Samuel Oluwafemi OYAMAKIN* and Tolulope Samuel Adeyina
Biometry Unit, Department of Statistics, Nigeria
*Corresponding author: Samuel Oluwafemi OYAMAKIN, Biometry Unit, Department of Statistics, Nigeria
Submission: February 04, 2020; Published: February 19, 2020
ISSN 2578-0190
Volume3 issues4
A nonlinear deterministic model was considered to study the dynamics transmission and control of LASSA fever virus. The total population was divided into seven mutually exclusive classes between human and rodents as susceptible human, exposed humans, infected human, removed human, susceptible rodents, exposed rodents and infected rodents. Existence and uniqueness of the solution of the model were determined, the basic reproduction number is derived the model threshold parameter was examined using next-generation operator method. The existence of disease-free equilibrium point and endemic equilibrium point was carried out. The model result shows that diseases free equilibrium is local asymptotically stable at Ro< 1 and unstable at Ro> 1, the model is globally asymptotically stable. Sensitivity analysis of the model parameters was carried out in order to identify the most sensitive parameters on the disease transmission using data from the Nigeria States.
Keywords: Nigeria; Infectious diseases; LASSA Virus; Transmission dynamics; Disease free equilibrium
This paper aimed at investigating LASSA through the dynamics of various factors that contribute in every stage of its transmission using a mathematical compartmental model based on data from the Nigerian states. LASSA Fever (LF) is an acute and sometimes severe viral hemorrhagic illness endemic in West Africa. The earliest cases of LASSA fever were thought to have occurred between 1920 and 1950, in Nigeria, Sierra Leone and Central African Republic and perhaps in other West African countries. However, the disease was first recognized in Nigeria in 1969, Ms. Lily (Penny) Pinneo (not real names), who was the first documented case of LF and from whom the first LASSA Fever Virus (LASV) was isolated in 1969. LASV is a single stranded RNA virus belonging to the Arenaviridae family of viruses. The virus is often named hemorrhagic fever virus because of the tendency to cause bleeding from body orifices. Sequencing of the small segment of the RNA of LASV has revealed the presence of four major lineages in West Africa with three in Nigeria (lineages I, II, and III) and one in the area comprising Ivory Coast, Sierra Leone, Liberia, and Guinea (lineage IV). Various viral strains have been associated with these major lineages with differences in their genome, serologic, and pathogenic characteristics. In Nigeria, mastomys natalensis have been identified by various names in some local languages including Eeku Asin (Yoruba), Jagba (Hausa), Nkapia or Nkakwu- (Igbo), and Isun (Kolokuma Ijaw) [1-10].
Infectious disease modelling
The central idea about transmission models, as opposed to statistical models, is a mechanistic description of the transmission of infection between two individuals. This mechanistic description makes it possible to describe the time evolution of an epidemic in mathematical terms and in this way connect the individual level process of transmission with a population level description of incidence and prevalence of an infectious disease. The rigorous mathematical way of formulating these dependencies leads to the necessity of analyzing all dynamic processes that contribute to disease transmission in much detail. Therefore, developing a mathematical model helps to focus thoughts on the essential processes involved in shaping the epidemiology of an infectious disease and to reveal the parameters that are most inﬂuential and amenable for control. Mathematical modeling is then also integrative in combining knowledge from very different disciplines like microbiology, social sciences, and clinical sciences. For many infections-such as inﬂuenza and smallpox-individuals can be categorized as either “susceptible,” “infected” or “recovered and immune.” The susceptible that are affected by an epidemic move through these stages of infection.
Compartmental models in epidemiology
Compartmental models are a technique used to simplify the mathematical modelling of infectious disease. The population is divided into compartments, with the assumption that every individual in the same compartment has the same characteristics. Its origin is in the early 20^{th} century, with an important early work being that of Kermack and McKendrick in 1927. The models are usually investigated through ordinary differential equations (which are deterministic), but can also be viewed in a stochastic framework, which is more realistic but also more complicated to analyze. Compartmental models may be used to predict properties of how a disease spreads, for example the prevalence (total number of infected) or the duration of an epidemic. Also, the model allows for understanding how different situations may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population (Figure 1).
Figure 1: Common compartmental models in infectious disease modeling.
Framework for modeling for LASSA transmission dynamics
LASSA fever model presented in this study incorporates a model for the reservoir, which is Susceptible, exposed and infectious. This is done based on the assumption that susceptible rodent becomes exposed and then progress to infectious stage when they share unprotected storage of garbage, food stuﬀ and water with infectious rodents or from inhalation of aerosols from urine and feces. These infected rats eventually tend to infect humans by direct contact or by inhalation of air follicles from dead infected rodents. It is also important to note that the existence of the exposed compartment is because of the incubation time that is the time it takes for an infected human to become infectious, which is twenty-one days for LASSA fever virus (Figure 2).
Figure 2: The SEIR model dynamics.
LASSA fever model presented in this study incorporates diagnostic factor, γ(ai)α1(ai)Eh(t,ai) (since early diagnostic enables treatment, that is 6 days after infection) and vaccination parameter (νh)which are critical for prediction and the need of extensions to enhance their predictive power for decision support; also a subdivision of rodent compartment called susceptible, exposed and infectious rodent with discrete age structure denoted by Sr(t,ej), Er(t,ej) and Ir(t,ej) respectively. This is done based on the assumption that susceptible rodent becomes exposed and then progress to infectious stage when they share unprotected storage of garbage, food stuff and water with infectious rodents or from inhalation of aerosols from urine and faeces [11-18].
Parameters
We assume that the population is closed i.e. it is assumed that individuals who recovered from LASSA fever will never go back to susceptible class again (they remain recovered for life). The assumptions above suggest that LASV can only be transmitted from: (i) human to human (ii) rodent to human (iii) rodent to rodent, and as a result, we have the following schematic diagram and system of nonlinear ordinary differential equation (Figure 3).
Figure 3: Schematic diagram of transmission dynamics of Lassa fever
Figure 4: Multiple bar chart of the behavior of R0 in every state at varying intervention levels
Figure 4 explains the behavior of R0 at varying intervention levels, explained earlier in Table 3. This implies that at low intervention, Edo and Ondo has their R0 greater than 1, which implies instability in the states. This could be accounted for by the proximity between the two states (Edo state shares boundary with Ondo state). Furthermore, R0 is stable at both moderate intervention and high intervention in other states. This means that at moderate intervention, Lassa fever dies off gradually over a while.
Attached separately to keep the formatting intact
Formulating the Compartment into Differential Equation
Existence and positivity of solutionSince the formulated models monitor changes in the human and rodent populations; the parameters are assumed to ber nonnegative for all t ≥ 0, therefore the system of equations in (1) to (7) was analyzed within a feasible region R of biological interest. Theorem 1: The feasible region R defined by {Sh(t,ai),Eh(t,ai),Ih (t,ai),Rh (t,ai),Sr(t,ej),Er(t,ej),Ir(t,ej) ∈ R^{7} :
with initial conditions Sh(0,ai) ≥ 0,Eh(0,ai) ≥ 0,Ih(0,ai) ≥ 0,Rh(0,ai) ≥ 0,Sr(0,ej) ≥ 0,Er(0,ej) ≥ 0,Ir(0,ej) ≥ 0 is positive invariant for system (1) through (7).
Proof: If the total population size is given by
Nh(t, ai) = Sh(t, ai) + Eh(t, ai) + Ih(t, ai) + Rh(t, ai) and the total size of rodent population is
Nr(t, ej) = Sr(t, ej) + Er(t, ej) + Ir(t, ej)
Then one sees from (1) through (7) that
Considering the above equations as a linear differential we have that:
Solving the differential inequalities (8) and (9) one after the other gives
so that
this implies
and
so that
this implies
Taking the limits as t → ∞ gives
and
Thus, the following feasible region
Stability of the disease-free equilibriumA disease-free equilibrium point is a steady-state solution where there is no LASSA fever infection. Solving the system of equation (3.1)-(3.7) in the absence of disease, we obtain the following equilibrium point,
We obtain a basic reproduction number R0 by expressing (1) to (7) as the difference between the rate of new infections in each infected compartment denoted by F and the rate of transfer between each infected compartment denoted by G
The Jacobian matrices J_{F} and J_{G} of F and G are found about π0 Where, T = J_{F}J-1_{G} From the Jacobian matrices J_{F} and J_{G} of F and G found about π0, we have R_{0}, the maximum eigenvalue of T
Local stability of a disease-free equilibriumTheorem 2: The disease-free equilibrium for the system (1) - (7) is locally asymptotically stable if R_{0}(ai)<1 and unstable if R_{0}(ai) >1. Proof:. The Jacobian of system (1) through (7) evaluated at the disease-free equilibrium point is
The ﬁrst, fourth and ﬁfth columns have diagonal entries, therefore the diagonal entries −µ_{h}(ai) twice and −µ_{r}(e_{j}) − δ_{j} (e_{j}) are three of the eigenvalues of the Jacobian matrix i.e λ1 =−µ_{h}(a_{i}), λ2 = −µ_{h}(a_{i}) and λ3 = −µ_{r}(e_{j}) − δ_{r}(e_{j}) . Thus, we ﬁnd the remaining eigenvalues by excluding the corresponding columns and rows. The remaining eigenvalues are obtained from the submatrix
The eigenvalues of the matrix J_{2}(π_{0}) are the roots of the characteristic equation
Further simplification of c _{0} in terms of R _{0}(a) gives
C0 = b_{1}b_{2} (1 − R_{0} (a))
The Routh-Hurwitz criterion follows since all the roots of the polynomial (6) have negative real part if and only if the coeﬃcient ci are positive and matrices Hi>0 for i = 0,1,2. From (7) it is shown that c_{1}>0 and c_{2}>0 since all b′ is are positive. Moreover, if R_{0}(a)<1 it is follows from (8) that c_{0}>0. All the eigenvalues of the Jacobian matrices J(π_{0}) have negative real part when R_{0}(a)<1 and the disease-free equilibrium point is locally asymptotically stable. However, when R_{0}(a)>1, we see that c_{0}<0 and by Descartes’ rule of signs ,there is exactly one sign change in the c_{2},c_{1},c_{0} of coeﬃcient of the polynomial (6). So, there is one eigenvalue with the positive real part and the disease-free equilibrium point is unstable. At this juncture, we can now infer from the equation (8) that there exists a c_{0} such that and since R_{0}(a) majorizes R_{0}(a_{i}) the system (1) - (7) is locally asymptotically stable if R_{0}(ai)<1 and unstable if R_{0}(a_{i})>1.
Existence of endemic equilibriumWe shall use the theorem below to show that the developed model (1) - (7) has an endemic equilibrium point Ee. The endemic equilibrium point is a positive steady state solution when the disease persists in the population. Theorem 3: LASSA fever model (1) - (7) has no endemic equilibrium when R0(a)<1 and a unique endemic equilibrium exist when R0(a)>1.
Proof: Let Ee = (sh*, Eh*, Ih*, Rh*, Sr*, Er*, Ir*, ) be a non-trivial equilibrium of system (1) - (7) i.e. all component of Ee are positive. At the steady state, we get
It is clearly seen that K_{1}>0. When R_{0}(a)<1 , one sees that K_{2}>0. However, when R_{0}(a)>1, then K_{2}<0 and endemic equilibrium exists. Finally, K_{3}<0 if d_{1}<(d_{2} + d_{3}).
Sensitivity analysisSensitivity analysis is a statistical technique that can be applied to mathematical or computational models to provide insight on how uncertainty in the input variables affect the model outputs and which input variables tend to drive variation in the outputs. For models such as this infectious disease model, where the output is intended to inform decision-makers, uncertainty in the output can be disconcerting, as a single value is not given. However, the benefit is that a range of output values reveals a suite of possible model outcomes. It is sometimes of value to put in place robust policies that avoid the worst that can happen, although the policies may not achieve optimal results. If mitigating policies and alternatives are included and parameterized within the model, then sensitivity analysis can provide estimates of how responsive key output measures are with respect to these inputs. Sensitivity analysis tells us how important each parameter is to disease transmission. Such information is crucial not only for experimental design, but also to data assimilation and reduction of complex nonlinear models. Sensitivity analysis is commonly used to determine the robustness of model predictions to parameter values, since there are usually errors in data collection and presumed parameter values [19-24]. It is used to discover parameters that have a high impact on and should be targeted by intervention strategies. Sensitivity indices allow us to measure the relative change in a variable when a parameter changes. The normalized forward sensitivity index of a variable with respect to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives. Here, we present the sensitivity analysis of our Model using the formula given as
From the equations above, the most sensitive parameters are η(a_{1}) and ε_{h} which are Effective transmission rate in susceptible humans by infected humans and Progression rate of humans from the exposed state to the infectious state respectively. This implies that an increase or decrease in these parameters will cause an instability in R_{0}, as a result of its direct effect on R_{0}.
Data presentation and analysisThe data below was extracted from the National Center of Disease Control’s weekly Epidemiological Report on LF up till February 2019. Susceptible is the population of the state where LASSA fever has Incidence, Exposed is a percentage of the susceptible which is a function of effective transmission rate, Infected is a population reported confirmed, while death, is the death of confirmed cases (Figures 4-9); (Tables 1-4).
Figure 5: Multiple bar chart illustrating the behavior of R0 for every state at varying intervention levels.
Figure 6: Simple bar chart illustrating R0 at low intervention for every state at varying intervention levels.
Considering Figure 6 above, at low intervention, Lassa fever becomes epidemic at Edo and Ondo state, while the Ro is less than 1for other states which implies stability.
Figure 7: Simple bar chart illustrating R0 at moderate intervention for every state at varying intervention levels.
Considering Figure 7 above, at moderate intervention, Lassa fever becomes stable with the Ro is less than 1 for all states which implies stability.
Figure 8: Simple bar chart illustrating R0 at maximum intervention for every state at varying intervention levels.
Considering Figure 8 above, at high intervention, Lassa fever becomes stable with the Ro as low as 0.11 for all states which implies stability.
Figure 9: Case fatality of LASSA Across The Nigerian State.
Figure 9 explains case fatality as the percentage of those that are infected that eventually die. It is important to note that Kano, Rivers, Cross Rivers, Borno, Kwara, Benue, Imo, Abia and Adamawa, has 100% fatality with low case of incidence, while compared to states like Edo states,Ogun state, with higher cases of incedence have low case fatality. This could only be pointed to the factor of proximity to diagnostic centers, which exists only in three states in Nigeria. Literarily cases that occur in states closer to diagnostics centers gets fast intervention, as the disease can easily be diagnosed, and detected.
Figure 10: The distribution of confirmed Lassa fever cases in Nigeria.
Figure 10 above validates the first discovery that proximity to diagnostic center affect the case fatality of Lassa fever patient. Population of confirmed cases are more in Edo and Ondo state, which are closer to Lagos Diagnostic center, while places like Plateau, Bauchi, Gombe, are closer to irrua Diagnostic center at Jos.
Table 1: Extracted Data from the National Center of Disease Control (NCDC) and Susceptible is the population of the state where Lassa fever has Incidence, Exposed is a percentage of the susceptible which is a function of effective transmission rate, Infected is a population reported confirmed, while death, is the death of confirmed cases.
Table 2: Parameter structure used in the models and their sources and explains the values of parameters used in the estimation of R_{0}.
Table 3: Basic Reproduction number obtained across the state at three different levels of parameter and iterates the estimates of R_{0} gotten from the data collected from NCDC and parameters in Table 2.
Table 4: Assumed Parameters of R0 at three different levels and above explains the varying degrees of interventions to reduce/stop Lassa fever which are captured by the above four parameters (Treatment rate of exposed humans α1(ai) Treatment rate of infected humans α2(ai), Diagnostic factor of exposed humans γ(ai) proportion of exective treatment of infected humans ψ(ai) )in all the compartments of the model. The values are varied at 0.1, which indicated little or no intervention, 0.5: which indicated medium intervention, while 0.9 indicates maximum intervention at these levels.
Considering the available data and the result of the sensitivity analysis, varying the values of four parameters which all signify effective and ineffective intervention, towards the eradication of the disease, showed a drastic reduction in R0, as the intervention level increased. It is also important to note that the lines with 100% fatality are states with first incidence of disease, especially places far away from diagnostic centers. This utmost depicts low diagnostics which is a function of proximity to test centers, which are in three places in Nigeria; National reference laboratory in Abuja, Lagos university teaching hospital (LUTH), Irrua specialist teaching hospital (ISTH), Institute of LASSA fever research and control (ILFRC), Irrua, Edo state, Nigeria.
The modification of the transmission dynamics between susceptible compartment to Exposed compartment by the removal of effective contact rate and interaction rate in reservoir has a positive effect on the R_0. At minimum intervention, Edo state and Ondo state has their R0>1 which implies instability. This means in time; LASSA fever will be epidemic in the two state. At mid intervention, R0<1 for all the states, which implies a steady state for the disease with minimal intervention. At maximum intervention R0 reduces to the barest below 0.2 in all states, which implies a steady state for the disease. Therefore, Proximity to diagnostic centers effective healthcare model has major part to play in eradicating LASSA fever in Nigeria.
© 2020 Samuel Oluwafemi OYAMAKIN. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.