Mathematical models for devising the optimal Ebola virus disease eradication

For feasibility and usefulness of our approach, the important attributes of the Ebola virus and EVD before constructing our models were be summarized and included (1) Origin: the bat species, fruit bats in particular, are considered to be the natural reservoir of Ebola virus. Severe forest loss in Africa in recent years has brought potentially infectious wild animals into closer contact with human settlements; this increases the risk of virus transmission from wild animals to people; (2) transmission: Ebola spreads mainly through body fluids. In each outbreak, the virus is first introduced into the human population through close contact with bodily fluids of infected wild animals or infected people [30]. The common symptoms of EVD include diarrhea, fatigue, fever, muscle pain, severe headache, abdominal or stomach pain, vomiting, and unexplained hemorrhage. The course of EVD shows three phases in each case: (1) Incubation phase: infected patients generally don’t show any particular symptoms of EVD and patients don’t transmit the virus in this period; (2) Early infective phase: infected patients begin showing explicit symptoms such as fever and fatigue, and the EVD medication developed by WMA can cure the disease; (3) Advanced infective phase: infected patients are close to death, and cannot be cured by existing EVD medication.

For the model for prediction of the numerical spread of Ebola and regional transmission of Ebola, given that births and deaths affect the total population and the duration of the outbreak is short, we won’t take the change of total population in consideration. Firstly, we assume the population of an infected region remains constant in the ongoing outbreak, which enables us to focus on the analysis of infected people. Since people in the incubation phase and early infective phase can be cured by EVD medication, and the incubation phase is assumed impossible to detect. Next, we assume that the vaccine could not provide 100% protection from the Ebola virus, which its true effective vaccination rate was at between 75 and 100% [31]. At the same time, it is expected that the treatment time of EVD will gradually reduce with improving medical condition in these countries.

A series of differential equations systems was built to study the trend of Ebola total cases and construct a basic model based on traditional SEIR model by fitting our model against the actual total cases data. However, the effect of hospital isolation and application of medication are usually neglected. So we build a modified epidemic model which takes hospital isolation, Ebola drug and vaccine into account, and later verified the result with Monte Carlo Algorithm. We are concerned about the numerical spread pattern of Ebola and the function of these resistance methods against the spread of the disease. We assume that (1) the population of an infected region remains constant in the ongoing outbreak; (2) vaccine recipients and recovered patients develop complete immunity to Ebola virus; and (3) 98 of every 100 people who are vaccinated will successfully develop immunity against the disease. In this study, we divide the total population into seven groups in our model including (1) Susceptible group (S), no immunity against the disease and are very likely to be infected in direct contact with the infective people. S denotes the number of people in the susceptible group; (2) Incubation group (E), infected and cured by taking Ebola drugs, but had not displayed any explicit symptom. E denotes the number of people in the incubation group; (3) Early stage infected group (I _E ), displayed explicit symptoms of EVD and could transmit the virus to susceptible people and be cured by the EVD drug. We assume this phase lasts 3 days; (4) Advanced infected group (I _L ), displayed explicit symptoms of EVD and could transmit the virus to susceptible people, but cannot be cured by the EVD drug. We assume this phase lasts 2.61 days; (5) Removed group (R), died of the disease or survived the disease; (6) Hospital isolation group (H), shifted from the infective group and isolated from the susceptible people, and Ebola drugs are only for patients in hospital isolation; (7) Immunity group (M), gain complete immunity against Ebola virus and could either get the effective immunity through vaccine injection or recovery from the disease. Other related parameters are shown in the Table 1. This part shows how we obtain the value of the critical parameter: the infection rate β. The model introduced below can be displayed in the flow chart shown in Fig. 1. Note that this model is simplified compared to our “numerical spread” model, due to the availability of data and the solvability of the differential equation system model. The simplified model is listed below:

The differential equation system model is numerically solvable given the values of the parameters (γ, σ and β, and note that the former two parameters are known). We apply the grid search algorithm to determine the value of β. The objective function for the algorithm iswhere f _i denotes the numerical solution of the model at date i as the total cases, and d _i denotes the true value of it. The population flow between these groups is shown in Fig. 2, and we present the differential equations system of our model as follows:

There are four important transmission paths, which are through household, community, hospital and unsafe funeral for calculating daily infection cases through each path. We find out the key paths in different situations and propose our suggestion to control regional transmission, and use a transmission model to study the regional spread of Ebola virus [18]. Here, we assume that people in isolation group cannot transmit Ebola virus to others, but it’s not true in reality. We break down the regional spread consider four Ebola transmission paths more practically shown as follows. (1) Transmission within households: from un-hospitalized infective people to his or her household; (2) Transmission in the general community: from unhospitalized infected people to other household in his or her community. (3) Transmission in hospitals: from hospitalized infected people to health care workers (HCWs), From hospitalized infected people to other not-Ebola-infected patients in hospitals and from hospitalized infective people to his or her uninfected household in the hospital who take care of them; (4) Transmission in unsafe funeral: To household of the dead people and other household. We illustrate the important transmission paths with Fig. 3. The letters on the arrow are the transmission rate from the infected people to the health people. And we denote the number of household of one un-hospitalized infected people, other household in the community of one un-hospitalized infected people and contacts with hospitalized infected people at time t as N _f(t) , N _a(t) and N _h(t) , shown in the Table 2.

An SEIR (susceptible-exposed-infectious-recovered) model was previously established for the Ebola endemic, which provided the estimates of reproduction numbers in Guinea, Sierra Leone and Liberia (Additional file 1: Table S1) [30]. However, it didn’t take control interventions into consideration. We take the method as estimate infection rate \({\hat{\text{a}}}\) in model; then consider the function of resistance methods in our modified model, which will offer more reference for government planning of Ebola eradication. Since the period that people will stay in the incubation phase and in the infective phase differed little, we take the incubation period and infective period (1/ό = 5.3 days and \(1/\tilde{a} = 5.61\)) days in previous estimates from an outbreak of EVD in Congo in 1995 with our model [32]. We assume the value of \({\hat{\text{a}}}\) remains constant in the studied period. In this study, we first fitted our model against the real EVD total cases data of Liberia from 2nd July 2014–28th August 2014. We obtain the value of infection rate \({\hat{\text{a}}}\) as 1.754 in our basic SEIR model and Monte Carlo algorithm given the value of other parameters in the differential equations system, which our estimating value of infection rate \({\hat{\text{a}}}\) is a little higher, but acceptable, comparing with other results of related research. For examples, Althaus CL estimated the value to be 1.59 and WHO Ebola Response Team estimated the value to be 1.51 (both for Liberia) [33].

Next, we predict the eradication process of Ebola. We set 1st Feb 2015 as t = 0, and use the Ebola disease statistics of t = 0 on WHO as the initial state of our model. As is stated before, we focus on the function of resistance methods, namely hospital isolation, Ebola medication and medical protection against the spread of the disease. Thus, we change the value of relevant parameters (vaccination rate, isolation rate and infection rate) to stimulate different eradication process under these settings. For analysis of vaccination rate, we set á = 0.2 and vary the vaccination rate è. The results in Fig. 4a show that the bigger the vaccination rate è is, the sooner the disease can be eradicated quite apparently and sensibly. Based on our prediction, which set è = 0.02 and á = 0.2, we can expect to eradicate the disease in 188 days (number of total cases <1). The actual eradication process should be shorter because when there are a small number of infected people, we can put all of them in hospital isolation and cure the with Ebola drugs. That is á = 1 when total cases number is small enough. This should also apply to our analysis of á and \({\hat{\text{a}}}\) later.

For analysis of isolation rate, we tested the effect of hospital isolation on eradicating Ebola. We set è = 0.01 and vary the isolation rate á. The general trend is very similar to our analysis about the vaccination rate in Fig. 4b, which the bigger the isolation rate á is, the sooner the disease can be eradicated; and the eradication process can be divided into three phases. When we set è = 0.02 and á = 0.1, the number of total cases decreases to 7706 in 4 days, increases to the second peak of 10,029 total cases in 23 days; we can expect to eradicate the disease in 285 days (number of total cases <1). The actual eradication process should be shorter as discussed above. Moreover, high isolation rate can resist the second rise (like á = 0.3). The determination of isolation rate á is subject to the medical condition of one country, i.e. the total number of hospital beds of Ebola. The demand for hospital beds by time is shown in the Fig. 4c. We assume there are 1000 isolated people in hospital on 1st Feb. We can notice from the figure above that high isolation rate means a faster rise and more quantity in demand of hospital beds. The numbers of maximum hospital beds needed are 3488 in 11 days, 2954 in 15 days and 2230 in 26 days for á = 0.3, á = 0.2, and á = 0.1, respectively. Therefore, each country can determine their isolation rate á based on their medical condition, i.e. the number of hospital beds.

Further, we set the value of the infection rate â as 1.754 in accordance with our model result. In fact, the infection rate can be controlled by public medical education and protection. We analysis the trend of total cases number with different \({\hat{\text{a}}}\) value (\({\hat{\text{a}}} = 1.75,\,1.55,\,1.35\)), we set á = 0.2 and è = 0.02. The result is shown in Fig. 4d. The general trend is also similar: (1) The smaller the infection rate \({\hat{\text{a}}}\), the sooner the disease can be eradicated: (2) The eradication process also can be divided into three phases as we have discussed above and low infection rate can resist the second rise.

We think that the average time from symptom onset to medical treatment will be shortened with the improvement of treatment, which will affect eradicating Ebola. Firstly, we simulate the average time from symptom onset to medical treatment by shortening the treatment time. Since treatment of EVD involves both of early and late stages in infection of EBOV, we set á = 0.2 and è = 0.02, and assume that treatment time will be shortened both in the early and late stages in infection of EBOV. Here, we simulate three models: one is the original model of the early treatment time (γE = 3 days) and late treatment time (γL = 2.61 days); if all the assuming is same, there are other two simulation results (Fig. 4e), which the early and late treatment time in the two models is reduced by 1/2 or one day. We investigate that the time of eradicating Ebola is shortened if the treatment time is shortened, and expect to eradicate EVD in 188, 152 and 120 days in the three models if the case is less than 1, respectively.

Imported or exported cases of Ebola from one country to another country will affect the eradicating Ebola of the two countries [34]. To simulate this situation, the second equation in the above model is changed and listed below 8.

During the outbreak of Ebola in Liberian in 2014, it is estimated that the imported and exported cases does not exceed 1% [34]. \(\Delta\) represents the ratio of the imported case (n < 0) or the exported case (n > 0). Here, we set á = 0.2 and è = 0.02 and simulate the three situations: one is no imported and exported case; one is only exported cases (1%) or imported cases (1%), which were shown in Fig. 4f. For eradicating Ebola, we can notice that the imported and exported effects are presented in two aspects: (1) the exported weakens the second peak, as the imported will strengthen the second peak; (2) the imported extends the outbreak time of EVD, and the exported can reduce its outbreak time. However, there is little difference among these three situations.

For each Ebola transmission path, at time t, we can calculate the force of infection as below:

For calculating the infection force of each Ebola transmission paths, we can get the infection cases, which are (1 − á)I(t) \(\hat{a}_{f}\) /N _f(t) , (1 − á)I(t) ó \(\hat{a}_{f}\) /N _a(t) , áI(t) \(\hat{a}_{h}\) /N _h(t) and ufI(t)( \(\hat{a}_{b}\) /N _f(t)  + ó \(\hat{a}_{b}\) /N _a(t) ) in the four transmission paths and the comprehensive information presented in Table 3.

We take \(\hat{a}_{f}\)= 0.12, ó = 0.79, \(\hat{a}_{h}\) = 0.44, \(\hat{a}_{b}\) = 0.12, and f = 0.7 [35]. And we assume an un-hospitalized infected people has an average of 5 households and an average of 5 other households, and they contact an average of 10 people at each time, thus N _f(t)  = N _a(t)  = 5, and N _h(t)  = 10 I(t ₀ ) = 2000 in one supposed region. Through our simulation, the number of infection cases was obtained though different paths in regional spread of Ebola by setting µ = 0.3 and vary isolation rate á with 0.3, 0.5 and 0.7, which shown in Table 4. When isolation rate is small, household transmission path is the most important way of Ebola spread. When isolation rate is big, hospital transmission path becomes the most important way. This means the most important path varies in different situations.