Research Article | Open Access
E. A. Bakare, E. B. Are, O. E. Abolarin, S. A. Osanyinlusi, Benitho Ngwu, Obiaderi N. Ubaka, "Mathematical Modelling and Analysis of Transmission Dynamics of Lassa Fever", Journal of Applied Mathematics, vol. 2020, Article ID 6131708, 18 pages, 2020. https://doi.org/10.1155/2020/6131708
Mathematical Modelling and Analysis of Transmission Dynamics of Lassa Fever
Sub-Saharan Africa harbours the majority of the burden of Lassa fever. Clinical diseases, as well as high seroprevalence, have been documented in Nigeria, Sierra Leone, Liberia, Guinea, Ivory Coast, Ghana, Senegal, Upper Volta, Gambia, and Mali. Deaths from Lassa fever occur all year round but naturally peak during the dry season. Annually, the number of people infected is estimated at 100,000 to 300,000, with approximately 5,000 deaths. There have been some work done on the dynamics of Lassa fever disease transmission, but to the best of our knowledge, none has been able to capture the seasonal variation of Mastomys rodent population and its impact on the transmission dynamics. In this work, a periodically forced seasonal nonautonomous system of a nonlinear ordinary differential equation is developed that captures the dynamics of Lassa fever transmission and seasonal variation in the birth of Mastomys rodents where time was measured in days to capture seasonality. It was shown that the model is epidemiologically meaningful and mathematically well posed by using the results from the qualitative properties of the solution of the model. A time-dependent basic reproduction number is obtained such that its yearly average is written as , when the disease does not invade the population (means that the number of infected humans always decreases in the seasons of transmission), and , when the disease remains constantly and is invading the population, and it was detected that . We also performed some evaluation of the Lassa fever disease intervention strategies using the elasticity of the equilibrial prevalence in order to predict the optimal intervention strategies that can be useful in guiding the local national control program on Lassa fever disease to make a proper decision on the intervention packages. Numerical simulations were carried out to illustrate the analytical results, and we found that the numerical simulations of the model showed that possible combined intervention strategies would reduce the spread of the disease. It was established that, to eliminate Lassa fever disease, treatments with ribavirin must be provided early to reduce mortality and other preventive measures like an educational campaign, community hygiene, isolation of infected humans, and culling/destruction of rodents must be applied to also reduce the morbidity of the disease. Finally, the obtained results gave a primary framework for planning and designing cost-effective strategies for good interventions in eliminating Lassa fever.
Lassa fever (LF) is an acute viral hemorrhagic illness that is common in West Africa. LF is caused by Lassa virus, a single-stranded RNA virus belonging to the family Arenaviridae. First discovered in 1969 when two missionary nurses died and named after the Lassa town in Borno State, Nigeria, where the first cases occurred, the disease is now endemic in many parts of West African countries including Nigeria, Sierra Leone, Liberia, and Guinea. Infections with Lassa virus are generally estimated to range from 100,000 to 300,000, with approximately 5,000 deaths each year . The “multimammate rat” (Mastomys natalensis) is regarded as the major reservoir host for the virus in West Africa. This rat species which is widely distributed throughout the region can shed the virus in its faeces or urine. Humans become infected during direct contact with the rodent reservoir, during consumption of food contaminated with rodent faeces and urine, or during direct contact with bodily fluids of infected humans. Signs and symptoms of Lassa fever typically occur 1–3 weeks after the patient comes in contact with the virus. For the majority of Lassa fever virus infections (approximately 80%), symptoms are mild and are undiagnosed. Mild symptoms include a slight fever, general weakness, and headache. In 20% of infected individuals, however, the disease may progress to more serious symptoms including hemorrhage (bleeding in gums, eyes, or nose), respiratory distress, repeated vomiting, facial swelling, pain in the chest, back, and abdomen, and shock, and the neurological problem has also been described including hearing loss and tremors [2–6]. Lassa fever is generally treated with the antiviral drug ribavirin, which has been very effective when given early in the course of the disease [7–9].
In Nigeria, sporadic outbreaks of Lassa fever have been documented since 1969. The infection is endemic in several states including Edo, Ebonyi, Onitsha, Jos, Taraba, Nasarawa, Yobe, Rivers, and Ondo states. In 2012, for example, 623 suspected cases (108 laboratory confirmed), including 70 deaths, were recorded from 19 states in Nigeria [10, 11]. A total of 11 confirmed cases of Lassa were recorded in Nigeria with high prevalence in Oyo State in 2014. Between January 1 and March 8, 2015, the Nigeria Centre for Disease Control  reported 21 cases of Lassa fever (4 lab confirmed) and 1 death due to Lassa [8, 13]. Between August 2015 and January 2016, there were 239 suspected cases of LF (44 lab confirmed), including 82 deaths, across 19 states including Anambra, Bauchi, Nasarawa, Niger, Delta, Ekiti, Ondo, Kogi, Ebonyi, Lagos, Osun, FCT, Taraba, Kano, Rivers, Edo, Plateau, Gombe, and Oyo states [12, 14]. Similarly, the years 2016, 2017, 2018, and 2019 were not spared from Lassa fever in Nigeria with outbreaks across several states (statistics can be viewed on the Nigeria Centre for Disease Control (NCDC) website).
The outbreak of Lassa fever is highest in humans during the dry season, following multimammate rodent reservoir breeding during the rainy season.
Studies have shown that seasonal timing of reproduction can affect the dynamics of host-pathogen systems [15–18]. Seasonality has to do with the systematic peaks of diseases at a certain time of the year, and one of its drivers is the birth rate pulses which will be considered in this work. Understanding how seasonally varying parameters act as a forcing mechanism and investigating their dynamical consequences are part of our interest in this work. In this case, we are interested in understanding how such periodically forced models permit us to better capture the observed pattern of recurrent epidemics different from the unforced models which predict damped oscillation toward the endemic equilibrium .
Several mathematical models have been used to capture the dynamics of physical, chemical, biological, economical, and many other complex systems. Various works have been done so far with the use of the mathematical model applied to epidemiology which include but are not limited to the following: [20–22]. Little attention has been paid to this disease in the past, leading to scanty information about its transmission dynamics. Despite this, few studies have attempted to study Lassa virus dynamics using the mathematical modelling approach. James et al.  studied the dynamics transmission of Lassa fever using the susceptible, infected, and removed (SIR) model. The authors obtained the disease-free equilibrium and endemic equilibrium states of the system of the differential equation describing the dynamics of the disease. From the stability analyses of the two equilibrium states, they were able to ascertain that if the death rate of the human reservoir is greater than the respective birth rate, then the disease could be control and eradicated. They further specified, from their analysis, when it is practically impossible to control the disease. From this, one may ask what happens to the spread if either of the conditions for disease eradication becomes the case. Ogabi et al.  proposed an SIR model for controlling Lassa fever in the northern part of the Edo State by considering two senatorial districts of the state with about two million people. Using the numerical approach, they analysed the relationship between the susceptible, infected, and removed classes with three health policies. These health policies which consist of three sets of parameters representing the birth rate, the natural death rate, the transmission rate, and the rate of recovery were employed by the authors to simulate their results. They were able to show that the reproduction number is affected by these parameters, which indirectly tells that “the health policies” can control the disease, given the role of the reproduction number in disease dynamics. They were able to show also that the disease can be controlled if the transmission rate becomes less than the recovery rate.
Bawa et al.  also did a study of Lassa fever dynamics by subdividing the rodent population into infant and adult classes. From their analyses of the disease-free and endemic equilibria, they established a global stability condition for the control of the disease, which is dependent on the reproduction number as obtained in their work. Mohammed et al.  in their work developed a transmission dynamic model for Lassa fever with human immigration. Model analysis was carried out to calculate the reproduction number, and sensitivity analysis of the model was also performed. Their results showed that the human immigration rate is the most sensitive parameter and then the human recovery rate is the second most sensitive parameter followed by the person-to-person contact rate. It was suggested that control strategies should target human immigration, effective drugs for treatment, and education to reduce person-to-person contact. Andrei et al. , in their paper, suggested that seasonal migratory dynamics of rodents played a key role in regulating the cyclic pattern of Lassa fever epidemics, but they had no explicit model. Joachim et al.  also suggested in their work that the use of continuous control or rodent vaccination is the strategy that can lead to Lassa fever elimination. Having gone through several works on Lassa fever disease and its transmission dynamics, we observed that none investigated the effect of the periodically forced per capita birth rate of Mastomys rodents on the transmission dynamics of Lassa fever. So, in this work, our aim is to investigate the effect of predictable variability in time-dependent per capita birth rate of Mastomys rats/rodents on the transmission dynamics of Lassa fever and to explore factors that contribute to continuous outbreak and how those factors can be curtailed in the presence of one or many intervention strategies that we will evaluate.
The rest of this paper is organized as follows: Model formulation and description are presented in Section 2. In Section 3, we displayed the nonautonomous periodically forced Lassa fever model. In Section 4, qualitative analysis of the periodically forced Lassa fever model is discussed. In Section 5, we present the existence of the endemic equilibrium point for the periodically forced Lassa fever model. In Section 6, we present the evaluation of the Lassa fever intervention strategies using the elasticity of the static quantity, while in Section 7, numerical results are shown. Conclusions and recommendations of this work are presented in Section 8.
2. Model Formulation and Analysis
We propose a Lassa fever model with compartmental design with standard incidence and variable total human and Mastomys rat/rodent population. stands for the number of susceptible humans in the population, is referred to as the number of exposed humans in the population, represents the number of asymptomatic humans in the population, is the number of infected and infectious humans, and is the number of recovered humans. The total human population at time t is given by , and it is grouped into susceptible humans , exposed humans , asymptomatic humans , infected humans , and temporary immune/recovered humans . Therefore,
The total Mastomys natalensis rat population at time t is given by , and it is subdivided into susceptible Mastomys natalensis rats and infected and infectious Mastomys natalensis rats . Hence,
The population of susceptible humans is produced by the human per capita birth rate followed by the rate at which humans lose their immunity σ. It is reduced by infection following contacts with the infected Mastomys natalensis rat at a rate ; by infection following contacts with asymptomatic humans at a rate ; and by infection following contacts with symptomatic infected humans at a rate . It is further reduced by the natural death rate of the human population .
Thus, the rate of change of the population of susceptible humans is given by
And the rate of change of the population of exposed humans is given bywhere θ is the rate of progression from exposed humans to asymptomatic humans and to infected humans and is the natural death rate.
The population of asymptomatic humans is generated following the rate at which exposed humans progress to the asymptomatic compartment, with the probability of an exposed individual becoming an asymptomatic case upon infection , which reduces by the transmission rate from an infected rat to asymptomatic humans and decreases by recovery at a rate ; it also decreases by Lassa fever-induced death at a rate and natural death rate .
Therefore, the rate of change of the population of asymptomatic humans is given by
The infected human population is defined aswhere is the probability of an exposed individual becoming asymptomatic upon infection, which decreases by the transmission rate from an infected rat to asymptomatic humans at a rate ; is the Lassa fever-induced death rate in the infectious and asymptomatic classes, while is the recovery rate of the infected class; and is the natural death rate.
The population of recovery/temporary immune is generated by the recovery rate of the asymptomatic humans and the recovery rate of the infected and asymptomatic humans. It is reduced by the loss of immunity at a rate σ and natural death at a rate .
Therefore, the rate of change of the recovered population is given by
The population of susceptible mastomys natalensis rat is generated by the time-dependent per capita birth rate of mastomys natalensis rat, . It is reduced by infection, following number of contacts with an infected mastomys natalensis rat at a rate . It is further reduced by the natural death rate of the susceptible mastomys natalensis rat population at a rate .
Therefore, the rate of change of the population of susceptible mastomys natalensis rat/rodent is given by
Hence, the rate of change of the population of infected mastomys natalensis rat is given bywhere is the natural death rate of mastomys natalensis rat.
The model schematic diagram is given in Figure 1 below:
3. The Lassa-Fever Nonautonomous Periodically Forced Model
The following is obtained from the model descriptions:wheresubject to the initial conditions
We discuss in details the basic properties of the periodically forced Lassa-fever model (10) which entails its basic mathematical analysis in the next section.
Suppose denotes the average, given that and we assume . The basic properties of the Lassa-fever model (10) which entails its basic mathematical analysis are presented in the next section.
4. Qualitative Analysis of the Periodically Forced Lassa Fever Model
4.1. Positivity of Solutions
We found it pertinent to prove that all the state variables of the periodically forced Lassa fever transmission dynamic model (10) are nonnegative at all time (10) for the model to be epidemiologically meaningful and mathematically well-posed.
Theorem 1. Given that the initial data , , , , , , , then the solutions of the periodically forced Lassa fever model (10) are non-negative for all .
Proof. Suppose . Since , , , , , , , then . If , then equals zero at .
It follows from the first equation of the system (10) thatHence,Such thatIt can also be proved that , , , , .
Remark 1. The solutions of the Lassa fever autonomous model with nonnegative initial data will remain nonnegative every time as .
4.2. Boundedness of the Solution
Theorem 2. Every solution of the Lassa fever non-autonomous model are bounded. Thus, if . Such that
Remark 2. This shows that the periodically forced Lassa fever model (10) is epidemiologically meaningful and mathematically well posed in the region . Hence, the total population of humans and the mastomys natalensis rats is bounded above and below.
Theorem 3. The region is positively invariant for the model (10) with nonnegative initial condition in .
Proof. The Lassa fever nonautonomous model (10) is analysed in a biologically feasible region as follows: the model equation (10) is divided into human population compartment and the Mastomys natalensis rat population . Hence, we consider the feasible region:withWe established the positive invariance of J which means solutions in J remains in J for every . The rate of change of the humans and the Mastomys natalensis rats population is given byWe then applied the standard comparison theorem by Lakshmikantham et al.  to prove thatIn particular,wheneverHence, J is the region that is positively invariant and it is sufficient to consider the dynamics of the flow generated by (10) in J.
Remark 3. J is the region where the model (10) is epidemiologically meaningful and mathematically well posed. Therefore, all solutions of the model (10) with initial condition in J stay in J at all time .
4.3. Equilibrium Solution and Analysis of the Lassa Fever Nonautonomous Model
The model equation (10) is analysed in this section so that we can obtain the equilibrium points of the system. In order to provide an answer for the long-term behaviour of the Lassa fever autonomous model, we obtain the equilibrium solution of the model (10) by setting the equations of model (10) to zero, with
We obtain the values of the variables denoted as , , , , , , which satisfy this criteria. It shows that there will be no trivial equilibrium and that the population will never goes into extinction if the birth rate of humans and the birth rate of mastomys natalensis rat are nonzero. Therefore, when the Lassa fever disease is not present in the population, the model equation (10) has a steady state, , which is the disease-free equilibrium (DFE). Hence, the DFE of the model is given by
4.4. The Basic Reproduction Number for the Lassa Fever Autonomous Model
The reproduction number is a very important threshold quantity in epidemiology which measures the average number of new cases in a completely susceptible population. We used the next-generation matrix method to calculate the . The next-generation method is the spectral radius of the next-generation matrix .
Proposition 1. Suppose of the Lassa fever autonomous model (10) which is computed as the largest positive eigenvalue of the next generation matrix is given by
Proof. Generation matrix. SupposeThe rate of appearance of the disease in a compartment is while is the transfer of individuals and mastomys natalensis rat into one compartment. Hence, is a Jacobian matrix evaluated at and the Jacobian matrix of V is evaluated at which yieldsThe product of matrix and givesLetTherefore the eigenvalue of is calculated to obtain for the Lassa fever autonomous model (10) is given byt ≥ 0The basic reproduction number according to  is given bywhile the average basic reproduction number in the presence of the periodic function is given byIt is essential to know that
Remark 4. This shows that persistence and/or extinction of Lassa fever is obtained by calculating the of the Lassa fever such that if , the disease will extinct from the population and if , the disease will persist in the population. We discuss in the section the existence of the disease-free equilibrium.
4.5. Existence of the Disease Free Equilibrium (DFE) of the Lassa Fever Nonautonomous Model
In a situation without the Lass fever, the Lassa fever free equilibrium of the Lassa fever autonomous model (10) becomes:
such that , This yields
When we subject each of the equation in the model (10) to zero, it yieldstherefore,we obtain .
Hence, the Lassa fever free equilibrium is given by
The Mastomys Natalensis rodent compartment is separated from the humans compartment; hence, the equation is linear with nonconstant coefficients. It can be solved clearly as given below:
We introduce the average of a periodic function over its period in order to define the Lassa fever reproduction number. Obtaining the reproduction number, we recall by definition, if is a periodic function with period Q, then the average of j is given by
By perturbation analysis:
for large sufficient q, the expression on the RHS goes to if and only ifwhich prompts us to define the following reproduction number:
We note that is periodic with period 365.25 days, hence and we summarised the whole concept here in the following result.
Lemma 1. The disease-free equilibrium (DFE) is locally asymptotically stable whenever , then
and if , then , and the DFE is unstable.
Lemma 2. Suppose the Lassa fever disease (LFD) persists in the community such that
If and , and the disease will extinct (which implies that as , when ). Therefore, this result holds for any periodically forced birth rate which satisfies the assumptions that is continuous for all and that a constants , such that and the average as uniformly in t, i.e., and all , , and finally, all the parameters in the model are nonnegative .
Proof. We prove this theorem by asserting firstly that for each initial condition in the positive orthant of the population of Mastomys natalensis rodent bounded above and below, in consequence, we show that in our model, the disease cannot drive the population into extinction. For each initial condition , some positive constants and such that , . To prove this, we initiate the following notation which says for a positive quantity , , supposeAlso,The total population satisfies the differential equation:LetSuppose , we haveLetwhich implies that .
Then . Hence, we say such that is bounded above. To prove that is bounded below, one considers the prevalence of Lassa fever in Mastomys Natalensis rat .
The prevalence of Lassa fever in the rat f satisfies the differential equation:It is of note that and , so that for , we obtainfor chosen sufficiently small then the following conditions hold (a) (b) (c) .
It is possible to choose such that , , and .
Case 1. If we say that is not bounded below then such that for .
Case 2. We show that is bounded below. Given for time is large sufficiently for the average to take over the dynamics of the equation. Such that for any , holds, so that for and the prevalence of Lassa fever decays exponentially.
Now, for . Suppose , we have for , such that which is a contradiction. Hence, having . We have that for such that . We infer that . Now we say such that . We have the inequality , by the uniform convergence hypothesis. Now we say , such that and , and find this to be possible since . We then have for such that which is contradiction. Therefore, and is bounded below as asserted.
Following , we recall that where .
This implies thatWe have shown before that in this case,Given such that such that , we obtainConsider the equation for , which is given bySolving the inequality above, we obtainTherefore, if , when as .
We assume for simplicity that□ThenTherefore, we sayFrom the compartment for the Mastomys Natalensis rat population, we obtain the single one compartment equation in given byUsing the above notations and concepts, we summarised Lemma 3 below.
Lemma 3. The disease-free equilibrium (DFE) is globally stable if goes to zero as t tends to infinity whenever .
Proof. First we define the region such that the equation (69) is considered in this region. We applied here the Poincare map to establish the existence of a periodic solution. It is obvious that by the Poincare map we definewhich impliessuch that is the value of solution at time .
We understand that the Poincare map is injective because of the properties of solutions of ODEs. Hence, we can further show that it is continuously differentiable. Hence, it is easy to show that . Since the number is an initial value of a periodic solution if and only if is a fixed point of the Poincare map. Hence, for one to establish the existence of a positive periodic solution of equation (69), it is expedient to show that the Poincare map has a fixed point. Hence, we defineWe obtain the derivative of the Poincare map which is given asHence, by differentiating equation (79) with respect to in order to obtain the derivative of the Poincare map. For this case, we obtain a differential equation in h such thatBy differentiating the which is the initial condition with respect to , we have that . For the derivative of the Poincare map, which gives the following expression, the differential equation for h can be obtained as follows: It is clear that and it is obvious that the Poincare map is increasing. Given that, if and are two initial conditions which satisfy , then we obtain This implies that the exponent since and that . Thus, for sufficiently smallIt means that is small enough for . For which implies that the function changes sign in the interval . Therefore, there should exist an such that it becomes zero, which means thatFor us to establish the uniqueness of a periodic solution, it is essential to assume that there are two kinds of periodic solutions . Hence, assume without loss of generality thatLet be a periodic solution which satisfies firstly the equation (69):For and , we obtain