ] 𝐴= [ 0(1−)𝑝 𝛾 𝜃 −𝑄 00 0 ] . (41) [ ] 𝑆=𝑥 , [ ] 00 𝛿 𝛿 −𝜇 0 0 [ 1 2 ] [ ] [ ] 00 0 𝜎 0 −𝜌 0 𝐸=𝑥 , [ ] 𝛼 𝛼 𝛼 𝛼 𝛼 0−𝜔 1 2 3 4 5 [ ] 𝐼 =𝑥 , 𝑎 3 𝑝𝜃 International Journal of Dieff rential Equations 7 The eigenvalues of 𝐴 are −𝜇 (twice), −𝜔 , 0,and theroots of we have polynomial (42) below: −𝑄 𝑄 𝑄 𝜌 1 2 3 𝑤 = , 3 2 (42) 𝑄 (𝜍 )=𝜍 +𝑑 𝜍 +𝑑 𝜍+𝑑 , 0 1 2 V =0, where 𝜌𝑄 𝑄 2 3 𝑤 = , 𝑑 =𝜌 + 𝑄 +𝑄 +𝑄 , 0 1 2 3 −𝜓 (𝑝𝜃 + (1 − )𝑝 𝑄 )(𝜌 + )𝜂𝜎 1 2 V = , 𝑑 =𝑄 (𝑄 +𝑄 )+𝑄 𝑄 +𝜌(𝑄 +𝑄 +𝑄 ) 1 1 2 3 2 3 1 2 3 𝜓 −𝜙 (1 − )𝑝 𝛾, (43) 𝑝𝑄𝜌𝛾 𝑤 = , 𝑑 =𝑄 𝑄 𝑄 +𝑄 𝑄 𝜌+𝑄 (𝑄 +𝑄 )𝜌 2 1 2 3 2 3 1 2 3 −𝜃𝜓 (𝜌 + 𝜂𝜎) 𝑄 ∗ 1 1 −𝛾(𝜃+( 𝑝 1 −𝑝)(𝑄 +𝜌+𝜂)𝜎 )𝜙 . V = , 2 3 𝛾𝜓 Our linearisation matrix 𝐴 will thus have zero as simple 𝑤 = , (46) eigenvalue. Statement (A1) is verified. We now show that (A2) is satisfied. −𝑄 𝑄 (𝜌 + 𝜂𝜎) 1 2 V = , The right eigenvector 𝑊=[𝑤 ,𝑤 ,𝑤 ,𝑤 ,𝑤 ] and the 1 2 3 4 5 𝛾𝜓 left eigenvector 𝑉=[ V , V , V , V , V , V ] associated with the 1 2 3 4 5 6 eigenvalue 0 such that 𝑉𝑊 = 1 are solutions of the system: 1 𝑄 𝜌𝛿 𝜌𝛿 3 1 2 𝑤 = ( − ), 𝜇 𝜓 𝜎 = [0, 0, 0, 0, 0, 0] , V =0, (44) = [0, 0, 0, 0, 0, 0] , 𝑤 =1, 𝑉𝑊 = 1. −𝜂𝑄 𝑄 𝑄 𝜓 1 2 3 1 V = , 𝛾𝜓 Setting 𝑤 = , 𝜓 = ( + (1 − 𝑝) 𝑄 )𝛾,𝜎 1 2 V =0. 𝑄 𝑄 𝜓 2 3 1 2 𝜓 =−𝜌(𝜌 + )𝜂𝜎 ( +𝑄 𝑝𝑄 𝜃) + (𝜌 2 1 3 𝛾𝜎 We notice that 𝐸 (𝑥 )=0, 𝑤 >0, +𝜂𝜎(𝑄 +𝜌))(𝑄 (𝑝 − 1) 𝑄 −𝑝𝑄 𝜃) , 0 2 2 3 1 2 2 𝐸 (𝑥 )=0, 𝑤 >0, 𝜓 =2𝑄 𝑄 𝑄 𝜌 (𝜌 + 𝜂𝜎) 𝜔 (𝜇 (𝑄 𝑄 +(𝜌+𝜎)(𝜃𝑝 0 3 3 3 1 2 3 1 2 𝐸 (𝑥 )=0, 𝑤 >0, (47) +(1−𝑝)𝑄 )+𝑝𝛾𝑄 𝜌) 𝜔 + 𝜌 (𝜇𝑄 𝑄 𝛼 0 4 4 2 3 2 3 2 𝐸 (𝑥 )=0, 𝑤 >0, +𝛾(𝑝𝑄 +(𝜃𝑝 +(1−𝑝)𝑄 )𝜇𝛼 0 5 5 3 3 2 4 (45) 𝐸 (𝑥 )=0, 𝑤 >0. +(𝜔+𝛼 )(𝑝𝑄 𝛿 + ( + (1 − 𝑝) 𝑄 )𝛿 )))) , 0 6 6 5 3 1 2 2 2 2 Besides, since 𝐸 (𝑥 ) and 𝐸 (𝑥 ) are positive, 𝑤 and 𝑤 𝜓 =𝛾Λ𝜎(−𝑝𝑄 𝜌 𝜃−𝑄 (𝑝𝜃 + (1 − )𝑝 𝑄 )𝜌 0 1 0 7 1 7 4 3 2 2 do not need to be positive according to Remark 1 in [22]. So, +𝑄 𝜂(−𝑝𝑄 − 𝑄 (𝑄 + 𝜌) ( + (1 − 𝑝) 𝑄 )) statement (A2) is verified. 1 3 2 3 2 eTh formulas of the constants 𝑎 and 𝑏 are ⋅𝜎−𝑄 𝑄 (𝑝𝜃 + (1 − )𝑝 𝑄 )𝜌(𝜌 + )𝜂𝜎 )(𝜔 2 3 2 𝑛 2 𝜕 𝑓 𝑘 ∗ 𝑎= ∑ V 𝑤 𝑤 (𝐸 ,𝜙 ), 𝑘 𝑖 𝑗 0 +𝛼 ) , 1 𝑖 𝑗 𝑘,𝑖,𝑗=1 (48) 𝜓 =𝑄 𝑄 𝜌(−𝑄 𝛼 +𝜇𝛼 )+𝛾𝜌(𝑝𝑄 (𝜇𝛼 +𝛿 𝛼 ) 𝑛 2 5 2 3 1 1 2 3 3 1 5 𝜕 𝑓 𝑏= ∑ V 𝑤 (𝐸 ,𝜙 ). 𝑘 𝑖 0 𝜕𝜙 + ( + (1 − 𝑝) 𝑄 )𝛾𝜌(𝛼𝜇 +𝛿 𝛼 )) , 𝑖 𝑘,𝑖=1 2 4 2 5 𝑝𝜃 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝑝𝜃 𝜃𝜌 𝑝𝜃 𝜇𝛼 𝑝𝜃 𝜔𝜓 𝑉𝐴 𝐴𝑊 𝑝𝛾 𝜇𝜓 8 International Journal of Differential Equations 0.5 1 0.45 0.9 0.4 0.8 0.35 0.7 𝜆 𝜆 0.3 0.6 Stable EE 0.25 0.5 Stable EE 0.2 0.4 0.15 0.3 Unstable EE 0.1 0.2 Unstable DFE Unstable DFE Stable DFE 0.1 Stable DFE 0.05 0 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 R 1 1.5 Reproduction number R Reproduction number R M M (a) (b) Figure 3: Forward bifurcation for 𝑅 = 3.11 in (a) and backward bifurcation for 𝑅 = 0.95 in (b) with 𝑅 = 0.68. 𝑀 𝑀 𝑀 0.014 After multiple derivations, we have 𝜓 0.012 𝑎= <0, 0.010 (49) 𝛾(𝑝 − 𝜃 + (1 − )𝑄𝑝 ) (𝜌 + 𝜂𝜎) 𝜔 𝑏= >0. 0.008 −𝜓 0.006 Since 𝑎<0 and 𝑏>0 ,byusing thefourthitemof eTh orem B.1, we can conclude that when 𝜙 changes from 0.004 negative to positive, 𝐸 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium 0.002 becomes positive and locally asymptotically stable and a forward bifurcation appears [21]. 0.000 0 200 400 600 800 1000 Time (days) 3.6. Bifurcation Analysis. The study of the DFE and EE led to the proof of the existence of a backward and a forward R = 0.60 bifurcation for our model. Graphically, they are, respectively, R = 0.96 R = 1.21 represented in Figures 3(a) and 3(b) where 𝑅 is chosen as the bifurcation parameter. We have shown that a forward Figure 4: Time series variation of the force of infection for different bifurcation exists for values of 𝑅 greater than one. This values of the reproduction number. meansthatEVD will persistaslongassecondary infections will occur and reducing 𝑅 to values less than one is enough to eradicate EVD. However, the existence of a backward zero. When the DFE is locally stable and the EE is unstable bifurcation makes it difficult to control the epidemic. In fact, (𝑅 <𝑅 <1), the force of infection reflects a persistent the coexistence of the DFE and the EE for 𝑅 in [𝑅 ,1] 𝑀 𝑀 𝑀 𝑀 infection. When the EE is unstable (𝑅 >1), the force of showsthatreducingthe number of secondaryinfections infection is maximal. This confirms the results obtained at the to less than one is not enough to eradicate EVD. Other bifurcation analysis and describes the unstable nature of EVD control measures like quarantine and contact tracing should which can easily become an explosive epidemic aer ft a small be implemented together with media campaigns to reach a increase in its force of infection as 𝑅 passes through 1. globally stable DFE and wipe out EVD. Figure 4 shows time series plots for the force of infection 𝜆 for varying initial conditions. The trajectories converge 4. Numerical Simulations to steady states depending on the initial conditions and the values of 𝑅 .Wecan observethatwhenthe DFEisasymp- In this section, we use Matlab to carry out simulations for totically stable (𝑅 <𝑅 ), the force of infection reaches our model. We rfi st verify our theoretic conclusions related ∗∗ Force of infection ∗∗ Force of infection Force of infection 𝜆(t) International Journal of Dieff rential Equations 9 to stability analysis of system (3)–(9) and then we vary our Table 2: Parameter values and their description. parameters values to better understand how media cam- Parameters Description (per day) Range Source paigns influence the prevalence and transmission of EVD. Λ Recruitment rate 20000 Estimated It is important to note that the figures chosen are for illustrative purposes only, as we endeavour to verify the Probability for a contact 𝛽 [0.02, 1] [7] to be infectious analytic results. 𝑐 Number of contacts [1, 500] Estimated Rate of exposed 4.1. Parameters’ Estimation. The parameters used in the sim- 𝛾 individuals becoming [0.04, 0.5] [5] ulations are either obtained from the literature or estimated. infectious Since the mean infectious period is set to be from 4 to 𝜇 Natural death rate 0.2 Estimated 10 days, the highest recovery rate 𝛿 is set to 1/4.The Proportion of recovery rate of asymptomatic individuals is assumed to be 𝑝 asymptomatic infected [0.15, 0.7] [7] greaterthanthe oneofthe symptomaticindividuals since individuals theformerhavestrongerresistancetoEVD.Without any Rate of asymptomatic reliable source for EVD media related data, we assume that 𝜃 individuals becoming 0.12 [6] individuals can send EVD related messages through media symptomatic independently of their disease status. At the beginning of the 𝜎 Disease related death rate [0.2, 0.9] [1] epidemic, there is neither a recovered nor an asymptomatic Disposal rate of dead infected individual since only symptomatic persons transmit 𝜌 0.497 [16] bodies thedisease.Wealsoassume that messages aretransmitted Recovery rate of through media at time 𝑡=0 ,atleast forpreventivepurpose. 𝛿 [0, 0.6] Estimated asymptomatic individuals The setting of the initial conditions is driven by the fact that the population of Nzer ´ ek ´ ore, ´ the region where this 2014 Recovery rate of 𝛿 [0, 0.25] [5] symptomatic individuals Ebola disease outbreak started in Guinea, is estimated to be 1,663,582 individuals [23]. We consider the introduction Rate of messaging by −1 𝛼 [0, 10 ] Estimated susceptible individuals of infectives in the population and high infectivity of dead bodies (𝜂 = 1.5 ). eTh initial conditions are then Rate of messaging by −1 𝛼 [0, 10 ] Estimated exposed individuals 𝑆 = 990000, Rate of messaging by −1 𝛼 infected asymptomatic [0, 10 ] Estimated 𝐸 = 8000, individuals Rate of messaging by 𝐼 =0, −1 𝑎0 𝛼 infected symptomatic [0, 10 ] Estimated individuals 𝐼 = 2000, (50) 𝑠0 Rate of messaging by −1 𝛼 [0, 10 ] Estimated recovered individuals 𝑅 =0, Outdating rate of media 𝜔 [0.2, 0.5] [17] 𝐷 =0, campaigns 𝑀 = 0.4. Den fi ition 6. The normalized forward sensitivity index of a Table 2 gives the description of parameters and their values. variable, 𝑢 , that depends differentiably on a parameter, 𝑝 ,is defined as 4.2. Sensitivity Analysis. In mathematical modelling, param- 𝜕𝑢 𝑝 eters whose values are not precisely known are oeft n used Υ fl × . (51) and may vary within some ranges. Numerical methods used to solve equations derived from models may introduce Media campaigns in this paper contribute to the limita- numerical errors in the results. eTh eect ff s of such errors tion of the disease transmission. eTh reproduction number or uncertainties in the model’s parameters are quantiefi d 𝑅 is an important concept when it comes to the disease through sensitivity analysis. eTh aim of sensitivity analysis transmission, because it helps to determine EVD incidence. is to quantify the inu fl ence of parameters variation on The normalized forward sensitivity indices of 𝑅 with calculated results [24]. respect to each parameter 𝑢 in expression (22) are given by Sensitivity indices allow us to measure the relative change in a state variable when a parameter changes. The normalized Υ fl × . (52) forward sensitivity index of a variable to a parameter is the 𝜕𝑢 𝑅 ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable Table 3 represents the numerical values of the sensitivity function of the parameter, the sensitivity index may be indices of the reproduction number 𝑅 for the parameters alternatively defined using partial derivatives (see [25]). used in the model. eTh most important parameters are 𝜕𝑅 𝜕𝑝 10 International Journal of Dieff rential Equations Table 3: Sensitivity indices for EVD reproduction number. 5. Discussion and Conclusion Parameter Sensitivity index To model the potential eeff ct of media campaigns on Ebola 𝜔 +1.5 transmission, we used a deterministic model, with compart- ments comprising individuals with different EVD infection 𝜇 +1.37 status, who send EVD related messages through media. eTh 𝛽+1 eect ff of media campaigns on people’s behaviour is repre- 𝑐+1 sented by a reduction factor which decreases the number of 𝜂 +0.613 new EVD cases. Stability analysis was presented in terms of 𝛾 +0.074 the model reproduction number 𝑅 .Itwas shownthatthe 𝜃 +0.037 disease-free and the endemic equilibria are locally stable if Λ −1.5 𝑅 <1 and 𝑅 >1, respectively. The inclusion of the 𝑀 𝑀 𝛼 −1.5 asymptomatic infected class resulted in the model exhibiting 𝜌 −0.613 a backward bifurcation, emphasizing the necessity of intense 𝜎 −0.347 efforts against EVD as a result of undetected asymptomatic cases. eTh existence of a backward bifurcation has important 𝑝 −0.05 implications in the design of policies and strategies to erad- 𝛿 −0.022 icate or control an epidemic. In the presence of a backward 𝛿 −0.002 bifurcation, classical policies on disease eradication need to be changed as EVD can persist even when the threshold parameter 𝑅 is less than one. thosewiththe highestabsolutevalues. Negative andpositive To be able to control EVD, governments and interna- correlations of the parameters to the reproduction number tional stakeholders should implement feasible campaigns are indicated by negative and positive signs. eTh parameters taking into account the social, economic, and mainly the Λ and 𝛼 have the largest absolute negative numerical values 1 cultural realities of the aeff cted countries. Interventions from with negative sensitivity index values. us, Th increasing their these campaigns should target the aeff cted populations and values will decrease EVD incidence. This result can be help them to understand the disease, comply with control explained by the fact that any increase in the two parameters measures, which sometimes seem severe, and change their leadstoanincreaseintheecffi acyofmediacampaigns.So,the behaviour in order to stop the disease transmission chain [1]. more the media campaigns are ecffi acious, the less the value The best way to contain this outbreak is to jointly implement of the reproduction number is. Another important parameter case isolation, contact tracing with quarantine, and sanitary with a negative index is 𝜌 , which is an expected result since funeral practices as suggested in [26]. burials of EVD dead bodies limit the disease transmission due This model is not without shortcomings. People’s reaction to infected corpses. The outdating rate of media campaigns to media campaigns does not always lead to a reduction in 𝜔 has the most positive influence on EVD reproduction the number of future contamination cases. So, a function rep- number. This means that the more frequently the messages resenting the influence of media campaigns on individuals’ spread by media on EVD are updated, the lower the number behaviour which takes into account the dieff rent cultural set- of new infections is. eTh larger the values of 𝜔 and 𝜇 are, the tings would be an innovative and informative addition to this less the ecffi acy of the messages is and the more the disease model. Aspects of quarantine, contact tracing, and case iden- spreads. The parameters 𝛽 and 𝑐 form the transmission rate ticfi ation initiatives are possible additions that can make this and their increase will directly contribute to an increase in the model more reliable. Despite these shortcomings, this model number of EVD cases. provides a good description of EVD outbreak. The model investigates a very important aspect in disease control in our times, that is, the use of social media in spreading messages. 4.3. Simulations Results and Interpretation. Figures 5(a) and 5(b) confirm the results on stability analysis. It follows that Appendix when 𝑅 <1, the epidemic dies out and, for 𝑅 >1,EVD 𝑀 𝑀 becomes endemic. This is a graphical description of the fact A. Differential Inequalities that the DFE is locally stable for 𝑅 <1 and the EE is locally stable whenever 𝑅 >1. 𝑀 Corollary A.1. Let 𝑥 and 𝑦 be real numbers, 𝐼=[𝑥 ,+∞), 0 0 0 The media campaigns reproduction number 𝑅 is made and 𝑎,𝑏 ∈𝐶(𝐼). Suppose that 𝑦∈𝐶 (𝐼) satisfies the following of parameters which differently influence its values in a inequality: variety of ways. The relationship between those parameters 𝑦 𝑥 ≤𝑎 𝑥 𝑦 𝑥 +𝑏 𝑥 , 𝑥≥𝑥 ,𝑦(𝑥 )=𝑦 . (A.1) ( ) ( ) ( ) ( ) 0 0 0 canbeevaluated throughcontour plots. We chosetwo parameters, 𝛼 and 𝜔 , whose influence on the reproduction Then, number is clearly significant as shown in the expression of 𝑦 (𝑥 )≤𝑦 exp [ ∫ 𝑎 (𝑡 )𝑑𝑡] 𝑅 .Figure6showsthat 𝛼 largely influences 𝑅 when 𝑀 1 𝑀 compared to 𝜔 .Increasingthe values of 𝛼 decreases 𝑅 . (A.2) 1 𝑀 𝑥 𝑥 u Th s, in order to eradicate EVD, the exchange of EVD related + ∫ 𝑏 (𝑠 )exp [ ∫ 𝑎 (𝑡 )]𝑡𝑑 ,𝑠𝑑 𝑥≥𝑥 . 𝑥 𝑠 messages is critical in eradicating the epidemic. 0 International Journal of Dieff rential Equations 11 14000 250000 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Time (days) Time (days) E R E R I D I D a a I I s s (a) (b) Figure 5: Population size at DFE (a) for 𝑅 = 0.61 and EE (b) for 𝑅 = 2.09 with Λ = 20000, 𝜇 = 0.02 , 𝛽 = (0.2, 0.105) , 𝑐 = (8, 12) , 𝜎 = 0.525 , 𝑀 𝑀 −6 −7 −7 −9 𝛾 = (0.2, 0.25) , 𝑝 = 0.17 , 𝜃 = 0.12 , 𝛿 = (0.199, 0.0313), 𝛿 = (0.0001, 0.0013), 𝜔 = 0.2 , 𝛼 = (1.8 × 10 , 1.2 × 10 ), 𝛼 =(2 × 10 ,2 × 10 ), 1 2 1 2 −6 −8 −7 −8 −7 −8 𝛼 =(5 × 10 ,5 × 10 ), 𝛼 =(8 × 10 ,8 × 10 ), 𝛼 = (9.99 × 10 , 9.9 × 10 ), 𝜂 = 1.5 ,and 𝜌 = 0.497 . 3 4 5 0.9 which yields 𝛼 𝑆 𝛼 𝑆 0.8 1 1 𝑀 (𝑡 ) ≥ exp [−𝜔𝑡 ] (𝑀 (0)− )+ . (A.5) 𝜔 𝜔 0.7 Before the disease is spread, we assume that 𝑀 is at the steady state level. So, 𝑀(0) = 𝛼 𝑆/𝜔 which is equivalent to 𝑀(0) = 0.6 𝑀 and (A.5) will give 𝜔 0.5 𝛼 𝑆 (A.6) 𝑀 (𝑡 ) ≥ . 0.4 Together with the assumption 0<𝑀≤1 ,wethushave 0.3 𝑀 ≤𝑀 (𝑡 )≤1. (A.7) 0.2 0.1 0.005 0.01 0.015 0.02 0.025 0.03 B. An Approach to Determine the Direction of Figure 6: Reproduction number contour plot. the Bifurcation eTh orem B.1. Consider a general system of ordinary dieff ren- tial equations with a parameter 𝜙 : If the converse inequality holds in (A.1), then the converse inequality holds in (A.2) too. =𝑓(𝑥,)𝜙 , Let us prove that 𝑀(𝑡) is bounded. (B.1) 𝑛 𝑛 2 𝑛 Proof. From (9), we have 𝑓: 𝑅 ×𝑅→ 󳨀 𝑅 ,𝑓 ∈𝐶 (𝑅 ×𝑅). (𝑡 ) Withoutlossofgenerality, it is assumedthat 0 is an (A.3) ≥𝛼 𝑆 (𝑡 )−𝜔𝑀 (𝑡 ). equilibrium for system (B.1) for all values of the parameter 𝜙 ; that is, 𝑓(0, 𝜙) ≡ 0 for all 𝜙. For system (3)–(9), 𝑀 (𝑡) ≥ 𝛼 𝑆−𝜔.𝑀 By applying Corollary A.1, we have Assume that 𝑀 (𝑡 )≥𝑀 (0)exp [ ∫ (−𝜔 )𝑑𝑢] (A1) 𝐴=𝐷 𝑓(0, 0) = ((𝜕𝑓 /𝜕𝑥 )(0, 0))is the linearisation 𝑥 𝑖 𝑗 (A.4) matrix of system (B.1) around equilibrium 0 with 𝜙 𝑡 𝑡 evaluated at 0. Zero is a simple eigenvalue of 𝐴 and all + ∫ 𝛼 𝑆 exp [ ∫ (−𝜔 )𝑑 V]𝑑,𝑧 ∀𝑡 ≥ 0, 0 𝑧 other eigenvalues of 𝐴 have negative real parts; 0.8 1.2 1.2 1.6 1.6 1.6 2 2 2 Population Population 𝑑𝑡 𝑑𝑀 𝑑𝑡 𝑑𝑥 12 International Journal of Dieff rential Equations (A2) matrix 𝐴 has a nonnegative right eigenvector 𝑤 References and a left eigenvector V corresponding to the zero [1] World Health Organisation, Ebola and Marburg virus dis- eigenvalue. Let 𝑓 be the 𝑘 th component of 𝑓 and ease epidemics: preparedness, alert, control, and evaluation, WHO/HSE/PED/CED/2014.05, 2014. 𝑛 2 𝜕 𝑓 𝑘 [2] Centers for Disease Control and Prevention, Morbidity and 𝑎= ∑ V 𝑤 𝑤 (0, 0), 𝑘 𝑖 𝑗 Mortality Weekly Report, https://www.cdc.gov/mmwr/pre- 𝑖 𝑗 𝑘,𝑖,𝑗=1 view/mmwrhtml/mm6325a4.htm. (B.2) 𝑛 2 [3] J.J.Muyembe-Tamfum, S. Mulangu, J. Masumu,J.M.Kayembe, 𝜕 𝑓 𝑏= ∑ V 𝑤 (0, 0). 𝑘 𝑖 A. Kemp, and J. T. Paweska, “Ebola virus outbreaks in Africa: 𝜕𝜙 𝑘,𝑖=1 past and present,” Onderstepoort Journal of Veterinary Research, vol. 79,no. 2, 8pages,2012. [4] E.M.Leroy,S.Baize,V.E.Volchkovetal.,“Humanasymp- The local dynamics of (B.1) around 0 are totally determined tomatic Ebola infection and strong inflammatory response,” The by 𝑎 and 𝑏 . Lancet, vol. 355, no. 9222, pp. 2210–2215, 2000. [5] J. Astacio, D. Briere, M. Guillen, J. Martinez, F. Rodriguez, and N. Valuenzuela-Campos, “Mathematical Models to study the (1)𝑎>0 , 𝑏>0 .When 𝜙<0 with |𝜙| ≪ 1 , outbreaks of Ebola,” http://ecommons.cornell.edu/bitstream/ 0 is locally asymptotically stable, and there exists a 1813/31962/1/BU-1365-M.pdf. positive unstable equilibrium; when 0<𝜙≪1 , [6] C. N. Haas, “On the quarantine period for ebola virus,” PLoS 0 is unstable and there exists a negative and locally Currents Outbreaks,2014. asymptotically stable equilibrium. [7] S. E. Bellan, J. R. C. Pulliam, J. Dusho,ff and L. A. Meyers, (2) 𝑎<0 , 𝑏<0 .When 𝜙<0 with |𝜙| ≪ 1 , 0 is unstable; “Ebola control: effect of asymptomatic infection and acquired when0<𝜙=1 , 0 is locally asymptotically stable, and immunity,” The Lancet ,vol.384,no. 9953,pp. 1499–1500, 2014. there exists a positive unstable equilibrium. [8] A. G. Baxter, “Symptomeless infection with Ebola virus,” Com- mentary,vol.355,article 2210,2014. (3)𝑎>0 , 𝑏<0 .When 𝜙<0 with |𝜙| ≪ 1 , 0 [9] World Health Organisation, Case Den fi ition Recommendations is unstable, and there exists a locally asymptotically for Ebola or Marburg Virus Diseases, WHO, Geneva, Switzer- stable negative equilibrium; when 0<𝜙≪1 , 0 is land, 2014. stable, and a positive unstable equilibrium appears. [10] Y. Xiao, T. Zhao, and S. Tang, “Dynamics of an infectious dis- eases with media/psychology induced non-smooth incidence,” (4) 𝑎<0 , 𝑏>0 .When 𝜙 changes from negative to Mathematical Biosciences and Engineering,vol.10, no.2,pp. positive, 0 changes its stability from stable to unsta- 445–461, 2013. ble. Correspondingly, a negative unstable equilibrium [11] Y. Xiao, S. Tang, and J. Wu, “Media impact switching surface becomes positive and locally asymptotically stable. during an infectious disease outbreak,” Scienticfi Reports ,vol.5, article 7838, 2015. Corollary B.2. When 𝑎>0 and 𝑏>0 ,the bifurcationat [12] J. M. Tchuenche and C. T. Bauch, “Dynamics of an infectious 𝜙=0 is subcritical or backward. disease where media coverage influences transmission,” ISRN Biomathematics,vol.2012, ArticleID581274, 10 pages, 2012. Competing Interests [13] A. Sharma and A. K. Misra, “Backward bifurcation in a smoking cessation model with media campaigns,” Applied Mathematical The authors declare that there are no competing interests Modelling,vol.39, no.3-4,pp. 1087–1098, 2015. regarding the publication of this paper. [14] K. A. Pawelek, A. Oeldorf-Hirsch, and L. Rong, “Modeling the impact of Twitter on influenza epidemics,” Mathematical Biosciences and Engineering. MBE, vol. 11, no. 6, pp. 1337–1356, Authors’ Contributions [15] S.D.DjiombaNjankou andF.Nyabadza, Modelling the potential Farai Nyabadza developed the model and supervised Sylvie role of control strategies on Ebola virus disease dynamics (Master Diane Djiomba Njankou during her M.S. which resulted in thesis in Mathematical Biology) [M.S. thesis], Stellenbosch Uni- this paper. Sylvie Diane Djiomba Njankou carried out the versity, Stellenbosch, South Africa, 2015. analysis and model simulations. [16] C. M. Rivers,E.T.Lofgren,M.Marathe,S.Eubank, andB.L. Lewis, “Modelling the impact of interventions on an epidemic of EbolainSierraLeone andLiberia,” PLOS Current Outbreaks, Acknowledgments vol. 6, 2014. 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