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Dynamical analysis and control strategies in modelling Ebola virus disease
Advances in Difference Equations volume 2019, Article number: 458 (2019)
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 two-patch model. We determine the basic reproduction number, the disease-free equilibrium, two boundary equilibria and the endemic equilibrium when the disease persists in the two sub-populations for specific conditions. Further, we introduce time-dependent 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.
1 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 non-segmented 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 socio-economic 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 inter-community 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 rain-forests, 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 health-related 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 meta-population 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 meta-population 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 two-patch 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 sub-populations 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 two-patch model, we introduce two time-dependent 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 two-patch model for Ebola virus disease (EVD). The model consists of two sub-populations of a large one. The recruitment in each sub-population is only in the susceptible class, and the migration between the sub-populations 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 sub-population 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 \(v_{i}\).
It has been shown that human individuals can be infected through the contaminated environment [51]. Thus, the Ebola virus is transmitted though both direct and indirect transmission. The susceptible individuals acquire the infection at rate \(\varLambda _{i}\) with
the form of incidence is bilinear. \(\beta _{I_{i}}I_{i} + \beta _{D_{i}}D _{i}\) is the direct infection from the infectious individuals and the dead bodies. \(\beta _{W_{i}}\) is the indirect infection from the Ebola virus in the environment to the susceptible individuals. \(\beta _{I _{i}}\) is the effective contact rate of the infectious individuals, \(\beta _{D_{i}}\) is the effective contact rate of the deceased individuals and \(\beta _{W_{i}}\) is the effective contact rate of the Ebola virus (in the environment).
Motivated by the fact that it has never been reported that an individual caught EVD for the second time, we assume that the infectious individuals recover at rate \(\phi _{i}\) attaining permanent disease-induced immunity. We assume that the Ebola virus finds its way into the environment only through the infectious and the deceased human individuals, which they shed into the environment especially through the urine and stool at rates \(\delta _{i}\) and \(\rho _{i}\), respectively. This assumption is supported by several references [52,53,54]. This happens in regions of poor sanitary facilities and/or in the regions where people do not observe appropriate hygiene practices. The Ebola virus in the environment decays and loses its infectiousness at a rate \(r_{i}\). The deceased human individuals lose their infectiousness through methods such as proper handling and burial at a rate \(\alpha _{i}\). The migration rates of the susceptible and the recovered individuals, between the two populations, are \(a_{i}\) and \(b_{i}\) for \(i=1,2\) respectively. This yields the following set of differential equations:
The total human population, the total deceased human individuals and the Ebola virus pathogens in the environment are given by \(N(t) = S_{1}(t) + I_{1}(t) + R_{1}(t) + S_{2}(t) + I_{2}(t) + R_{2}(t)\), \(D(t) = D_{1}(t) + D_{2}(t)\) and \(W(t) = W_{1}(t) + W_{2}(t)\), respectively.
Once \(S_{i}\), \(I_{i}\), \(D_{i}\), \(W_{i}\), \(i=1,2\) are obtained from Eqs. (2)1, (2)2, (2)4, (2)5, the functions \(R_{i}\), \(i=1,2\) are readily given by Eq. (2)3. Thus without loss of generality, we are led to the following reduced system:
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 non-negative for all \(t >0\).
A complete proof for Theorem 1 has been outlined in Appendix 1.
Theorem 2
System (3) is a dynamical system on the biologically feasible compact set
where \(b = b_{1} + b_{2}\), \(\mu _{0} = \min (\mu _{1},\mu _{2})\), \(\overline{v} = \max (v_{1},v_{2})\), \(\overline{\delta } = \max (\delta _{1},\delta _{2})\), \(\overline{\rho } = \max (\rho_{1},\rho _{2})\), \(\overline{\alpha} = \max (\alpha _{1},\alpha _{2})\), \(\underline{\alpha} = \min (\alpha _{1},\alpha _{2})\), \(\underline{r}= \min (r_{1},r_{2})\).
A complete proof for Theorem 2 has been outlined in Appendix 2.
3 Analysis of the model
3.1 The disease-free equilibrium and basic reproduction number
The global behaviour for this model crucially depends on the basic reproduction number, that is, the average number of secondary cases produced by a single infective individual which is introduced into an entirely susceptible population. System (3) has an evident equilibrium \(\mathcal{E}^{0} = (S_{1}^{0},0,0,0,0,S_{2}^{0},0,0,0,0)\) when there is no disease, where
We calculate the basic reproduction number, \(\mathcal{R}_{0}\), using the next generation method approach developed in van den Driessche and Watmough [55]. Following [55], the non-negative matrix F and the non-singular matrix V for the new infection terms and the remaining transfer terms are respectively given at the disease-free equilibrium by
Then the reproduction number \(\mathcal{R}_{0}\) of system (3) is the spectral radius of the next generation matrix \(FV^{-1}\),
where \(\mathcal{R}_{i}\) (\(i=1,2\)) is the basic reproduction number of each sub-population i given by
Following the interpretation in [56], \(\mathcal{R}_{i}\) for patch i, for \(i=1,2\), is the sum of three infection contributions:
\(\mathcal{R}_{I_{i}}\) 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 \(W_{i}\).
Biologically \(\mathcal{R}_{1}\) and \(\mathcal{R}_{2}\) measure the average number of secondary infections generated by an Ebola virus in patch 1 and patch 2 respectively, when introduced in a disease-free population. The reproductive number \(\mathcal{R}_{0}\) controls the number of equilibria of system (3).
If \(\mathcal{R}_{0} \leq 1\), then the disease-free 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 disease-free 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.
Let \(\mathcal{Y}(t) = (I_{1},D_{1},W_{1},I_{2},D_{2},W_{2})\). Since
it follows that
where F and V are defined in (6). It is worth noting that F and \(V^{-1}\) are non-negative. By the Perron–Frobenius theorem [60], the non-negative matrix \(V^{-1}F\) has a non-negative left eigenvector \(u \geq 0\) with respect to \(\rho (V^{-1}F) = \rho (FV^{-1}) = \mathcal{R}_{0}\), that is, \(u^{T}V^{-1}F = \mathcal{R}_{0}u^{T}\). Motivated by [59], we consider the Lyapunov function
Differentiating \(\mathcal{L}\) along solutions of (3), we have
It can be easily verified that the largest invariant subset of \(\mathcal{G}\) where \(\dot{\mathcal{L}} = 0\) is the singleton \(\lbrace \mathcal{E}^{0} \rbrace \). Therefore, by LaSalle's invariance principle [61], \(\mathcal{E}^{0}\) is globally asymptotically stable in \(\mathcal{G}\) when \(\mathcal{R}_{0} \leq 1\).
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
If \(\mathcal{R}_{0} > 1\), system (3) is uniformly persistent, namely, there exists a constant \(\zeta > 0\) such that
where ζ is independent of the initial data in \(\mathcal{G}\).
We prove that system (3) is uniformly persistent with respect to \((X_{0},\partial X_{0})\), where
By the form of (3), it is very easy to see that both X and \(X_{0}\) are positively invariant and \(\partial X_{0}\) is relatively closed in X. Furthermore, system (3) is point dissipative. Let
We now show that
It is worth noting that
so we only need to prove
Assume \(\lbrace (S_{i}(0),I_{i}(0),D_{i}(0),W_{i}(0)) \rbrace \in M _{\partial }\), for \(i=1,2\). It suffices to show that
Suppose not, then there exist an \(i_{0}\), \(i_{0}=2\) and \(t_{0} \geq 0\) such that
without loss of generality, assume
then we have
It follows that there is an \(\eta > 0\) such that \(W_{2}(t) > 0\), for \(t_{0} < t < t_{0} + \eta \). This means that
does not belong to \(\partial X_{0}\), \(i =1,2\) for \(t_{0} < t < t_{0} + \eta \), which contradicts the assumption that
Thus (15) holds.
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 component-wise positive equilibrium of system (3). The proof is complete. □
3.2 Existence of equilibria
System (3) has one disease-free 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 sub-population and dies out in the second sub-population, the disease persists in the second sub-population and disappears in the first sub-population, and when the disease persists in the two sub-populations.
3.2.1 The first boundary equilibrium
The patch-1 system is obtained by making \(I_{2} = D_{2} = W_{2} = 0\), and it is given by
Let \(\mathcal{E}^{*}_{1} = (S_{1}^{*},I_{1}^{*},D_{1}^{*},W_{1}^{*},S _{2}^{*},0,0,0)\) be the first boundary endemic equilibrium with \(I_{1},D_{1}, W_{1} > 0\). Then the endemic equilibrium \(\mathcal{E} ^{*}_{1}\) can be obtained by setting to zero the right hand side of Eqs. (21), giving
It is straightforward to prove that, when \(\mathcal{R}_{1} > 1\) and \(\mathcal{R}_{2} \leq 1\), system (3) has exactly one non-trivial boundary endemic equilibrium \(\mathcal{E}^{*}_{1} = (S _{1}^{*},I_{1}^{*},D_{1}^{*},W_{1}^{*},S_{2}^{*},0,0,0)\) where \(S_{1} ^{*}\), \(I_{1}^{*}\), \(D_{1}^{*}\), \(W_{1}^{*}\), \(S_{2}^{*}\) are given by
We now present the local stability of \(\mathcal{E}^{*}_{1}\) when the reproduction number is close to 1. We make use of the centre manifold theory [66], but we first present the following lemma.
Lemma 1
Consider the following general system of ordinary differential equations with parameter ϕ:
where 0 is an equilibrium of the system, that is, \(f(0,\phi )=0 \) ∀ϕ, and assume:
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 and ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts
A2) Matrix A has the right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue. Let \(f_{k}\) be the kth component of f and
The local dynamics of (24) around zero are totally governed by a and b.
\(a>0\), \(b>0\). 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.
\(a<0\), \(b<0\). 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.
\(a>0\), \(b<0\). 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.
\(a<0\), \(b>0\). When ϕ changes 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 1-only 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.
The proof of Theorem 6 is outlined in Appendix 3.
3.2.2 The second boundary equilibrium
Let \(\mathcal{E}^{**}_{2} = (S_{1}^{**},0,0,0,S_{2}^{**},I_{2}^{**},D _{2}^{**},W_{2}^{**})\) be the second boundary endemic equilibrium with \(I_{2},D_{2},W_{2} \neq 0\). Then the endemic equilibrium \(\mathcal{E} ^{**}_{2}\) (steady state with \(I_{2},D_{2},W_{2} > 0\)) can be obtained by setting the right hand side of equations in system (3) to zero, that is,
It is straightforward to prove that, when \(\mathcal{R}_{2} > 1\) and \(\mathcal{R}_{1} \leq 1\), system (3) has exactly one non-trivial boundary endemic equilibrium \(\mathcal{E}^{**}_{2} = (S _{1}^{**},0,0,0,S_{2}^{**},I_{2}^{**},D_{2}^{**},W_{2}^{**})\) where \(S_{1}^{**}\), \(S_{2}^{**}\), \(I_{2}^{**}\), \(D_{2}^{**}\), \(W_{2}^{**}\) are given by
Theorem 7
The unique patch 2-only 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.
The proof for Theorem 7 follows the same procedure as outlined in Sect. 3.2.1 on the proof of the stability of patch 1-only boundary equilibrium, and the bifurcation parameter is computed by setting \(\mathcal{R}_{2} = 1\). □
3.2.3 The interior endemic equilibrium
\(\mathcal{E}^{*} = (\overline{S}_{1},\overline{I}_{1},\overline{D} _{1},\overline{W}_{1},\overline{S}_{2},\overline{I}_{2},\overline{D} _{2},\overline{W}_{2})\) be the interior endemic equilibrium when both infectious of the two sub-populations co-exist, like, \(I_{i} \neq 0\), \(D _{i} \neq 0\), \(W_{i} \neq 0\), \(i=1,2\). To find this endemic equilibrium, we equate system (3) to zero and rewrite it as follows:
Solutions of Eqs. (28) are defined by
Note that, when \(\mathcal{R}_{1}^{\star } > 1\) and \(\mathcal{R}_{2} ^{\star } > 1\), one has \(\mathcal{R}_{1} > 1\) and \(\mathcal{R}_{2} > 1\). Then system (3) has a non-trivial interior endemic equilibrium \(\mathcal{E}^{*} = (\overline{S}_{1},\overline{I}_{1},\overline{D} _{1},\overline{W}_{1},\overline{S}_{2},\overline{I}_{2},\overline{D} _{2},\overline{W}_{2})\) when \(\mathcal{R}_{1}^{\star } > 1\) and \(\mathcal{R}_{2}^{\star } > 1\).
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
System (3) has:
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 sub-population and dies out in the second population.
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 sub-population and dies out in the first population.
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 sub-populations.
4 Optimal control in educational campaigns
In this section, an optimal control problem is formulated by incorporating two intervention strategies into our basic model (3). Optimal control has been applied in the study of infectious diseases such as in [67,68,69]. Stopping the EVD transmission chain is still the main strategy to limit the spread of the disease. This can be achieved through educational campaigns that encourage behavioural changes like safe burial practices, self-isolation and proper hygiene [70]. To include educational campaigns into patch 1 and patch 2 as controls, \(u_{1}\) and \(u_{2}\) respectively, we extend system (3). The controlling effort \(u_{1}\) represents a time-dependent control in educational campaigns with respect to patch 1 and \(u_{2}\) representing a time-dependent control in educational campaigns with respect to patch 2. Educational campaigns, in this case, are aimed at encouraging the uninfected to have some protective behaviours. We assume that \(u_{1} > u_{2}\) since some neighbouring countries or communities might have lesser efforts compared to their neighbours in terms of control efforts. The extended model is given by
A successful control strategy is one that reduces the number of individuals infected with Ebola while minimising the costs associated with these efforts. Thus, our goal is to find a pair of controls \((u_{1}^{*},u_{2}^{*})\) that minimise the number of infected individuals and the cost of educational campaigns. Our objective functional is therefore formulated as
subject to the constraints given by (31) and where \(A_{1}\), \(A_{2}\), \(B_{1}\), \(B_{2}\) are positive balancing coefficients transferring integrals into monetary quantity over a finite period of time T weeks. The first two terms in (32) with coefficients \(A_{1}\) and \(A_{2}\) represent the weights of the individuals \(I_{1}\) and \(I_{2}\), respectively. The coefficients \(B_{1}\) and \(B_{2}\) represent the cost associated with the efforts in educational campaigns for patch 1 and 2, respectively. We seek the pair \((u_{1}^{*},u_{2}^{*}) \in \mathcal{U} \) such that
subject to the state system given by (31), where
is the control set. Next, we derive the optimality system.
4.1 The optimality system
Using the Pontryagin maximum principle [71], we now derive the necessary conditions that an optimal control and corresponding states must satisfy. This principle converts (31)–(33) into a problem of minimising pointwise a Hamiltonian \(\mathcal{H}\), with respect to \((u_{1}(t),u_{2}(t))\):
where \(\lambda _{i}\), \(i=1,2,\ldots,10\). are the adjoint variables. In the following theorem, we present the adjoint system and control characterisation.
Theorem 10
Given an optimal control \((u_{1}^{*},u_{2}^{*})\) and corresponding state solutions \(S_{i}\), \(I_{i}\), \(R_{i}\), \(D_{i}\), \(W_{i}\) of the corresponding state system (31) that minimises \(\mathcal{J}(u_{1},u_{2})\) over \(\mathcal{U}\), there exist adjoint variables (functions), \(\lambda _{i}(t)\), for \(i = 1,2,\ldots,10\), satisfying
with terminal conditions
Furthermore, the optimal controls \(u_{1}^{*}\) and \(u_{2}^{*}\) are represented by
The existence of optimal control follows from Corollary 4.1 of [72] since the integrand of \(\mathcal{J}\) is a convex function of \((u_{1},u_{2})\) and the state system satisfies the Lipshitz property with respect to the state variables. The following can be derived from the Pontryagin maximum principle [71]:
with \(\lambda _{i}(T) = 0\) for \(i=1,2,\ldots,10\) evaluated at the optimal controls and corresponding states, which results in the adjoint system (34). The Hamiltonian \(\mathcal{H}\) is minimised with respect to the controls at the optimal controls, so we differentiate \(\mathcal{H}\) with respect to \(u_{1}\) and \(u_{2}\) on the set \(\mathcal{U}\), respectively, giving the following optimality conditions:
Hence, we obtain
Taking into account the bounds on the controls, we obtain the desired characterisations. □
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.
To illustrate the results of the foregoing analysis, we have simulated system (3) using the parameters in Table 1. Parameters in Table 1 are taken from the recent works on the West African Ebola outbreaks. We then assume some of the parameters in the realistic range for illustrative purposes. We are using the same table of parameters for both patches. Among the estimated parameters, are the balancing coefficients which have been arbitrarily chosen for illustration purposes. These weight parameters determine the importance of variables in the objective functional [73]. Thus \(A_{1} =100\), \(A_{2} = 100\), \(B_{1} = B_{2} = 90\).
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).
Figures 1\((a)\) and 1\((b)\) generally show that the controls have an impact on the control of EVD. Figure 1\((a)\) shows that the controls are highly effective in patch 1 since in the presence of the control the infected population is very low for the whole period under study. Figure 1\((b)\) depicts that the controls are effective only after 9 weeks. Figure 1\((c)\) and 1\((d)\) represents the controls \(u_{1}\) and \(u_{2}\) respectively. Both controls are at the upper bound for the whole period under study, 32 weeks. Thus under the given scenario, \(a_{2} > a_{1}\), both controls are feasible and effective for the whole period under study, hence more effort should be devoted to both controls in implementation.
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).
Figure 2\((a)\) and 2\((b)\) shows that the controls are effective in the reduction of EVD. Figure 2\((a)\) shows that the control \(u_{1}\) becomes effective after 14 weeks and Fig. 2\((a)\) shows that control \(u_{2}\) becomes effective after 2 weeks. It is worth noting that Fig. 2\((a)\) and 2\((b)\) shows that the controls are significant to a certain extent in the reduction of EVD. Figure 2\((c)\) and 2\((d)\) shows that the controls may not be sustainable for certain time intervals. In Fig. 2\((c)\) the control \(u_{1}\) is at the lower bound for 14 weeks. Thus, the control \(u_{1}\) is initiated after 14 weeks under this scenario. Control \(u_{2}\) starts at the upper bound and stays at the upper bound for the rest of the period under study. In this scenario, \(a_{2} = a_{1}\), more effort should be devoted to control \(u_{2}\). It is worth noting that control \(u_{2}\) is the less efficient one, hence devoting more effort to control \(u_{2}\) in controlling EVD might not be ideal.
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).
Figures 3\((a)\) and 3\((b)\) depict that the controls are only effective up to a certain extent. Figure 3\((a)\), shows that the controls are effective during the last 28 weeks of the whole of 32 weeks under study. Figure 3\((b)\) shows that the controls are only effective in the first 28 weeks only. Figure 3\((c)\) and 3\((d)\) also shows that the controls may not be sustainable for certain time intervals. Figure 3\((c)\) shows that the control \(u_{1}\) starts at the lower bound and is initiated after 26 weeks. Figure 3\((d)\) shows that control \(u_{2}\) is at the upper bound for the rest of the period under study. Under this scenario, \(a_{1} > a_{2}\), it is worth noting that more effort should be devoted to control \(u_{2}\). Under the given scenario control, \(u_{1}\), which is the most effective, it is not feasible.
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 equation-based two-patch 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 disease-free 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 two-patch 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.
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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.
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Appendix 1
Proof of Theorem 1
Rewriting system (3) in expanded form, we have the following system (system (3) is still the same as system (40)):
It is worth noting that, since (40)1 and (40)5 are first order linear equations in \(S_{1}\) and \(S_{2}\), respectively, it is easy to see that \(S_{1}(t) > 0\) if and only if \(S_{2}(t) > 0\). With this remark in mind, we shall prove below that \(S_{1}(t) > 0\) for \(t \geq 0\). To this end, put \(t_{1}^{0} = \sup \lbrace t > 0, S_{1}(t) > 0 \rbrace \) and \(t_{2}^{0} = \sup \lbrace t > 0, S_{2}(t) > 0 \rbrace \).
If \(t_{1}^{0} = +\infty \) and \(t_{2}^{0} = +\infty \), we use the above-mentioned remark to conclude that both \(S_{1}(t)\) and \(S_{2}(t)\) are positive for all \(t \geq 0\). If \(t_{1}^{0} < \infty \) and \(t_{2}^{0} < \infty \), we are going to prove that this leads to a contradiction. By a continuity argument, the solution functions \(S_{1}(t)\) and \(S_{2}(t)\) change sign at least once in the intervals \(J_{1} = [t_{1}^{0},+\infty )\) and \(J_{2} = [t_{2}^{0},+\infty )\), respectively.
Denote by \(t_{1}^{m} \in J_{1}\) and \(t_{2}^{m} \in J_{2}\) the first real numbers such that \(S_{1}(t_{1}^{m}) =0\) and \(S_{2}(t_{2}^{m}) =0\), respectively. We then have
Without loss of generality, suppose that \(t_{1}^{m} \leq t_{2}^{m}\). Then, from system (40), we have
Equation (42) implies that there exists a positive number \(t_{1}^{m_{1}} > t_{1}^{m}\) such that
Putting Eqs. (41) and (43) together and using the continuity of \(S_{1}(t)\), we conclude that \(t_{1}^{m}\) is an extremum of \(S_{1}(t)\). Moreover, since \(S_{1}(t)\) is a differentiable function on \(\mathbb{R}\), one has \(S'(t_{1}^{m}) = 0\). This is a contradiction to (42). Therefore, \(t_{1}^{0} = +\infty \), which implies that \(t_{2}^{0} = +\infty \) as well.
To prove that \(I_{1}(t),D_{1}(t),W_{1}(t),I_{2}(t),D_{2}(t),W_{2}(t) \geq 0\) for all \(t \geq 0\), whenever
we rewrite the corresponding equations in (40) in the form
Since \(S_{1}(t) > 0\), \(S_{2} (t) > 0\) as established above, \(\mathcal{M}\) is a Metzler matrix. Thus (44) is a monotone system. Therefore \(\mathbb{R}^{6}_{+}\) is invariant under the flow of system (44). This completes the proof of positivity of the solutions. □
Appendix 2
Proof of Theorem 2
Adding the total human population we have
Applying the Gronwall inequality to Eq. (46), we have
which implies that \(0 \leq N(t) \leq \frac{b}{\mu _{0}}\) for all \(t \geq 0\) if \(N(0) \leq \frac{b}{\mu _{0}}\). Furthermore, it follows from Eq. (3)3 that we have the relation
to which another application of the Gronwall inequality yields the bounds
where \(D(0) \leq \frac{\overline{v}b}{\underline{\alpha }_{1}\mu _{0}}\).
The boundedness of \(W(t)\) is proved similarly. □
Appendix 3
Proof of Theorem 6
To apply Lemma 1, the following simplifications and change of variables are carried out. Let \(S_{1}=x_{1}\), \(I_{1}=x_{2}\), \(D_{1}=x _{3}\), \(W_{1}=x_{4}\), \(S_{2}=x_{5}\), \(S_{1}^{0} = x_{1}^{0}\). Furthermore, by using the vector notation \(\mathcal{X} = (f_{1},f_{2},f_{3},f_{4},f _{5})\), system (21) can be written in the form \(\frac{d \mathcal{X}}{dt} = F(x)\), with \(F = (f_{1},f_{2},f_{3},f_{4},f_{5})^{T}\), such that
The Jacobian matrix of system (50) at \(\mathcal{E} ^{0}\) is given by
from which it can be shown that
with \(S_{1}^{0} = x_{1}^{0}\) as defined in Eq. (5).
Now, we consider \(\varrho _{1} \beta _{I_{1}}= \beta _{D_{1}}\) and \(\varrho _{2} \beta _{W_{1}} = \beta _{D_{1}}\) regardless of whether \(\varrho _{1},\varrho _{2} \in (0,1)\) or \(\varrho _{1},\varrho _{2} \geq 1\). Taking \(\beta _{D_{1}}\) as the bifurcation parameter and considering that \(\mathcal{R}_{1} = 1\) and solving for \(\beta _{D_{1}}\), we have
The transformed system (50), with \(\beta _{D_{1}} = \beta ^{*}\), has a hyperbolic equilibrium point (like, the linearised system has a simple eigenvalue with zero real part, and all other eigenvalues have negative real part), so that the centre manifold theory [66] can be used to analyse the dynamics of (50) near \(\beta _{D_{1}} = \beta ^{*}\). It can be shown that the Jacobian of system (50) has a right eigenvector associated with the zero eigenvalue given by \(p = (p_{1},p_{2},p_{3},p _{4},p_{5})^{T}\), where
The left eigenvectors of \(J(\mathcal{E}^{0})\) associated with the zero eigenvalue at \(\beta _{D_{1}} = \beta ^{*}\) is given by \(q = (q_{1},q_{2},q_{3},q_{4},q_{5})^{T}\), where
Computation of the bifurcation parameters a and b. For the sign of a, it can be shown that the associated non-vanishing partial derivatives of F are given by
From (56) it follows that
This excludes the possibility of a backward bifurcation since \(a<0\). For the sign of b, it is associated with the following non-vanishing partial derivatives of F:
It follows from the expressions in (58) that
Thus, \(a<0\) and \(b>0\) and applying Lemma 1 item (iv), we have established our result. The proof is complete. □
Appendix 4
Proof of Theorem 6
To apply Lemma 1, it is convenient to make the following change of variables:
let \(S_{1}=x_{1}\), \(I_{1}=x_{2}\), \(D_{1}=x_{3}\), \(W_{1}=x_{4}\), \(S_{2}=x_{5}\), \(I _{2}=x_{6}\), \(D_{2}=x_{7}\), \(W_{2}=x_{8}\), \(S_{1}^{0} = x_{1}^{0}\), \(S_{2} ^{0} = x_{2}^{0}\). Further, by using the vector notation \(\mathcal{X} = (f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},f_{7},f_{8})\), system (3) can be written in the form \(\frac{d\mathcal{X}}{dt} = F(x)\), with \(F = (f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},f_{7},f_{8})^{T}\), such that
The Jacobian matrix of system (60) at \(\mathcal{E} ^{0}\) is given by
from which it can be shown that \(\mathcal{R}_{0} = \max \lbrace \mathcal{R} _{1}, \mathcal{R}_{2} \rbrace \), where
Now, we consider \(\varrho _{1} \beta _{I_{1}}= \beta _{D_{1}}\), \(\varrho _{2} \beta _{W_{1}} = \beta _{D_{1}}\), \(\varrho _{3} \beta _{I_{2}}= \beta _{D_{2}}\), \(\varrho _{4} \beta _{W_{2}} = \beta _{D_{2}}\) regardless of whether \(\varrho _{1},\varrho _{2},\varrho _{3},\varrho _{4} \in (0,1)\) or \(\varrho _{1},\varrho _{2},\varrho _{3},\varrho _{4} \geq 1\). Taking \(\beta _{D_{1}}\) as the bifurcation parameter and considering that \(\mathcal{R}_{0} = 1\) and solving for \(\beta _{D_{1}}\), we have
Note that the linearised system of the transformed equation (60) with the bifurcation point \(\beta ^{*}\) has a zero eigenvalue. Hence, the centre manifold theory [66] can be used to analyse the dynamics of system (60) near \(\beta _{I_{1}} = \beta ^{*}\). It can be shown that the Jacobian of system (60) has a right eigenvector associated with the zero eigenvalue given by \(p = (p_{1},p_{2},p_{3},p_{4},p_{5},p_{6},p_{7},p _{8})^{T}\), where
with \(\mathcal{H}\) as defined in Eq. (54).
The left eigenvectors of \(J(\mathcal{E}^{0})\) associated with the zero eigenvalue at \(\beta _{D_{1}} = \beta ^{*}\) is given by \(q = (q_{1},q_{2},q_{3},q_{4},q_{5},q_{6},q_{7},q_{8})^{T}\), where
Computation of the bifurcation parameters a and b. For the sign of a, it can be shown that the associated non-vanishing partial derivatives of F are given by
From (65) it follows that
This excludes the possibility of a backward bifurcation since \(a<0\). For the sign of b, it is associated with the following non-vanishing partial derivatives of F:
It follows from the expressions in (67), and after some algebraic manipulations, we have
Thus, \(a<0\) and \(b>0\) and applying Lemma 1 item (iv), we have established our result. The proof is complete. □
Suppose that \(\beta _{D_{2}}\) is chosen as a bifurcation parameter. By using the same approach as was used when \(\beta _{D_{1}}\) was taken as the bifurcation parameter in the proof of Theorem 6, we arrive at the same conclusion, that is, \(a < 0\) and \(b > 0\).
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Mhlanga, A. Dynamical analysis and control strategies in modelling Ebola virus disease. Adv Differ Equ 2019, 458 (2019). https://doi.org/10.1186/s13662-019-2392-x
DOI: https://doi.org/10.1186/s13662-019-2392-x


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King Bach: Every Dating Affair and Girlfriend This Viner Has So Far
Quick Information
- Ex-Girlfriend Lele Pons (2016), Liane Valenzuela (2014-2015), Amanda Cerny (Rumored)
- Official Site andrewbachelor.com
- Eye Color Dark Brown
- Hair Color Black
- Hair Short
- Weight 74 kg (163 lbs)
- Height/ How Tall? 5'8" (1.73 m)
- College Florida State University(2010)
- High School Coral Springs Charter School
- Career 2010-present
- Net Worth $3 Million (Estimated)
- Gay No
- Ex-Wife No
- Date of Birth Jun 26, 1988
- Age 35 Years, 8 Months
- Divorce/Split Yes (Once)
- Relationship Status Single
- Children/Kids Not Yet
- Sister/s Christina Bachelor
- Mother Ingrid Mourice
- Father Byron Bachelor
- Ethnicity/Race African-Jamaican
- Zodiac Sign Cancer
- Birth Name Andrew B. Bachelor
- Profession Actor, Businessman
- Nationality Canadian
It's no secret that the on-screen industry has always attracted highly ambitious people.
Andrew Bachelor, aka King Bach, wanted to be the biggest movie star in the world ever since he watched his first movie.
As of now, Bach is not just an actor but also an online personality, producer, content writer, and Ceo of RuJohn Foundation. And not to mention, this multitalented figure is also a Viner, indubitably, the most successful Viner of all time with more than 9.6 million followers. He also has the most views movie on Netflix.
Well, the actor has even found fame beyond the internet and among the ladies. Accordingly, it is indeed difficult for the 32-year-old Viner not to get tangled up in dating rumors in between his journey.
King Bach Girlfriend
One of the names that are on the list is Liane Valenzuela, who is a Vine star, actress, singer, dancer, and model.
Bach started dating the talented beauty back in January 2014. They were pretty much open about their relationship on social media handles.
They even appeared in a couple of projects together. However, the duo parted their ways in February 2015.
Who is King Bach Dating?
Right after breaking up with Liane, Bach's dating rumors with the model and TV personality, Amanda Cerny, hit the headlines.
Caption: Bache and Amanda have also worked together on several Vines (Published on Aug 29, 2017)
Later, the online media went as far as to claim that they were married for about a year.
Back in July 2015, Bach, however, uploaded a tweeted photo with a caption that referred to Amanda as his friend.
Caption: The picture shared by King Bach with Amanda Cerny on Twitter with the caption "#FriendshipGoals" in July 2015 (Photo Credit: King Bach's Twitter)
Despite the rumors, the pair hasn't made any official statement regarding their possible relationship so far. But, in 2017, Amanda had few loving words to offer to King on her Twitter. Here's the tweet.
I miss you @KingBach wish you were here with me on this adventure. Xo we will forever work hard and live our lives to the fullest. Love you! — Amanda Cerny (@AmandaCerny) August 28, 2017
And, responding to the tweet, Bach expressed his mutual feelings to Amanda. Are they dating or just friends?
Moreover, Bach's dating rumors with YouTube/Vine Star Lele Pons swirled since May 2016 when the duo uploaded a video titled "Couple's therapy."
Later, in September 2016, Pons and Amanda, who were friends, accused each other of public shaming one another.
Caption: King Bach's reported Girlfriends had a public fallout. Photo Credits: Amanda Cerny's Instagram
Reportedly, Amanda first accused Pons of deleting her famous online content without her knowledge, and then she uploaded an Instagram picture blurring Pons' face.
Who is King Bach Girlfriend?
This summer, rumors swirled around the internet that King was in a relationship with Jessica Allain, an actress. This all started when King shared a post on her Instagram where the couple was seen relaxing in Lake Tahoe. The caption read: "Laughter is the best medicine. I just gave her an extra dose."
Jessica, on the other side, shared the same post where her caption read: "Best dau off!"
But the rumors turned down as IMDb reported that the duo was shooting a horror flick directed by Deon Taylor.
Anyway, it's not known if the famous Viner, King Bach, is currently single or is dating someone in 2020. But it is for sure that King Bach doesn't have a wife.
King Bach's Wiki-like Bio
American actor, online personality, and businessman Andrew Byron Bachelor, a.k.a. King Bach, was born on June 26, 1988, in Toronto, Canada, to accountant parents. He inherits Jamaican roots as his parents Ingrid Maurice and Byron Bachelor, were from Jamaica. He spent his childhood in a strict Christian household in West Palm Beach, Florida.
He has a sister whose name is Christina Bachelor.
The actor graduated with a Bachelor Degree of Science in Business Management from Florida State University in 2010. Later, King enrolled at New York Film Academy right after his graduation.
While studying at New York Film Academy, he also joined a comedy troupe and a drama club.
Before joining YouTube and then Vines, Bach dropped out while studying in the last semester at New York Film Academy to pursue his future in the video making world.
King was also an excellent athlete in his early days, but his passion for acting drew him towards the light of stardom.
Remarkably, Andrew has shown his acting and comedy skills not only in his VInes videos but also in TV series such as "Black Jesus," "The Mindy Project," and in movies like "Fifty Shades of Black, and "Meet the Blacks."
Net Worth
Bachelor's net worth is estimated to be $3 million. He is currently working on a gay parody movie,' Get Out 2'; Although the movie revolves around the same story as the original, there's a hilarious twist in the remake.
Byron Bachelor was successful in grabbing the title of the most followed person on Vine in 2015. He had earned 112.3 million followers peaking at the top of the app for the number of followers.

subset: commoncrawl


text:
From: | Raymond Brown <ray.brown@...> |
Date: | Sunday, March 18, 2001, 19:27 |
At 12:18 pm +1100 18/3/01, D Tse wrote:>> >> And the urge to change names to attract tourists was not confined to 19th >> cent. Wales. In England the seaside resort hitherto known simply as >> "Weston-on-Sea" became, and still remains, "Weston-super-mare" (tho, alas, >> the last word is pronounced as tho it were a female horse!) > >A female horse with superpowers!..and to us non-rhotics it sounds as tho the super-powered female horse is also equipped with a couple of six-shooters and wearing a cowboy hat :) -------------------------------------------------------------------- At 9:20 pm -0600 17/3/01, Eric Christopherson wrote: [snip]> >And would "super mare" even mean "near the sea" in Latin?No - Weston-juxta-mare would've made more sense.>I don't know much >(classical) Latin, but in my mind it conjures up images of a town floating >on the water ;)Quite correct - I mean the picture in your mind :) The actual town is only next to sea at high tide; at low tide it seems like a mile away :) ------------------------------------------------------------------- At 4:56 am -0500 18/3/01, Andreas Johansson wrote:>Ray wrote: >>But to get back to length of Welsh names. They are no longer than many >>good ol' English place names like, e.g. Ashby-de-la-Zouch, >>Chapel-en-le-Frith, Piddletrenthide, Chorton-cum-Hardy, >>Bourton-on-the-Water, Mapleborough Green etc etc etc. > >Speaking of English place-names, Bury St. Edmunds is pretty cool.Yep - and the one that always amused me when I was a kid was "Barrow-in-Furness" which I sort of heard as "barrow in furnace", and pictured our wheel-barrow being incinerated :) Ray. ========================================= A mind which thinks at its own expense will always interfere with language. [J.G. Hamann 1760] =========================================


text:
Greek Literature God
Also known as Kadmos
Founder of Thebes and demi-god of alphabets
A prince of divine blood, he was a brother of Europa whose life was dogged by dragons. First he killed one, which happened to be Ares's special favorite. Oops. Then, following the advice of Athena, he got involved in the ridiculous enterprise of sowing dragon's teeth. Everyone knows these turn into giants. Still it all ended well as the giants all killed each other, leaving five survivors who became the first Thebans.
Before settling into kingship, Cadmus thought it prudent to placate Ares, who was still very cross about the dragon-slaying incident. So he spent eight years as his servant. Or should that be serpent? Whoever wrote the job application might have got his wires crossed.
Cadmus obviously got into the God of War's good books, as he ended up marrying his daughter Harmonia. She gave birth to all sorts of legendary people, including Agave, Ino and Semele, and it was a happy ever after thing from then on. Although not for their daughters. Cadmus himself spent a lot of his time inventing new letters of the alphabet, or rather watching while his scribes did it. The Cadmiean Alphabet is still in use today, although not by anyone we exchange emails with.
Cadmus and Harmonia are now enjoying life in the Elysian Fields for ever. In, as it happens, the form of serpents.
Cadmus Facts and Figures
Name: Cadmus
Pronunciation: Coming soon
Alternative names: Kadmos
Gender: Male
Type: God
Celebration or Feast Day: Unknown at present
In charge of: Words and Writing
Area of expertise: Writing, Literature, Words
Good/Evil Rating: Unknown at present
Popularity index: 26235
Cite this article
Here's the info you need to cite this page. Just copy the text in the box below.
Article last revised on September 06, 2018 by Rowan Allen.
Editors: Peter J. Allen, Chas Saunders
References: Coming soon.


text:
stand-out talk at the outside the box conference by thor magnusson, talking about making creative tools in the digital realm. ostensibly about the nature of digital music instruments, it really dug into how their design is far from a neutral act, and how in use our minds often extend to think through them.
there's an academic paper from thor that deals with a lot of this at: http://www.ixi-audio.net/thor/EpistemicTools_OS.pdf
Through the analysis of material epistemologies it is possible to describe the digital instrument as an epistemic tool: a designed tool with such a high degree of symbolic pertinence that it becomes a system of knowledge and thinking in its own terms.
i'd be the first to admit its a much nicer way in to have the presentation before chewing on such texts directly, but its good stuff.


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If adding a cell modem is dealing with a drama queen of a hardware component, then choosing from among the many types of modules available turns the designer into an electronics Goldilocks. There are endless options for packaging and features all designed to make your life easier (or not!) so you-the-designer needs to have a clear understanding of the forces at work to come to a reasonable decision. How else will Widget D'lux® finally ship? You are still working on Widget D'lux®, aren't you?
OK, quick recap from last time. Cell modems can be used to add that great feature known as The Internet to your product, which is a necessary part of the Internet of Things, and thus Good. So you're adding a cell modem! But "adding a cell modem" can mean almost anything. Are you aiming to be Qualcomm and sue Apple build modems from scratch? Probably not. What about sticking a Particle Electron inside to bolt something together quickly? Or talk to Telit and put a bare modem on a board? Unless you're expecting to need extremely high volume and have a healthy appetite for certification glee, I bet you've chosen to get a modem with as many existing certifications as possible, which takes us to where we are today. Go read the previous post if you want a much more elaborate discussion of your modem-packaging options and some of the trade offs involved.
The Cell Module Spectrum
A cursory web search turns up an unhelpful number of options for cell modem modules. What parameters matter? Why would you choose one over another? A little market survey can place most cell modules on a spectrum of choices, which is the subject of this post. It's worth noting there's one more discussion about cell technologies that we haven't had yet, but that will come another day. For now we'll presuppose that all of these options meet our needs. With that caveat out of the way, here we go!
Why did I choose that X axis? Well the spectrum really comes down to how many layers of sugar a vendor wraps around the network connection. It is my experience that every method of networking devices together fundamentally sucks, but the precise way in which they suck is what varies between technologies. Cell modems are no different, but like WiFi and ZigBee and Bluetooth and everything else each provider packages the relevant APIs in different ways. So in that sense the spectrum is sort of from "easy to develop" to "hard to develop".
How Do Networks Work?
Cell modems have one extra layer that some of those other technologies avoid or intermediate; a cell modem connects to The Internet. To be clear that's quite a simplification, as a modem doesn't actually quiiiite connect directly to The Internet. What? Confused yet? Modems connect to a cell carrier which in turn internally routes data that can sometimes connect to the Internet. That's true of, say, an ESP8266 too, but in that case the ESP connects to your home WiFi, your WiFi connects to a modem which is screwed into a coax connection that eventually leads to your ISP. You pay your ISP to provide Internet to that modem on that cable, the WiFi has an Internet connection, done. The difference here is subtle but important. In the ESP example, the relationship between your widget and the Internet is the same as your laptop and the Internet. Comcast doesn't care what you plug in and in fact they can't see what's connected. Not so with a cell modem.
The cell modem is connected to the carrier directly. This relationship is governed by the SIM card inserted into the device. The cell modem only knows how to connect to the carrier tied to the inserted SIM card and its network connection is completely governed by that carrier. There are some configurations where the modem's network connection actually won't terminate in the public Internet. I swear the specificity of this digression will be relevant in a bit, so bear with me.
Anyway, there is another implication of this relationship, and that's that every cell modem will be connected to a network. So paragraphs ago when we mentioned the vendor can provide a nice candy coating around the network related APIs? That can wrap all the around past the hardware and encompass the other end of the connection too in someone else's computer the cloud. The spectrum goes from "do everything yourself" to "you send some data and it magically appears on a webhook in our cloud."
An Aside on SIM Pricing and Plans
Before we break down what the elements of the spectrum mean we should take a moment and talk about cell plans. For small devices like these the plans available are designed for what is referred to as "machine to machine" messaging. Even if the modems have the bandwidth capabilities to stream 8K HDR cooking shows the carriers expect them to send, say, one message an hour, or one message a day. Instead of having deep data, text, and voice allowances M2M plans tend to have low per-device maintenance and registration costs with tiny data allowances. The expectation is that the purchaser is managing a fleet of thousands/tens of thousands/hundreds of thousands of devices but that each one sends and receives very little data, and plans are priced accordingly. DigiKey resells M2M data SIMs recently so you can use their pricing as a guideline, though I find it to be significantly different from agreements I've seen for shipping products. For another reference point, NimbeLink lists prices too. Like all things this is a negotiable element and scale helps.
A final note; you can always use a normal consumer cell data SIM instead of a specific M2M SIM. If you're building just one of something this is probably a fine option if you can find a plan with cheap overhead. There are a few US carriers who have relatively low per-SIM costs and work well. I've used US Mobile in the past, and if you're a Google Fi customer their data-only SIMs may work though obviously without SMSs.
Other Considerations
Vendors selling cell modules expect them to be deployed en mass to a large fleet of end devices — this goes with the expected use cases for those M2M SIM cards we just talked about. Some vendors will package and sell "fleet management" capabilities with their solutions even if they don't provide much additional sugar. Typical fleet management tools will let you see basic telemetry for each node such as which devices are active, connection quality, and modem version. Some will also let you track resource consumption per-device to help manage spend. This telemetry is typically separate from any data your application may report. Fleet management tools may expose this bulk metadata via APIs so that they can be plugged into your application level data stores to include in other health metrics.
In the previous post we made an offhand comment about keeping your modem up to date, but that was no joke. Every carrier will want you to keep your modems up to date or they may evict you from the network. This is a protective measure to make sure that buggy devices don't ruin things for everyone but it means that your device must provide it. Some vendors provide this as a remote OTA capability as part of their fleet management solution. If that's not an option your user firmware may need to download and apply modem updates itself. This can be especially complex because, as with all OTAs, if there is a problem your device may be knocked off the network permanently. To reiterate, this OTA is separate from the ability to update user firmware remotely and one does not necessarily imply the other.
Breaking Down the Modem Spectrum
Whew! Caveats done, it's finally time to talk through the spectrum in more detail. For that I've prepared… yet another table! Keep in mind that this is not an exhaustive list, and not meant to be interpreted as an endorsement of any particular choice. It's merely a representative selection of the market as it stands at time of writing.
Whole Fat
In the "whole fat" category we have offerings that take care of nearly everything. These are the vendors who will sell you a cell plan, wrap every network API, and make sending and receiving data as easy as possible. Often that means the on-device APIs look something like "message.send()" and data will magically appear in the backend wholly formed. No complex network state error handling to write. No framing and unframing to debug. And their infrastructure just begs to have some of your server code running in it. Two exemplars here are Particle and Electric Imp. Both of these companies sell products on basically the same premise: "building network connected hardware is hard, so use our products to make it easy."
Unsurprisingly solutions in this segment of the market tend to be on the expensive side in exchange for ease of use. But even the pricing is easier, they may abstract the cell plan out entirely so the developer never has to talk to carrier or any other organization. You get a single entity to deal with one pricing model.
Obviously these can be great choices for getting a new project out the door quickly but they also involve more platform lock in. You're buying everything in one place, from cloud infrastructure to fleet management to electrical interfaces to a vendor relationship. Nothing carries over if you try to pull out, even plan pricing will need to be negotiated from scratch. It's a great deal for the vendor but might end up being pretty expensive in the long run for you the hardware manufacturer, depending on what your business model is.
There's also an exchange of control and visibility for an SLA. If an internal component of a full fat vendor's infrastructure goes down, becomes unreliable, or changes substantially talking to a Field Application Engineer may be the only recourse. And if things really go down hill and you want to leave, refer to the previous point about lock in.
Ahh wonderful 2%, the great taste of full fat without the serving of guilt that comes along with it. The 2% "medium fat" option sits in a sweet spot between "too many batteries included" and "make it from sticks on a deserted island". Here there is a more gentle interface to the underlying hardware than the fat-free choices below but substantially less hand holding the full fats. There are two main areas where the medium fat options differ from the others, included network infrastructure and on device APIs.
Each of the full fat vendors would prefer if you buy the entire stack from modem hardware through cloud services from them. That arrangement is really the way their offerings are designed to work, that's the special sauce. Though it might be possible to use the full fat modules without the entire backend by making web requests directly on the device, it would negate much of the benefit of the higher cost. The medium fat vendors mostly don't offer that choice; they give you a modem that can open a socket and it's up to the developer to do the rest. What they do add is fleet management. Both of my favorite examples, LinkLabs and Digi offer interfaces for commissioning and managing large groups of devices, and both support remote modem OTA.
A momentary aside about the LinkLabs offering. Remember I bent over backwards to describe how a cell modem talks to the carrier to mediate its network connection? The LinkLabs module routes all of its traffic this way and provides no other choices. To interface with the Internet all traffic goes through Verizon's infrastructure then through LinkLabs' before it hits the public network.
The other benefit to going medium fat is nicer device side network APIs. If you dig deep enough most modems speak ye' olde AT commands under the hood. With a fat-free option the developer needs to write code that interacts with the modem directly themselves. This is certainly possible, after all AT commands aren't terribly complex, but it does force the developer to spend some quality time with an ugly modem data sheet and it doesn't insulate the main firmware from the quirks of a particular modem. The medium fat options include the same benefit as the full fat options in the form of a friendly neighborhood coprocessor which does all the messy modem support itself, ideally hiding behind a UART and a vendor-provided C library.
Another aside, this time about the Digi modules. Digi might be a familiar name because they make the ubiquitous and often confused pentagonal XBee modules. Their cell modules fit in exactly the same form factor and use the same pinout. It's possible to swap one of their WiFi or ZigBee modules out for cell and continue using it with no code changes, except for the APIs that don't make sense anymore.
Fat Free
If all of those options were the complex ones, fat-free isn't. Fat free is simple! Do you want a cell modem? Do you want nothing else? Great! Choose a fat free cell module! Though even here the lines can be a little blurry.
The fattier cell modules above usually gain their nice features by adding a second microcontroller to the package instead of just providing a modem in a PCBA. This lets them include whatever shims and padding are required to make the software and electrical interfaces more palatable. But even without this ride-along micro some electrical and mechanical cross compatibility is present in these modems.
Both our example fat-free module vendors Nimbelink and Multitech sell modules that are footprint-compatible with other options in the range, but they aren't quite as universal as something like Digi's XBee line. Here it's more like electrical comparability than software, meaning things like power pins at the same locations, level shifters on IOs, and UART TX/RX pairs on the same pins. Definitely an improvement over redesigning a board from scratch for a new cell modem, but also still requiring the firmware to adapt.
What Now?
So where does all that leave us? The right choice is all about expectations. (And money, it's always about money.) How many of your project or product do you expect to make? How many other people or employees do you expect to be able to staff on the project? Are you able to pass recurring costs on to a customer? Do you expect to sell and support the product for a long time? Internationally? (We haven't even taken that can of worms down off the shelf!)
If you're making a few of something, or may not be able to staff a large team, or want to launch quickly, or can pass the cost to your customers, then a full fat option might be right for you! If you want the absolute lowest cost and most control, maybe fat-free is the way to go. And if you want a little sugar, but not too much, and can stomach a slightly higher price point for a nicer experience overall, then the 2% camp is where it's at.
Thanks to [Yoo Yoo] for noticing DigiKey's cell plan options!


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Faith United Church of Christ
About Us
Faith UCC is not just a church. Rather, we are a dynamic and energetic spiritual community dedicated toward walking the Jesus Way in all do and are. The heartbeat of our faith revolves love, inclusion, equality, authenticity, and justice. We seek to walk humbly in this beautiful world, while engaging areas of injustice that try to hide the world's beauty.
We pray. We protest. We march. We love. And above all, we welcome.
And did I mention we laugh - a lot? Well, we do!
If this sounds like something you are interested in experiencing as part of your spriritual journey, put on your flip flops, grab your coffee, and come on a Sunday at 10:30am for a service.
The Mission of Faith United Church of Christ is to be a progressive spiritual voice rooted in love. Through intentionally inclusive fellowship, challenging spiritual education and lively advocacy around issues of justice, we seek to inspire spiritual journeys as we follow the Jesus way of love and peace in the world.
While recognizing that every member has the freedom to discern the truths of God, Faith UCC ascribes to the spiritual path outlined by the Center for Progressive Christianity in its "8 Points of Progressive Christianity." The eight points follow.
1. Believe that following the path and teachings of Jesus can lead to an awareness and experience of the Sacred and the Oneness and Unity of all life;
2. Affirm that the teachings of Jesus provide but one of many ways to experience the Sacredness and Oneness of life, and that we can draw from diverse sources of wisdom in our spiritual journey;
3. Seek community that is inclusive of ALL people, including but not limited to:
• Conventional Christians and questioning skeptics,
• Believers and agnostics,
• Women and men,
• Those of all sexual orientations and gender identities,
• Those of all classes and abilities;
4. Know that the way we behave towards one another is the fullest expression of what we believe;
5. Find grace in the search for understanding and believe there is more value in questioning than in absolutes;
6. Strive for peace and justice among all people;
7. Strive to protect and restore the integrity of our Earth;
8. Commit to a path of life-long learning, compassion, and selfless love.


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A frigid air mass remains in place over Texas with devastating, widespread power outages underway. As a final round of winter weather sweeps through on Thursday, here's what you need to know:
Recommended Videos
- As of late Thursday afternoon, snow was beginning to wind down across South & Central Texas
- Snowfall is expected to end completely by Thursday night
- Areas near San Antonio could saw up to 5 inches of snow on Thursday, with much higher totals out west - closer to Del Rio
- Even though we'll have freezing mornings Friday and Saturday, we'll finally start to thaw in the afternoons
- By the weekend, high temperatures will be back in the upper-50s and 60s!
Stay Warm
Amid state-wide power outages, CPS says that power may not fully be restored until the weekend. If you find yourself without power, take these steps to insure your safety.
- Dress in layers.
- Block cracks and crevices with thick towels to keep out drafts. Stay in one room with all family members & pets to conserve warmth.
- Call friends and family to check and see if they have power and water. If you plan to visit them, do so midday or in the afternoons when the roads are slightly warmer, and USE CAUTION. Road temperatures are well below freezing and icy.
- Close blinds overnight to keep in the warmth. Open blinds during the day to let the sun come through.
- If you choose to warm-up in your car, NEVER do so in the garage. This can be deadly and may lead to carbon monoxide poisoning.
- Carbon monoxide poisoning can also be caused by operating generators inside. DO NOT DO THIS.
Planning Forecast
Stay Informed
As always, Your Weather Authority team will keep you updated. You can get the very latest forecast anytime by bookmarking our weather page and downloading the KSAT Weather Authority App - available for both Apple and Android devices.


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The CS Local North East, Yorkshire and the Humber Culture and Engagement network would like to invite you to join Karen Notaro and Ruth Patel for this online event. Karen and Ruth will take you through how employee experience has evolved in the Ministry of Justice. This will be an interactive session exploring how to create more good days at work.
Having a great experience at work is shown to be not just good for employees, but benefits organisations with innovation, productivity and retentions. So how can we create a positive experience at work for our people?
The event will look at universal themes of a positive employee experience, the importance of connecting people with purpose and their journey through adapting the approach to employee engagement since the pandemic.
When is the event taking place?
Date:Wednesday 15 September 2021
Time: 1100 am - 1200 pm.
How do I register to attend?
To attend please register via this eventbrite link using your civil service work and email address.
The event will be hosted on Microsoft Teams. The Teams link to join will be sent to you the day before the event. If you are unable to access Microsoft Teams on a work device you may wish join using a personal device.


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Genetic analysis of the critically endangered Trinidad Piping guan (Pipile pipile): Implications for phylogenetic placement and conservation strategies
Robinson, Louise A.Advisors
McDowall, IanHosie, Charlotte A.
Publication Date
Show full item recordAbstract
Classified as critically endangered since 1994, the Trinidad Piping guan (Pipile pipile) is an endemic species estimated to number less than 200 individuals. Known to locals of Trinidad as the 'Pawi' this bird has been the subject of substantial hunting pressures and much of the species habitat has been destroyed through deforestation. Although officially protected since 1958, occasional recreational hunting of this elusive species still occurs. Due to difficulties locating and capturing the species, no genetic research has previously been performed using samples obtained from Trinidad. All previous research studies have been conducted using biological materials obtained from captive birds outside Trinidad and island data has never been obtained or compared. The genetic diversity of the remaining population was therefore examined through the investigation of mitochondrial haplotypes, pairwise comparison and SNP analysis. With the intention of assisting the protection of this endangered species by the location of remaining areas of habitation, methods of genetic identification were established for the Trinidad Piping guan utilising non-invasive feather samples. Species specific primers were created in the regions of the ND2 and cyt b genes of the mitochondrial genome to identify Pipile pipile. Species detection was further verified with the use of PCR-RFLP of the same gene regions digested with BsaXI, EcoRV and BsrDI. This combined approach allowed the separation of closely related taxa based on single inter-species SNPs. Confirmation of species identification was subsequently performed through the use of forensically informative nucleotide sequencing. The established methodologies were used in the current study to correct the classification of a UK breeding population of Piping guans thought to be Pipile pipile and to identify Trinidad field samples. These detection methods have implications for ecological studies through the location of populations from trace evidence collected in the field. In addition this method could be used to assist Trinidadian police forces in the identification of bushmeats or simply act as a deterrent to hunters. The sequence data obtained in the present study were also used to re-assess the phylogeny of Piping guans. As genetic sequence from a true island bird was previously unstudied, differences between phylogenies created using non-island and island bird data sets were examined. Combined analysis was performed on 1884bp of the ND2 and cyt b genes and placement of Trinidad Piping guan was found to differ from that which has been previously published.Publisher
University of ChesterType
Thesis or dissertationLanguage
Funding from the University of Chester Gladstone Fellowship scheme and the World Pheasant Association.Collections
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