Vertical dynamics of Ebola with media impact

1 Introduction

Ebola virus disease (EVD) having fatality rate around 50%–90% (Kucharski and Edmunds, 2014) is a serious major threat to human population. Ebola virus diseases (EVD), formerly known as Ebola haemorrhagic fever get spread by infection from virus of the family Filoviridae, genus Ebolavirus. As per WHO Ebola situation report March 2015, around 10,311 deaths with total 24,872 infection cases of Ebola were reported from western African countries. In 1976, two simultaneous outbreaks of Ebola were occurred in South Sudan and Democratic Republic of Congo. Thereafter, outbreak was occurred in a village near river Ebola, and then it was started to identify as EVD. As a Zoonotic disease Ebola virus can transmit amongst animal and human population.

Various study of Ebola virus transmission amongst animal population suggests that fruit bats are the reservoir hosts for the Ebola virus. Baize et al. (2014) observed that from handling bushmeat and contacts with fruit Bats, the virus can transmit to humans and nonhuman primates like animals, apes, monkeys. Pourrut et al. (2005) and, Martin-Moreno et al. (2014) studied that after the first infection in human, transmission of the virus amongst human population occur through contact with broken skin or mucous membranes of the blood or body fluids and surfaces of materials contaminated with such fluids of EVD infected people and dead but not buried bodies of those who have died because of EVD (Kupferschmidt, 2014).

Chowell et al. (2015) studied that the early diagnosis of pre-symptomatic individuals helps to decrease basic reproduction number. Effective isolation may improve result as isolated class has lower transmission rate. Individual stays long in isolated class, which increases possibility of disease spread if effective precautions are not taken.

As per Legrand et al. (2007), Adebamowo et al.(2014) and Pandey et al.(2014), the incubation period from infection with the virus to development of symptoms is 2–21 days. Feldmann and Geisbert (2011) studied that humans remain uninfectious until they develop symptoms like fever, severe headache, muscle pain, weakness, fatigue, diarrhoea, vomiting, stomach pain, and unexplained hemorrhage (bleeding or bruising).

Studies of Rodriguez et al. (1999), Bausch et al. (2007), and Center for Disease Controls and Prevention (2015) Morbidity and Mortality weekly report suggest that semen from man survived from Ebola (Convalescent Ebola patients) also contains virus up to nine months of infection and its spread is also possible through sexual activities. As per WHO (2014), the transmission of Ebola virus can occur via up to 7 weeks after recovery from an Ebola virus infection from the semen of a convalescent men. Center for Disease Control and Prevention (2015), Morbidity and mortality weekly report also indicates that there were no traces of virus spread due to vaginal fluids during intercourse from a woman who had Ebola. Bausch et al. (2007) also studied that Ebola virus can survive in breast milk. This leads to study of vertical transmission of EVD. Studies indicates that sexual and congenital transmission are relative congruent. If mother get infection due to sexual activity, baby will not survive. If mother do not get infection then virus will direct move to placenta. Baby gets infected and directly moves toward infectious class.

As per WHO factsheet (2016), there is no vaccines or medicine available for cure of Ebola. Only supportive care-rehydration with oral or intravenous fluids and treatment of specific symptoms helps to improve survival. As per Center for Disease Controls and Prevention report (2016), recovery from Ebola depends on good supportive clinical care and the patient’s immune response. People who recover from Ebola infection develop antibodies that last for at least 10 years. Chowell et al. (2015), Dhillon et al. (2015) and Chowell and Nishiura (2014) studied the effect of quick diagnosis and Isolation with rapid testing onset of symptoms on disease spread. Galvani et al. (2014) observed that in absence of medicine and vaccines, early diagnosis, isolation of suspected, awareness campaigns and sanitary burial practices are effective controls to stop disease spread.

Liu et al.(2007), Pang and Cui (2009), Safi and Gumel (2011) observed that during outbreak of infectious disease like Ebola, the informative reporting by media highly affects belief of people about risk and threat of disease. Detail media coverage develops sense about precautions like wearing mask, avoiding travel and visit of public places during disease outbreaks. Wang and Xiao (2014) observed that effective news coverage with detail and faster flow of information increase the knowledge about disease in public domain. Sun et al. (2011) studied that, during early stage of outbreak when medicine or vaccines are not developed effective media coverage including warning and advisory helps in curb of disease. The transmission dynamics of EVD is studied by different researchers like Smailhodvic et al. (2014), Love et al. (2014), Basch et al. (2014) and Househ (2015) with various strategies. Mathematical modeling is an effective tool to analyse the epidemiological characteristics of infectious diseases. Shen et al. (2015a,b) studied model involving different intervention strategies like effects of early diagnosis, isolation, improved media coverage, restricted burial procedures and vaccination on the epidemic of Ebola infection. Shau and Dhar (2015) studied SEQIRHS model and observed that media coverage did affect basic reproduction number but it helps to lowering infectious individuals at endemic stage.

The paper is organised as follow. In Section 2, mathematical model with notations, assumptions, and detail discussion of dynamics of disease spread with population flow amongst compartments followed by a suggested system of ordinary differential equations is formulated. In Section 3, basic reproduction number, disease free equilibrium and endemic equilibrium points with and without media effects are derived. In Section 4, stability of model has been discussed. In Section 5, numerical simulation has been carried out and sensitivity of model is discussed using graphical analysis of various parameters for both with and without media effects. For the model, the graphical comparisons of the various populations for both with and without media effects are elaborated. Results obtained support calculations made in Sections 2 and 3. In Section 6, conclusions emphasize the role of media, isolation of exposed, proper burial process and safe sexual activities to control disease spread.

2 Mathematical model

The mathematical model is developed with following notations (Table 1).

A non-linear dynamical system of differential equations is suggested to study the spread of EVD with vertical transmission and media effect. Total population is divided in to seven compartments viz. Susceptible, Exposed, Infectious, Hospitalised/Isolated, Convalescent, Dead (not properly buried dead bodies of Ebola victims), and Recovered adults.

To prepare the model, the following possibilities of disease spread are considered.

• (1)

  Vertical Transmission of infection to new-borns.

• (2)

  Nonsexual transmission amongst total population.

• (3)

  Heterosexual transmissions amongst adults (from man to women).

• (4)

  Transmission from Dead body of EVD victims.

To formulate the model following assumptions are taken in account.

It is assumed that the infected, exposed, hospitalised/isolated individuals are not taking part in sexual activities and hence not responsible for disease spread via sexual activity. It is also assumed that, recovered individuals should not persist infection due to immunity. As dead bodies of EVD victims are responsible for disease transmission the dead individuals are also considered as part of total population till these bodies are not properly buried. For simplicity, rate of bury of EVD victims dead bodies is considered as equal to natural mortality rate.

The flow of population amongst the compartments is shown in Fig. 1.

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Fig. 1

Disease dynamics in population.

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New recruitments in observed population are entering in system at fixed number $\overline{B}$ . The convalescent (not fully recovered) sexually active man (contains virus in their semen) transmits the Ebola virus to susceptible women at rate $\overline{\psi }{\overline{\beta }}_C\frac{\overline{C}}{\overline{N}}$ or direct placenta of women which leads birth of infected new-born at rate ${\overline{\delta \beta }}_C\frac{\overline{C}}{\overline{N}}$ and such susceptible individuals moves toward exposed class or new-born babies towards infected class respectively.

Susceptible individuals get infection by contacts of body fluid of infected individuals at rate $\overline{\beta }\frac{\overline{I}}{\overline{N}}$ , by Leakage in hospitalisation/isolation at rate $\overline{\epsilon }\overline{\beta }\frac{\overline{H}}{\overline{N}}$ and by contacts of dead bodies of EVD victims at rate ${\overline{\beta }}_D\frac{\overline{D}}{\overline{N}}$ and hence they move towards exposed class. Exposed individuals either move toward infected class at rate $\overline{\sigma }$ or hospitalised/isolated class at rate $\overline{\theta }$ . Infected individuals either move toward hospitalised/ isolated class at rate $\overline{\eta }$ or move toward Dead bodies’ class at rate ${\alpha }_1$ or get partial recovery and move towards convalescent class at rate ${\overline{\gamma }}_1$ . Hospitalised/Isolated individuals get move toward Dead bodies’ class at rate ${\alpha }_2$ or get partial recovery and move towards convalescent class at rate ${\overline{\gamma }}_2$ . Convalescent individuals get full recovery and move toward recovered class at rate ${\overline{\gamma }}_3$ . The mortality rate $\overline{\mu }$ affects the population from each compartment.

Thus, the dynamics of the disease can be expressed by the system of non-linear differential equations as

As solutions of (1)–(7) are positive real numbers, solution set of (1)–(7) is subset of ${\hslash }_{+}^7$ .

Also ${\left(\frac{d\overline{S}(\overline{t})}{d\overline{t}}\right)}_{\overline{S}=0}>0\text{,}{\left(\frac{d\overline{E}(\overline{t})}{d\overline{t}}\right)}_{\overline{E}=0}>0\text{,}{\left(\frac{d\overline{I}(\overline{t})}{d\overline{t}}\right)}_{\overline{I}=0}>0\text{,}{\left(\frac{d\overline{H}(\overline{t})}{d\overline{t}}\right)}_{\overline{H}=0}>0\text{,}{\left(\frac{d\overline{C}(\overline{t})}{d\overline{t}}\right)}_{\overline{C}=0}>0\text{,}{\left(\frac{d\overline{D}(\overline{t})}{d\overline{t}}\right)}_{\overline{D}=0}>0\text{,}\text{and}{\left(\frac{d\overline{R}(\overline{t})}{d\overline{t}}\right)}_{\overline{R}=0}>0$ verifies that all solutions trajectories of (1)–(7) initiating in ${\hslash }_{+}^7$ will remain inside ${\hslash }_{+}^7$ .

Adding Eqs. (1)–(7), the total change in population can be expressed as $\frac{d\overline{N}(\overline{t})}{d\overline{t}}=\overline{B}+\bar{\delta }{\beta }_C\frac{\overline{C}}{\overline{N}}\overline{S}-\overline{\mu }\overline{N}-{\overline{\alpha }}_1\overline{I}-{\overline{\alpha }}_2\overline{H}$ . i.e. $\frac{d\overline{N}(\overline{t})}{d\overline{t}}⩽\overline{B}-\overline{\mu }\overline{N}$ .

Hence, by Brikhoff and Rota’s theorem, $\overline{N}(\overline{t})⩽\frac{\overline{B}}{\overline{\mu }}$ as $\overline{t}\to \infty$ .

Thus, $\overline{\mathrm{\Omega }}=\left\{(\overline{S}\text{,}\overline{E}\text{,}\overline{I}\text{,}\overline{H}\text{,}\overline{C}\text{,}\overline{D}\text{,}\overline{R})\in {\hslash }_{+}^7:\overline{S}\text{,}\overline{E}\text{,}\overline{I}\text{,}\overline{H}\text{,}\overline{C}\text{,}\overline{D}\text{,}\overline{R}⩾0\text{,}\overline{S}+\overline{E}+\overline{I}+\overline{H}+\overline{C}+\overline{D}+\overline{R}⩽\frac{\overline{B}}{\overline{\mu }}\right\}$ is positively invariant region, in which existence, uniqueness of solution and continuation of solutions hold.

Now, reducing the system of equations in to non-dimensional form (Shau and Dhar, 2015) using $S=\frac{\overline{S}}{\overline{N}}$ , $E=\frac{\overline{E}}{\overline{N}}$ , $I=\frac{\overline{I}}{\overline{N}}$ , $H=\frac{\overline{H}}{\overline{N}}$ , $R=\frac{\overline{R}}{\overline{N}}$ , $D=\frac{\overline{D}}{\overline{N}}$ , $N=\frac{\overline{N}}{{\overline{N}}_0}$ , $t=\frac{\overline{\mu }}{\overline{t}}$ and $S+E+I+H+D+R=1$ .

Hence, required non-dimensional system can be given by,

3 Disease free equilibrium (DFE) and basic reproduction number

The DFE of the system (8)–(14) can be given by $E_0=(0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}1)$ .

To study local stability of DFE, Basic Reproduction number is required.

Apply next generation matrix method suggested by van den Driessche and Watmough (2002) at DFE. Let $X^{\prime }=\frac{\mathit{dX}}{\mathit{dt}}=\mathfrak{I}(X)-\upsilon (X)$ , with $X=(E\text{,}I\text{,}H\text{,}C\text{,}D)$ where $\mathfrak{I}(X)$ is the appearance rate of new infected individuals in different class and $\upsilon (X)$ represents the rate of infection transmission in the system, which is given by $$\mathfrak{I}(X)=\left[\begin{array}{c}\hfill [\beta (I+\epsilon H)+{\beta }_{D}D+{\beta }_C(\psi +\delta )C](1-E-I-H-C-D-R)\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right]$$ and $$\upsilon (X)=\left[\begin{array}{c}\hfill (\theta +\sigma +1)E+\left(\frac{1}{N}-1+{\alpha }_{1}I+{\alpha }_{2}H+\delta {\beta }_{C}C(1-E-I-H-C-D-R)\right)E\hfill \\ \hfill -\sigma E+(1+{\alpha }_1+\eta +{\gamma }_1)I+\left(\frac{1}{N}-1+{\alpha }_{1}I+{\alpha }_{2}H+\delta {\beta }_{C}C(1-E-I-H-C-D-R)\right)I\hfill \\ \hfill -\eta I-\theta E+(1+{\alpha }_2+{\gamma }_2)H+\left(\frac{1}{N}-1+{\alpha }_{1}I+{\alpha }_{2}H+\delta {\beta }_{C}C(1-E-I-H-C-D-R)N\right)H\hfill \\ \hfill -{\gamma }_{1}I-{\gamma }_{2}H+(1+{\gamma }_3)C+\left(\frac{1}{N}-1+{\alpha }_{1}I+{\alpha }_{2}H+\delta {\beta }_{C}C(1-E-I-H-C-D-R)N\right)C\hfill \\ \hfill -{\alpha }_{1}I-{\alpha }_{2}H+D+\left(\frac{1}{N}-1+{\alpha }_{1}I+{\alpha }_{2}H+\delta {\beta }_{C}C(1-E-I-H-C-D-R)N\right)D\hfill \\ \hfill \hfill \end{array}\right]$$

Defining, $F=\left[\frac{\partial {\mathfrak{I}}_i(X_0)}{\partial X_j}\right]=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill \beta \hfill & \hfill \beta \epsilon \hfill & \hfill (\psi +\delta ){\beta }_C\hfill & \hfill {\beta }_D\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right]\text{and}V=\left[\frac{\partial {\upsilon }_i(X_0)}{\partial X_j}\right]=\left[\begin{array}{ccccc}\hfill \sigma +\theta +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\sigma \hfill & \hfill {\alpha }_1+{\gamma }_1+\eta +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\theta \hfill & \hfill -\eta \hfill & \hfill {\alpha }_2+{\gamma }_2+1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -{\gamma }_1\hfill & \hfill -{\gamma }_2\hfill & \hfill {\gamma }_3+1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -{\alpha }_1\hfill & \hfill -{\alpha }_2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right]$

The basic reproduction number $R_0$ is spectral radius of ${\mathit{FV}}^{-1}$ . $$R_0=\frac{({\beta }_1+{\beta }_{D1}{\alpha }_1)+({\beta }_1\epsilon +{\beta }_{D1}{\alpha }_2)b_3+(\psi +\delta ){\beta }_{C1}{b}_5}{b_1{b}_{10}}$$ where $b_1=\frac{1+{\alpha }_1+{\gamma }_1+\eta }{\sigma }\text{,}{b}_{10}=\sigma (\sigma +\theta +1)$ .

4 Stability

5 Numerical simulation and observation

Numerical simulation is carried out with the parametric values of disease parameters as tabulated in Table 2 in appropriate units. For tabulated parameters, the basic reproduction number $R_0=0.966032$ . Next sensitivity of $R_0$ with different disease parameters is verified to understand disease spread possibilities.

In Fig. 2, the effects of change in contact rates $\overline{\beta }\text{,}{\overline{\beta }}_C$ and ${\overline{\beta }}_D$ on $R_0$ are shown. It is observed that increase in these contact rates will also increase $R_0$ and hence disease will persist for long time in population. Also it can be seen that sensitivity of $R_0$ with $\overline{\beta }$ is much higher than ${\overline{\beta }}_C$ and ${\overline{\beta }}_D$ .

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Fig. 2

Effect of change in contact rates on $R_0$ .

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In Fig. 3, the effects of change in recovery rates ${\overline{\gamma }}_1\text{,}{\overline{\gamma }}_2$ and ${\overline{\gamma }}_3$ on $R_0$ are described. It is observed that increase in recovery rates will decrease $R_0$ , which finally makes stable and control system.

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Fig. 3

Effect of change in recovery rates on $R_0$ .

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In Fig. 4, the effects of changes in parameters $\overline{\delta }\text{,}\overline{\psi }\text{,}\overline{\epsilon }$ on $R_0$ are plotted. It can be observed that increase in probability of a female individual get infected due to sexual activity $\overline{\delta }$ , probability of virus enter direct in placenta due to sexual activity $\overline{\psi }$ and probability of disease spread due to leakage in isolation $\overline{\epsilon }$ will also increase $R_0$ .

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Fig. 4

Effect of change in $\overline{\delta }\text{,}\overline{\psi }\text{,}\overline{\epsilon }$ on $R_0$ .

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Fig. 5 indicates the effects of change in parameters $\overline{\theta }\text{,}\overline{\sigma }\text{,}\overline{\eta }$ on $R_0$ . It can be observed that increase in early hospitalisation of exposed individuals with rate $\overline{\theta }$ and hospitalisation of infected with rate $\overline{\eta }$ will help to control disease resulting decrease of $R_0$ . Increase in movement of exposed towards infected class with rate $\overline{\sigma }$ will also spread Ebola at a faster rate.