Dynamical analysis and control strategies in modelling Ebola virus disease
 333 Downloads
Abstract
Ebola virus disease (EVD) is a severe infection with an extremely high fatality rate spread through direct and indirect contacts. Recently, an outbreak of EVD in West Africa brought public attention to this deadly disease. We study the spread of EVD through a twopatch model. We determine the basic reproduction number, the diseasefree equilibrium, two boundary equilibria and the endemic equilibrium when the disease persists in the two subpopulations for specific conditions. Further, we introduce timedependent controls into our proposed model. We analyse the optimal control problem where the control system is a mathematical model for EVD that incorporates educational campaigns. The control functions represent educational campaigns in their respective patches, with one patch having more effective controls than the other. We aim to study how these control measures would be implemented for a certain time period, in order to reduce or eliminate EVD in the respective communities, while minimising the intervention implementation costs. Numerical simulations results are provided to illustrate the dynamics of the disease in the presence of controls.
Keywords
Ebola Reproduction number Sensitivity analysis Educational campaign Optimal control1 Introduction
Ebola virus disease (EVD) is a highly infectious disease caused by the Ebola virus from the family of Filoviridae viruses. The Ebola virus is believed to be found in mammals of the family of Pteropodidae (fruit bats) [1]. It is a virus having a filamentous, enveloped and a nonsegmented negative sense with the entry of the virus to living cells facilitated by its enveloped glycoprotein [2, 3]. The current Ebola virus disease (EVD) outbreak in West Africa is now the largest documented [4]. Ebola originated in Zaire and Sudan in 1976 with several strains known at present. The outbreak in 2015 affected mainly Guinea, Liberia and Sierra Leone (West Africa) causing more than 28 000 infections and over 11 100 deaths by December 2015 [5]. The epidemic continued increasing due to socioeconomic disadvantage and shortages in the health systems of the three mainly affected countries (Guinea, Liberia and Sierra Leone) [4, 6]. Currently, in 2018 the EVD has resurfaced in the Province of Kivu in the Democratic Republic of Congo (DRC). In Kivu, there is a military conflict and there are thousands of displaced refugees [7, 8]. The affected area has a lot of trade among the different communities within, and this leads to massive intercommunity movements. Due to the running battles in the Province of Kivu, health care workers and those who might be able to assist in the fight against EVD find it difficult to reach all the communities [9]. Thus, the respective communities in the affected areas of the DRC have greatly varying intervention strategies and some are not perfect. In the Province of Kivu, rebels are against the presence of the health caregivers and UN peacekeepers, hence EVD might be difficult to curtail in the Democratic Republic of Congo [9, 10].
Partially eaten fruits and pulp dropped by the bats are then eaten by the land mammals such as monkeys, apes, baboons and gorillas. This chain of events then forms a possible transmission of the Ebola virus from bats to other mammals [1]. Infection with humans can occur through direct contact with blood or body fluids (like saliva, sweat, faeces, breast milk and semen), objects like needles and syringes that have been contaminated with the virus and infected fruit bats or other mammals [11, 12]. The survival of the virus in the environment, due to poor hygienic and sanitary conditions, is probably another source of Ebola infection in many places in Africa [13, 14]. In Africa, and particularly in the regions that were affected by Ebola outbreaks, people who live close to the rainforests, hunt bats and monkeys and harvest forest fruits for food [15, 16]. Africans are sincere to the point that even a transmittable disease would not stop them from showing compassion for their relatives at home, caressing them and shaking hands, as this is part of their beliefs and customs. In addition, in the course of the funerals, they bath and clothe up their lifeless relatives [17]. They even share without appropriate washing the clothes of their lifeless relatives. Thus, the virus can be spread directly or indirectly.
Without effective vaccines and treatment, educational campaigns may be implemented as a countermeasure. One of the primary reasons for the spread of the virus is the poorly functioning health systems in the part of Africa where the disease occurs. Other reasons are illiteracy, poverty and lack of enough information on the mode of spread of the virus [18]. People who care for infected individuals have an increased risk of transmission. Recommended measures when caring for the infected include medical isolation through proper use of gloves, masks, gowns, boots and goggles as well as sterilisation of equipment and surfaces. Certain control strategies are employed when local, regional and international associates are informed of an establishment of a possible Ebola epidemic. The controls among them are and not restricted to the following; assessment of the worldwide health risk, establishment of social mobilisation and health education curriculum to listen and address public concerns. The application of such control procedures leads to the curtailing of the current EVD epidemic of 2014, 2015. Due to the weaker economies of the affected countries, it was a limiting factor in the fight against EVD [17]. Resource allocation needs to be optimal and the control strategies need to be implemented in such a way as to derive maximum benefits. It is worth noting that optimal control is best described by mathematical modelling. Optimal control theory has proven to be a successful tool in understanding ways to curtail the spread of infectious diseases by devising the optimal disease intervention strategies. The method consists of minimising the cost of infection or the cost of implementing the control, or both. For more on optimal control theory in epidemiology, we refer the reader to Malik and Sharomi, 2015 [19].
Mathematical modelling remains a powerful tool in describing the dynamics of disease outbreaks and making predictions. It also enables us to estimate the long term effects of interventions, integrates evidence from different scientific disciplines and investigates how healthrelated practices evolve in complex systems. Statistical and biological studies have been applied in the study of the complexity of the Ebola virus life ecology [20, 21], and are worthy of attention. Some dynamic models have been proposed to study and to try to understand the dynamics of EVD [22, 23, 24, 25, 26, 27, 28, 29, 30]. Recently, Berge et al. [29] proposed a deterministic mathematical model for the transmission dynamics of EVD which involved a synergy between the epizootic phase, enzootic phase and the endemic phase. It included the direct and indirect modes of contamination, between and within the three different types of populations consisting of humans, animals and fruit bats. Work done by Berge et al. [29] was an extension of their previous work done in 2015 [26], in which a simple mathematical model was developed, which incorporated both the direct and the indirect Ebola virus transmission in such a way that there is a provision of Ebola viruses. Models have also been developed to try and understand various intervention strategies in trying to curtail the spread of EVD [31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. The impact of vaccination and vaccines was investigated in [35, 36, 39, 40] and the issue of quarantining analysed in [35, 37] through mathematical models. Further, Berge et al. [31] developed a mathematical model to understand the impact of contact tracing as a control strategy of EVD. In recent times, optimal control problems have generated a lot of interest from researchers. Furthermore, various techniques and intervention strategies have been applied to study optimal control problems related to EVD [41, 42, 43, 44, 45, 46]. In particular, Area et al. [44] proposed an EVD mathematical model with the vaccination of the susceptible population, with the aim of controlling the spread of the disease and analysis of two optimal control problems related to the transmission of EVD with vaccination. A deterministic SEIR type model with additional hospitalisation, quarantine and vaccination was developed and studied by Ahmad et al. [43], to understand the disease dynamics.
Although substantial work has been done on the study of EVD transmission dynamics, not much consideration of metapopulation study has been done; see [47, 48]. The population is subdivided into several discrete patches which are supposed to be well mixed. Then, in each patch, the population is subdivided into compartments corresponding to different epidemic status. For more on metapopulation patch models, see [49]. In this work, motivated by the usefulness of and the current investigation on modelling EVD, we intend to systematically investigate the modelling and analysis of a twopatch model for the transmission dynamics of EVD. We make use of the model studied by Berge et al. [26] which takes into account both, the direct and indirect modes of transmission. Thus, in our model, we distinguish two patches. We assume the susceptible and the recovered individuals are the only ones who migrate, hence the two subpopulations can vary. This is consistent with the fact that in African refugee camps refugees move very little between two camps and only persons that bring them help move within the camps. These limited migrations on the susceptible individuals have taken place in places such as the Democratic Republic of Congo [9, 10]. We also support our assumption that the majority of the infectious individuals do not migrate, due to their state of health. This can also be supported by the fact that in periods of epidemics, countries tighten people’s movements and their respective boarders through screening and detection. Both the people who are allowed to cross a border and those who move between two refugee camps/communities are assumed to be susceptible and recovered. A mathematical model describing the optimal control of an epidemic with educational campaigns have been discussed before [50], hence in our twopatch model, we introduce two timedependent controls in educational campaigns, which have different effectiveness.
The paper is structured as follows. The EVD transmission model is formulated in Sect. 2 and the analytical results of the model are presented in Sect. 3. In Sect. 4 optimal control theory has been applied to the model formulated in Sect. 2. Simulation results and projection profiles of EVD are presented in Sect. 5. A summary and concluding remarks complete the paper.
2 Model formulation
We consider a twopatch model for Ebola virus disease (EVD). The model consists of two subpopulations of a large one. The recruitment in each subpopulation is only in the susceptible class, and the migration between the subpopulations is by the susceptible and recovered. Having the infectious class not migrating, we assume that infection does not take place during the migration process. Each human subpopulation is divided into four classes: the susceptibles \(S_{i}(t)\), infectious \(I_{i}(t)\), recovered \(R_{i}(t)\) and deceased human individuals \(D_{i}(t)\). The Ebola virus in the environment is denoted by \(W_{i}(t)\). For all the parameters and compartmental classes, we will consider the case for \(i=1,2\), representing patch 1 and 2, respectively. The susceptible human population is replenished by constant recruitment at rate \(b_{i}\). Death for a reason that is not related to EVD is proportional to the population size and with a constant rate \(\mu _{i}\). The additional death due to disease affects only the class \(I_{i}(t)\), at a rate \(v_{i}\).
2.1 Basic properties of the model
In this section, we study the basic properties of the solutions of system (3) which are essential in the proofs of stability.
Theorem 1
Let the initial data be\(S_{i}(0) >0\), \(I_{i}(0)>0\), \(D_{i}(0)>0\), \(W_{i}(0)>0\), \(i=1,2\). Then the solutions\(S_{i}(t)\), \(I_{i}(t)\), \(D_{i}(t)\), \(W_{i}(t)\)for system (3) are nonnegative for all\(t >0\).
A complete proof for Theorem 1 has been outlined in Appendix 1.
Theorem 2
A complete proof for Theorem 2 has been outlined in Appendix 2.
3 Analysis of the model
3.1 The diseasefree equilibrium and basic reproduction number

\(\mathcal{R}_{I_{i}}\) is the contribution of the infectious human \(I_{i}\).

\(\mathcal{R}_{D_{i}}\) is the contribution of the infected corpses \(D_{i}\).

\(\mathcal{R}_{W_{i}}\) is the contribution due to the environmental contamination by the virus \(W_{i}\).
If \(\mathcal{R}_{0} \leq 1\), then the diseasefree equilibrium \(\mathcal{E}^{0}\) is the only equilibrium in \(\mathcal{G}\). Using Theorem 2 in van den Driessche and Watmough (2002) [55], the following result is established.
Theorem 3
The diseasefree equilibrium (DFE) \(\mathcal{E}^{0}\)of system (3) is locally asymptotically stable (LAS) if\(\mathcal{R} _{0} < 1\)and unstable otherwise.
We now utilise the approach of Lyapunov functions [57, 58, 59] in the analysis of the global asymptotic stability.
Theorem 4
If\(\mathcal{R}_{0} \leq 1\), the DFE is globally asymptotically stable (GAS) in\(\mathcal{G}\). If\(\mathcal{R}_{0} > 1\), the system is uniformly persistent.
Proof
If \(\mathcal{R}_{0} > 1\), then, by continuity, \(\dot{\mathcal{L}} > 0\) in a neighbourhood of \(\mathcal{E}^{0}\) in the interior of \(\mathcal{G}\). Solutions in the interior of \(\mathcal{G}\) sufficiently close to \(\mathcal{E}^{0}\) move away from the DFE, implying that the DFE is unstable. This completes the proof. □
The result in Theorem 4 shows that \(\mathcal{R}_{0} = 1\) is a sharp threshold for disease dynamics: the disease will die out when \(\mathcal{R}_{0} \leq 1\), whereas the disease will persist when \(\mathcal{R}_{0} > 1\). We now investigate uniform persistence; we claim the following result.
Theorem 5
Proof
It is worth noting that \(\mathcal{E}^{0}\) is globally stable for system (3). It is clear that there is only an equilibria \(\mathcal{E}^{0}\) in \(M_{\partial }\), by the aforementioned claim, it then follows that \(\mathcal{E}^{0}\) is an isolated invariant set in X, \(W^{S}(\mathcal{E}^{0}) \cap X_{0} = \emptyset \). Clearly, every orbit in \(M_{\partial }\) converges to \(\mathcal{E}^{0}\), \(\mathcal{E} ^{0}\) is acyclic in \(M_{\partial }\). Using Theorem 4.6 in Thieme [62], we conclude that system (3) in uniformly persistent with respect to \((X_{0},\partial X_{0})\).
By the result of [63, 64, 65], system (3) has an equilibrium \(\mathcal{E}^{*} = (\overline{S}_{i},\overline{I}_{i}, \overline{D}_{i},\overline{W}_{i}) \in X_{0}\), \(i=1,2\). We further claim that \(\overline{S}_{i} > 0\), \(i=1,2\). Suppose that \(\overline{S}_{i} = 0\), \(i=1,2\), from the second equation of (3), we can get \(\overline{I}_{i} = \overline{D}_{i} = \overline{W}_{i} = 0\), \(i=1,2\). It is a contradiction. Then \((\overline{S}_{i},\overline{I}_{i}, \overline{D}_{i},\overline{W}_{i}, i=1,2)\), is a componentwise positive equilibrium of system (3). The proof is complete. □
3.2 Existence of equilibria
System (3) has one diseasefree equilibrium, \(\mathcal{E} ^{0} = (S_{1}^{0},0,0,0,S_{2}^{0},0,0,0)\) and three endemic equilibria of the forms \(\mathcal{E}^{*}_{1} = (S_{1}^{*},I_{1}^{*},D_{1}^{*},W _{1}^{*},S_{2}^{*},0,0,0)\), \(\mathcal{E}^{*}_{2} = (S_{1}^{**},0,0,0,S _{2}^{**},I_{1}^{**},D_{1}^{**},W_{1}^{**})\) and \(\mathcal{E}^{*} = ( \overline{S}_{1},\overline{I}_{1},\overline{D}_{1},\overline{W}_{1}, \overline{S}_{2},\overline{I}_{2},\overline{D}_{2},\overline{W}_{2})\), corresponding respectively to states where the disease persists in the first subpopulation and dies out in the second subpopulation, the disease persists in the second subpopulation and disappears in the first subpopulation, and when the disease persists in the two subpopulations.
3.2.1 The first boundary equilibrium
Lemma 1
A1) \(A=D_{x}f(0,0)= ( \frac{\partial f_{i}}{\partial x_{j}}(0,0) )\)is the linearisation of system (24) around the equilibrium 0 andϕevaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts
 i.
\(a>0\), \(b>0\). When\(\phi < 0\)with\(\vert \phi \vert \ll 1\), and there exists a positive unstable equilibrium, when\(0 < \phi \ll 1\), 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.
 ii.
\(a<0\), \(b<0\). When\(\phi <0\)with\(\vert \phi \vert \ll 1\), 0 unstable; when\(0< \phi \ll 1\), 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.
 iii.
\(a>0\), \(b<0\). When\(\phi <0\)with\(\vert \phi \vert \ll 1\), 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when\(0 < \phi \ll 1\), 0 is stable, and a positive unstable equilibrium appears.
 iv.
\(a<0\), \(b>0\). Whenϕchanges from negative to positive, 0 changes its stability from stable to unstable. Corresponding a negative unstable equilibrium becomes positive and locally asymptotically stable.
Theorem 6
The unique patch 1only boundary equilibrium, \(\mathcal{E}^{*}_{1}\), of system (3), is locally asymptotically stable when\(\mathcal{R}_{1} > 1\)but only if\(\mathcal{R}_{1}\)is close to 1.
3.2.2 The second boundary equilibrium
Theorem 7
The unique patch 2only boundary equilibrium, \(\mathcal{E}^{*}_{2}\), of system (3), is locally asymptotically stable when\(\mathcal{R}_{2} > 1\)but only if\(\mathcal{R}_{2}\)is close to 1.
3.2.3 The interior endemic equilibrium
We now investigate the local stability of the interior endemic equilibrium (29), and we shall make use of Lemma 1. We claim the following result.
Theorem 8
The unique interior equilibrium, \(\mathcal{E}^{*}\), of system (3), is locally asymptotically stable when\(\mathcal{R}_{0} > 1\)but only if the corresponding reproduction number is close to 1.
The proof of Theorem 8 is outlined in Appendix 4.
The existence of the endemic equilibrium of system (3) is summarised in the following theorem.
Theorem 9
 1.
A boundary endemic equilibrium of the form\(\mathcal{E}^{*}_{1} = (S _{1}^{*},I_{1}^{*},D_{1}^{*},W_{1}^{*},S_{2}^{*},0,0,0)\)whenever\(\mathcal{R}_{1} > 1\)and\(\mathcal{R}_{2} \leq 1\). This means that the disease is endemic in the first subpopulation and dies out in the second population.
 2.
A boundary endemic equilibrium of the form\(\mathcal{E}^{**}_{2} = (S _{1}^{**},0,0,0,S_{2}^{**},I_{2}^{**},D_{2}^{**},W_{2}^{**})\)whenever\(\mathcal{R}_{1} \leq 1\)and\(\mathcal{R}_{2} > 1\). This means that the disease is endemic in the second subpopulation and dies out in the first population.
 3.
An interior endemic equilibrium of the form\(\mathcal{E}^{*} = ( \overline{S}_{1},\overline{I}_{1},\overline{D}_{1},\overline{W}_{1}, \overline{S}_{2},\overline{I}_{2},\overline{D}_{2},\overline{W}_{2})\)whenever\(\mathcal{R}_{1} > 1\)and\(\mathcal{R}_{2} > 1\)which corresponds to the case when the disease persists in the two subpopulations.
4 Optimal control in educational campaigns
4.1 The optimality system
Theorem 10
Proof
5 Numerical simulations
In this section, we now provide some numerical simulations. The existence of an optimal control is provided and the behaviour of the optimality system made of 10 ordinary differential equations is evaluated through numerical simulations done with Matlab. The optimality system is solved using an iterative method with the Runge–Kutta fourth order scheme. Starting with a guess for the adjoint variables, the state equations are solved forward in time. Then these state values are used to solve the adjoint equations backward in time, and the iterations continue until convergence.
Model parameters, \(i =1,2\), for patch 1 and 2 respectively. The time unit is in weeks
Definition  Symbol  Baseline values  Source 

Effective contact rate of human individuals  \(\beta _{I_{i}} \)  0.16  [74] 
Effective contact rate of deceased human individuals  \(\beta _{D_{i}} \)  0.489  [74] 
Effective contact rate of Ebola virus  \(\beta _{W_{i}}\)  0.062  [74] 
Rate of migration from first subpopulation to the second subpopulation (susceptible)  \(a_{i}\)  (0,0.3)  Assumed 
Rate of migration from first sub population to the second sub population (recovered)  \(b_{i}\)  (0,0.3)  Assumed 
Natural death rate of human individuals  \(\mu _{i}\)  (0,1)  [75] 
Recovery rate of human individuals  \(\phi _{i}\)  0.018 (0.16–0.202)  [76] 
Diseaseinduced death rate of human individuals  \(v_{i}\)  0.5  [74] 
Decay rate of Ebola virus in the environment  \(r_{i}\)  (0,∞)  
Shedding rate of infectious human individuals  \(\delta _{i}\)  (0,∞)  Assumed 
Shedding rate of deceased human individuals  \(\rho _{i}\)  (0,∞)  Assumed 
Burial rate of deceased human individuals  \(\alpha _{i}\)  (0,∞) 
We first consider a scenario where we have net migration on \(a_{2}\), thus, \(a_{2} > a_{1}\) implying that we have \(\mathcal{R}_{1} > \mathcal{R}_{2} > 1\). We take \(a_{1} = 0.03\), \(a_{2} = 0.3\) and we assume that all other parameters take the same values for the two patches (see Table 1).
We now consider a scenario where we have 0 net migration, thus \(a_{2} = a_{1}\), implying that we have \(\mathcal{R}_{1} = \mathcal{R} _{2} > 1\). We take \(a_{1} = 0.165\), \(a_{2} = 0.165\) and we assume that all other parameters take the same values for the two patches (see Table 1).
Lastly, we consider a scenario where we have a net migration in \(a_{1}\), thus \(a_{1} > a_{2}\), implying that we have \(\mathcal{R}_{2} > \mathcal{R}_{1} > 1\). We take \(a_{1} = 0.3\), \(a_{2} = 0.03\) and we assume that all other parameters take the same values for the two patches (see Table 1).
6 Discussion
Several countries in West Africa, in particular, Sierra Leone, Guinea and Liberia experienced morbidity and mortality during the Ebola epidemic from 2013–2015. At the time of this epidemic there was no known vaccine or drug, so effective disease control required coordinated efforts that include both standard strategies, such as hospitalisation, as well as community efforts, such as safe burial practices, proper hygiene in hospitals, etc. Not only are such efforts difficult to implement in practice, but there is also added complexity with connectivity between populations that have different policies in place. These complexities may affect some communities, that is, neighbouring communities might have lesser efforts compared to their neighbours in terms of control strategies. Currently, starting in August 2018 EVD is terrorising North Kivu in the Democratic Republic of Congo, a place with high mobility among the traders from different communities [7, 8].
In this manuscript, we formulated and analysed a differential equationbased twopatch model, with the patches connected by migration. EVD features and dynamics within a population such as the infectious environment, deceased individuals and the infectious individuals are incorporated in both patches. The susceptible and recovered individuals are the only ones who migrate. The basic reproductive number, \(\mathcal{R}_{0}\), for the model has been computed. Our results show that \(\mathcal{R}_{0}\), can provide a sharp threshold for the disease dynamics when \(\mathcal{R}_{0} \leq 1\) the diseasefree equilibrium is globally stable indicating that the disease would die out, but when \(\mathcal{R}_{0} > 1\) the disease persists. Subsequently, we have performed an optimal control study on this twopatch model to effectively design control strategies to control EVD. The control \(u_{1}\) represents educational campaigns in patch 1 and the control \(u_{2}\) represents educational campaigns in patch 2, with \(u_{1} > u _{2}\). We considered two communities with the same control strategy, which differ in their levels of efficacy. Our major aim is to assess the spread of the EVD epidemic within two communities connected by migration, with these communities employing the same control strategy which only differ in efficiency. The technical tool used to determine the optimal strategy is the Pontryagin maximum principle. Numerical simulations results show that the implementation of the optimal control has a huge impact on the reduction of the infected individuals in both patches, and that the outcome of the control from each patch may be different due to their different characteristics. We also realised that, given our controls \(u_{1} > u_{2}\), it is advisable to have \(a_{2} > a_{1}\), as in the net migration is into patch 1, so as to control EVD faster and with maximum effort for fewer resources. It is worth noting that for the scenario \(a_{2} > a_{1}\) we have \(\mathcal{R}_{1} > \mathcal{R}_{2} > 1\). Thus, in the case of an EVD epidemic, we can manage to move the people who are not infected from places that have inefficient intervention strategies to places with higher efficient intervention strategies, regardless of the relative magnitude of the respective reproduction numbers in both communities. Generally, the study finds that EVD can be controlled if optimal educational campaigns are implemented, although this might not be appropriate for certain time intervals.
However, just like any other model, we cannot say the model is complete, it can be extended to include the aspect of the refugees, rebels and health care workers, like the case in the Democratic Republic of Congo.
Notes
Acknowledgements
A. Mhlanga, the author, would like to thank the editors and the anonymous referees for their helpful comments and suggestions. He would also like to acknowledge with thanks the support of the Department of Mathematics, University of Zimbabwe.
Availability of data and materials
The data used to support the findings of this study are included within the article and cited accordingly.
Authors’ contributions
The author did all the work. All authors read and approved the final manuscript.
Funding
None declared.
Ethics approval and consent to participate
None declared.
Competing interests
The author declares that they have no competing interests.
Consent for publication
None declared.
References
 1.Leroy, E.M., Kumulungui, B., Pourrut, X., Rouquet, P., Hassanin, A., Yaba, P., Delicat, A., Paweska, J.T., Gonzalez, J.P., Swanepoel, R.: Fruit bats as reservoirs of Ebola virus. Nature 438, 575–576 (2015) CrossRefGoogle Scholar
 2.Takada, A., Robison, C., Goto, H., Sanchez, A., Murti, K.G., Whitt, M.A., et al.: A system for functional analysis of Ebola virus glycoprotein. Proc. Natl. Acad. Sci. 94(26), 14764–14769 (1997) CrossRefGoogle Scholar
 3.WoolLewis, R.J., Bates, P.: Characterization of Ebola virus entry by using pseudo typed viruses: identification of receptordeficient cell lines. J. Virol. 72, 3155–3160 (1998) Google Scholar
 4.Lough, S.: Lessons from Ebola bring WHO reforms. CMAJ, Can. Med. Assoc. J. 187(12), E377–E378 (2015) CrossRefGoogle Scholar
 5.WHO: Ebola response roadmap situation report. World Health Organization. http://www.who.int/csr/disease/Ebola/situationreports/en/ (2015). Accessed 5 Feb 2015
 6.Lewnard, J.A., Ndeffo Mbah, M.L., AlfaroMurillo, J.A., Altice, F.L., Bawo, L., Nyenswah, T.G., et al.: Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis. Lancet Infect. Dis. 14(12), 1189–1195 (2014) CrossRefGoogle Scholar
 7.NPR.org: Ebola in a conflict zone. Retrieved 2 August 2018 Google Scholar
 8.Relief Web: Congo Ebola outbreak compounds already dire humanitarian crisis. Retrieved 3 August 2018 Google Scholar
 9.Reuters: Editorial, Reuters (20180904). Rebels ambush South African peacekeepers in Congo Ebola zone. Retrieved 4 September 2018 Google Scholar
 10.VOA: Rebel attack in Congo Ebola zone kills at least 14 civilians. Retrieved 23 September 2018 Google Scholar
 11.Peters, C., Peters, J.: An introduction to Ebola: the virus and the disease. J. Infect. Dis. 179, 9–16 (1999). https://doi.org/10.1086/514322 CrossRefGoogle Scholar
 12.CDC: CDC report to Ebola virus disease 2014. Technical report (2014) Google Scholar
 13.Bibby, K., Casson, L.W., Stachler, E., Haas, C.N.: Ebola virus persistence in the environment: state of the knowledge and research needs. Environ. Sci. Technol. Lett. 2, 2–6 (2015) CrossRefGoogle Scholar
 14.Piercy, T.J., Smither, S.J., Steward, J.A., Eastaugh, L., Lever, M.S.: The survival of filoviruses in liquids, on solid substrates and in a dynamic aerosol. J. Appl. Microbiol. 109(5), 1531–1539 (2010) Google Scholar
 15.Leroy, E.M., Rouquet, P., Formenty, P., Souquière, S., Kilbourne, A., Froment, J.M., Bermejo, M., Smit, S., Karesh, W., Swanepoel, R., Zaki, S.R., Rollin, P.E.: Multiple Ebola virus transmission events and rapid decline of central African wildlife. Science 303(5656), 387–390 (2006) CrossRefGoogle Scholar
 16.Leroy, E.M., Kumulungui, B., Pourrut, X., Rouquet, P., Hassanin, A., Yaba, P., Délicat, A., Paweska, J.T., Gonzalez, J.P., Swanepoel, R.: Fruit bats as reservoirs of Ebola virus. Nature 438, 575–576 (2005) CrossRefGoogle Scholar
 17.Butler, D.: Six challenges to stamping out Ebola. http://www.nature.com/ (2015)
 18.Chan, M.: Ebola virus disease in West Africa—no early end to the outbreak. N. Engl. J. Med. 371, 1183–1185 (2014) CrossRefGoogle Scholar
 19.Sharomi, O., Malik, T.: Optimal control in epidemiology. Ann. Oper. Res. 251(1–2), 55–71 (2015). https://doi.org/10.1007/s1047901518344 MathSciNetCrossRefzbMATHGoogle Scholar
 20.Groseth, A., Feldmann, H., Strong, J.E.: The ecology of Ebola virus. Trends Microbiol. 15, 408–416 (2007) CrossRefGoogle Scholar
 21.Judson, S.D., Fischer, R., Judson, A., Munster, V.J.: Ecological contexts of index cases and spillover events of different Ebola viruses. PLoS Pathog. 12(8), e1005780 (2016) CrossRefGoogle Scholar
 22.Area, I., Losada, J., Ndaïrou, F., Nieto, J.J., Tcheutia, D.D.: Mathematical modeling of 2014 Ebola outbreak. Math. Methods Appl. Sci. 40, 6114–6122 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
 23.Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A.: On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015, 278 (2015). https://doi.org/10.1186/s1366201506135 MathSciNetCrossRefzbMATHGoogle Scholar
 24.Imran, M., Khan, A., Ansari, A., Shah, S.: Modeling transmission dynamics of Ebola virus disease. Int. J. Biomath. 10(4), 1750057 (2015). https://doi.org/10.1142/S1793524517500577 MathSciNetCrossRefzbMATHGoogle Scholar
 25.Ivorra, B., Ngom, D., Ramos, A.M.: A mathematical model to predict the risk of human diseases spread between countriesvalidation and application to the 2014–2015 Ebola virus disease epidemic. Bull. Math. Biol. 77(9), 1668–1704 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
 26.Berge, T., Lubuma, J.M.S., Moremedi, G.M., Morris, N., KonderaShava, R.: A simple mathematical model for Ebola in Africa. J. Biol. Dyn. 11(1), 42–74 (2017). https://doi.org/10.1080/17513758.2016.1229817 MathSciNetCrossRefGoogle Scholar
 27.Chavez, C., Barley, K., Bichara, D., Chowell, D., Diaz Herrera, E., Espinoza, B., Moreno, V., Towers, S., Yong, K.E.: Modeling Ebola at the Mathematical and Theoretical Biology Institute (MTBI). Not. Am. Math. Soc. 63(4), 367–371 (2016) MathSciNetzbMATHGoogle Scholar
 28.Agusto, F.B.: Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 283, 48–59 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
 29.Berge, T., Bowong, S., Lubuma, J., Manyombe, M.L.M.: Modeling Ebola virus disease transmissions with reservoir in a complex virus life ecology. Math. Biosci. Eng. 15(1), 21–56 (2018). https://doi.org/10.3934/mbe.2018002 MathSciNetCrossRefzbMATHGoogle Scholar
 30.Funk, S., Camacho, A., Kucharski, A.J., Eggo, R.M., Edmunds, W.J.: Realtime forecasting of infectious disease dynamics with a stochastic semimechanistic model. Epidemics 22, 56–61 (2018) CrossRefGoogle Scholar
 31.Berge, T., Ouemba Tasse, A.J., Tenkam, H.M., Lubuma, J.: Mathematical modelling of contact tracing as a control strategy of Ebola virus disease. Int. J. Biomath. 11(7), 1850093 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
 32.Guo, Z., Xiao, D., Li, D., Wang, X., Wang, Y., Yan, T., Wang, Z.: Predicting and evaluating the epidemic trend of Ebola virus disease in the 2014–2015 outbreak and the effects of intervention measures. PLoS ONE 11(4), e0152438 (2016). https://doi.org/10.1371/journal.pone.0152438 CrossRefGoogle Scholar
 33.Salem, D., Smith R.: A mathematical model of Ebola virus disease: using sensitivity analysis to determine effective intervention targets. In: SummerSimSCSC 2016, Montreal, Quebec, Canada, July 24–27 2016. Society for Modelling & Simulation International (SCS) (2016) Google Scholar
 34.Weitz, J.S., Dushoff, J.: Modeling postdeath transmission of Ebola: challenges for inference and opportunities for control. Sci. Rep. 5, 8751 (2015). https://doi.org/10.1038/srep08751 CrossRefGoogle Scholar
 35.Tulu, T.W., Tian, B., Wu, Z.: Modeling the effect of quarantine and vaccination on Ebola disease. Adv. Differ. Equ. 2017, 178 (2017). https://doi.org/10.1186/s136620171225z MathSciNetCrossRefzbMATHGoogle Scholar
 36.Berge, T., Chapwanya, M., Lubuma, J., Terefe, Y.A.: A mathematical model for Ebola epidemic with self protection measures. J. Biol. Syst. 26(1), 107–131 (2017). https://doi.org/10.1142/S0218339018500067 MathSciNetCrossRefzbMATHGoogle Scholar
 37.Denes, A., Gumel, A.B.: Modeling the impact of quarantine during an outbreak of Ebola virus disease. Infect. Dis. Model. 4, 12–27 (2019) Google Scholar
 38.Kucharski, A.J., Eggo, R.M., Watson, C.H., Camacho, A., Funk, S., Edmunds, W.J.: Effectiveness of ring vaccination as control strategy for Ebola virus disease. Emerg. Infect. Dis. 22(1), 105–108 (2016) CrossRefGoogle Scholar
 39.Bodine, E.N., Cook, C., Shorten, M.: The potential impact of a prophylactic vaccine for Ebola in Sierra Leone. Math. Biosci. Eng. 15(2), 337–359 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
 40.Kelly, J., et al.: Projections of Ebola outbreak size and duration with and without vaccine use in Équateur, Democratic Republic of Congo, as of May 27, 2018. PLoS ONE 14, e0213190 (2019) CrossRefGoogle Scholar
 41.Jiang, S., Wang, K., Li, C., Hong, G., Zhang, X., Shan, M., Li, H., Wang, J.: Mathematical models for devising the optimal Ebola virus disease eradication. J. Transl. Med. 15, 124 (2017). https://doi.org/10.1186/s1296701712246 CrossRefGoogle Scholar
 42.Diane, S., Njakou, D., Nyabadza, F.: An optimal control model for Ebola virus disease. J. Biol. Syst. 24(1), 1–21 (2016) MathSciNetGoogle Scholar
 43.Muhammad, D.A., Muhammad, U., Adnan, K., Mudassar, I.: Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination. Infect. Dis. Poverty 5, 72 (2016). https://doi.org/10.1186/s4024901601616 CrossRefGoogle Scholar
 44.Area, I., et al.: Ebola model and optimal control with vaccination constraints. J. Ind. Manag. Optim. 14(2), 427–446 (2018) MathSciNetzbMATHGoogle Scholar
 45.Rachah, A., Torres, D.F.M.: Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discrete Dyn. Nat. Soc. 2015, Article ID 842792 (2015). https://doi.org/10.1155/2015/842792. MathSciNetCrossRefzbMATHGoogle Scholar
 46.Takaidza, I., Makinde, O.D., Okosun, O.K.: Computational modelling and optimal control of Ebola virus disease with nonlinear incidence rate. J. Phys. Conf. Ser. 818(1), 012003 (2017) CrossRefGoogle Scholar
 47.Meakin, S., Tildesley, M., Davis, E., Keeling, M.: A metapopulation model for the 2018 Ebola outbreak in Equateur province in the Democratic Republic of the Congo. Cold Spring Harbor Laboratory: bioRxiv 465062. https://doi.org/10.1101/465062 (2018)
 48.Ivorra, B., Ngom, D., Ramos, A.M.: Version 4: BeCoDiS: an epidemiological model to predict the risk of human diseases spread between countries. Validation and application to the 2014 Ebola Virus Disease epidemic. Preprint arXiv.org; Cornell Universiy Library, Date:. 1410. 1–32. (2014). http://arxiv.org/abs/1410.6153
 49.Njagarah, J.B., Nyabadza, F.: A metapopulation model for cholera transmission dynamics between communities linked by migration. Appl. Math. Comput. 241, 317–331 (2014). https://doi.org/10.1016/j.amc.2014.05.036 MathSciNetCrossRefzbMATHGoogle Scholar
 50.Castillo, C.: Optimal control of an epidemic through educational campaigns. Electron. J. Differ. Equ. 2006, 125 (2006) MathSciNetzbMATHGoogle Scholar
 51.Piercy, T.J., Smither, S.J., Steward, J.A., Eastaugh, L., Lever, M.T.: The survival of filoviruses in liquids, on solid substrates and in a dynamic aerosol. J. Appl. Microbiol. 109(5), 1531–1539 (2010) Google Scholar
 52.Francesconi, P., Yoti, Z., Declich, S., Onek, P.A., Fabiani, M., Olango, J., Andraghetti, R., Rollin, P.E., Opira, C., Greco, D., Salmaso, S.: Ebola hemorrhagic fever transmission and risk factors of contacts, Uganda. Emerg. Infect. Dis. 9(11), 1430–1437 (2003) CrossRefGoogle Scholar
 53.Chowell, G., Nishiura, H.: Transmission dynamics and control of Ebola virus disease (EVD): a review. BMC Med. 12, 196 (2014) CrossRefGoogle Scholar
 54.Youkee, D., Brown, C.S., Lilburn, P., Shetty, N., Brooks, T., Simpson, A., Bentley, N., Lado, M., Kamara, T.B., Walker, N.F., Johnson, O.: Assessment of environmental contamination and environmental decontamination practices within an Ebola holding unit, Freetown, Sierra Leone. PLoS ONE (2015). https://doi.org/10.1371/journal.pone.0145167 CrossRefGoogle Scholar
 55.Van den Driessche, P., Watmough, J.: Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
 56.CastilloChavez, C., Feng, Z., Huang, W.: On the computation of \(R_{0}\) and its role on global stability. In: Mathematical Approaches for Emerging and Re Emerging Infectious Diseases: An Introduction, Minneapolis, MN, 1999. IMA Math. Appl., vol. 125, pp. 229–250. Springer, New York (2002) CrossRefGoogle Scholar
 57.Korobeinikov, A.: Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. 21, 75–83 (2004) zbMATHCrossRefGoogle Scholar
 58.McCluskey, C.C.: Lyapunov functions for tuberculosis models with fast and slow progression. Math. Biosci. Eng. 3, 603–614 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
 59.Shuai, Z., Heesterbeek, J.A.P., Van den Driessche, P.: Extending the type reproduction number to infectious disease control targeting contact between types. J. Math. Biol. 67, 1067–1082 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
 60.Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) zbMATHCrossRefGoogle Scholar
 61.La Salle, J.P.: The Stability of Dynamical Systems. CBMSNSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 12 (1976) CrossRefGoogle Scholar
 62.Thieme, H.R.: Persistence under relaxed pointdissipativity with an application to en epidemic model. SIAM J. Math. Anal. 24, 407–435 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
 63.Zhao, X.Q.: Uniform persistence and periodic coexistence states in infinitedimensional periodic semi flows with applications. Can. Appl. Math. Q. 3, 473–495 (1995) zbMATHGoogle Scholar
 64.Wang, W.D., Zhao, X.Q.: An epidemic model in a patchy environment. Math. Biosci. 190, 97–112 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
 65.Wang, W.D., Fergola, P., Tenneriello, C.: Innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003) MathSciNetzbMATHGoogle Scholar
 66.CastilloChavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
 67.Levy, B., Edholm, C., Gaoue, O., KaonderaShava, R., Kgosimore, M., Lenhart, S., Lephodisa, B., Lungu, E., Marijani, T., Nyabadza, F.: Modeling the role of public health education in Ebola virus disease outbreaks in Sudan. Infect. Dis. Model. 2(3), 323–340 (2017) Google Scholar
 68.Mallela, A., Lenhart, S., Vaidya, N.K.: HIVTB coinfection treatment: modelling and optimal control theory perspectives. J. Comput. Appl. Math. 307, 143–161 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
 69.Kirschner, D., Lenhart, S., Serbin, S.: Optimal control of the chemotherapy of HIV. J. Math. Biol. 35(7), 775–792 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
 70.WHO: Ebola and Marburg disease epidemics: preparedness, alert, control and evaluation. World Health Organization, WHO/HSE/PED/CED/2014.05 (2014) Google Scholar
 71.Pontryagin, L.S., Boltyanskii, V.T., Gamkrelidze, R.V., Mishchevko, E.F.: The Mathematical Theory of Optimal Processes. Gordon & Breach, New York, 4 (1985) Google Scholar
 72.Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975) zbMATHCrossRefGoogle Scholar
 73.Lenhart, S., Workman, J.T.: Optimal Controls Applied to Biological Models. Chapman & Hall/CRC, London (1997) zbMATHGoogle Scholar
 74.Rivers, C.M., Lofgren, E.T., Marathe, M., Eubank, S., Lewis, B.L.: Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLOS Curr. Outbreaks. Edition 1. https://doi.org/10.1371/currents.outbreaks.fd38dd85078565450b0be3fcd78f5ccf (2014)
 75.Ndanguza, D., Tchuenche, J.M., Haario, H.: Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo. Afr. Math. 24, 55–68 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
 76.WHO Ebola Response Team: Ebola virus disease in West Africathe first 9 months of the epidemic and forward projections. N. Engl. J. Med. 371(16), 1481–1495 (2014). https://doi.org/10.1056/NEJMoa1411100 CrossRefGoogle Scholar
 77.Bibby, K., Casson, L.W., Stachler, E., Haas, C.N.: Ebola virus persistence in the environment: state of the knowledge and research needs. Environ. Sci. Technol. Lett. 2, 2–6 (2015) CrossRefGoogle Scholar
 78.The Centers for Disease Control and Prevention: Ebola (Ebola virus disease). http://www.cdc.gov/Ebola/resources/virusecology.html (Page last reviewed August 1, 2014)
 79.Fasina, F.O., Shittu, A., Lazarus, D., Tomori, O., Simonsen, L., Viboud, C., Chowell, G.: Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014. Euro Surveill. 19(40), 20920 (2014). http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=20920 CrossRefGoogle Scholar
 80.Towers, S., PattersonLomba, O., CastilloChavez, C.: Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. PLOS Curr. September 18 (2014) Google Scholar
 81.Li, M.Y., Graef, J.R., Wang, L., Karsai, J.: Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191–213 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.