Article Contents
Article Contents

Ebola model and optimal control with vaccination constraints

• The Ebola virus disease is a severe viral haemorrhagic fever syndrome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. In this paper, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyse two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyse the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vaccination, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlimited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time.

Mathematics Subject Classification: Primary: 49J15, 92D30; Secondary: 34C60, 49N90.

 Citation:

• Figure 1.  Flowchart presentation of the compartmental model (1) for Ebola

Figure 2.  (a) Cumulative confirmed cases: in dashed circle line the real data from WHO and in continuous line the values of $I(t) + R(t) + D(t) + H(t) + B(t) + C(t) - \mu(N - S(t) - E(t))$ from (1) with the parameter values from Table 1. (b) Cumulative confirmed cases given in (2), when available an unlimited supply of vaccines, also with the parameter values from Table 1

Figure 3.  Individuals $S(t)$, $E(t)$, $I(t)$ and $R(t)$. In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case with no vaccination with the parameter values from Table 1

Figure 4.  Individuals $D(t)$, $H(t)$, $B(t)$ and $C(t)$, with the parameter values from Table 1. In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case of no vaccination

Figure 5.  Optimal control and number of vaccines with the parameter values from Table 1, when an unlimited supply of vaccines is available

Figure 6.  (a) Cumulative confirmed cases. (b) Optimal control for the case of limited total number of vaccines. Dashed line for $W(90)\leq 10000$ and continuous line for $W(90)\leq 20000$

Figure 7.  Individuals $S(t)$, $E(t)$, $I(t)$ and $R(t)$. The dashed line represents the case where $W(90) \leq 10000$ and the continuous line represents the case where $W(90) \leq 20000$

Figure 8.  Individuals $D(t)$, $H(t)$, $B(t)$ and $C(t)$. The dashed line represents the case where $W(90) \leq 10000$ and the continuous line represents the case where $W(90) \leq 20000$

Figure 9.  Optimal control $u(t)$ and number of vaccines $W(t)$ for $W(90)\leq 10000$, $W(90)\leq 11000$, $W(90)\leq 13000$, $W(90)\leq 15000$, $W(90)\leq 16000$, $W(90)\leq 18000$ and $W(90)\leq 20000$

Figure 10.  Optimal control $u(t)$ and number of vaccines $W(t)$ for $W(90)\leq 10000$. In dashed line the case $w_2=50$ and in continuous line the case $w_2 = 500$

Figure 11.  (a) Cumulative confirmed cases, (b) completely recovered, (c) optimal control. In (a), (b) and (c) the following mixed constraints are considered: $S(t) u(t) \leq 1000$ for all $t \in [0, 10]$, $S(t) u(t) \leq 1200$ for all $t \in [0, 15]$, and $S(t) u(t) \leq 900$ for all $t \in [0, 16]$

Table 1.  Parameter values for model (1), corresponding to a basic reproduction number $R_0 = 2.287$. The values of the parameters come from [7,14,22,24,28,34,35]

 Symbol Description Value $\sigma$ per capita rate at which exposed individuals become infectious $1/11.4$ $\mu$ death rate $14/1000$ $\beta_i$ contact rate of infective and susceptible individuals $0.14$ $\beta_d$ contact rate of infective and dead individuals $0.40$ $\beta_h$ contact rate of infective and hospitalized individuals $0.29$ $\beta_r$ contact rate of infective and asymptomatic individuals $0.185$ $\gamma_1$ per capita rate of progression of individuals from the infectious class to the asymptomatic class $1/10$ $\epsilon$ fatality rate $1/9.6$ $\delta_1$ per capita rate of progression of individuals from the dead class to the buried class $1/2$ $\delta_2$ per capita rate of progression of individuals from the hospitalized class to the buried class $1/4.6$ $\gamma_2$ per capita rate of progression of individuals from the hospitalized class to the asymptomatic class $1/5$ $\tau$ per capita rate of progression of individuals from the infectious class to the hospitalized class $1/5$ $\gamma_3$ per capita rate of progression of individuals from the asymptomatic class to the completely recovered class $1/30$ $\xi$ incineration rate $14/1000$
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