Category: Corona Virus Vaccine

Let us emphasize that the spectral matrix collocation approach based on the SCPSK may not yield convergence on a long time interval ([t_a,t_b]). One remedy is to use a large number of bases on the long domains accordingly to reach the desired level of accuracy. Another approach is to divide the given interval into a sequence of subintervals and employ the proposed collocation scheme on each subinterval consequently.

Towards this end, we split the time interval ([t_a,t_b]) into (Nge 1) subdomains in the forms

$$begin{aligned} K_n:=[t_{n},t_{n+1}],quad n=0,1,ldots , N-1. end{aligned}$$

Here, we have (t_0:=t_a) and (t_N:=t_b). The uniform time step is taken as (h=t_{n+1}-t_n=(t_b-t_a)/N). Note that by selecting (N=1), we turn back to the traditional spectral collocation method on the whole domain ([t_a,t_b]). Therefore, on each subinterval (K_n) we take the approximate solution of the modelEq. (1) to be in the formEq. (25) as

$$begin{aligned} x^n_{mathcal {J}}(t):=sum limits _{j=0}^{mathcal {J}} omega ^n_j,mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^n_mathcal {J},quad tin K_n, end{aligned}$$

(29)

where we utilized the notations

$$begin{aligned} varvec{W}_{mathcal {J}}^n:=left[ omega ^n_0quad omega ^n_1quad ldots quad omega ^n_{mathcal {J}}right] ^T,quad varvec{U}_{mathcal {J}}(t):=left[ mathbb {U}_0(t)quad mathbb {U}_1(t)quad ldots quad mathbb {U}_J(t)right] , end{aligned}$$

as the vector of unknown coefficients and the vector of SCPSK bases respectively. Once we get the all local approximate solutions for (n=0,1,ldots ,N-1), the global approximate solution on the given (large) interval ([t_a,t_b]) will be constructed in the form

$$begin{aligned} x_{mathcal {J}}(t)=sum limits _{n=0}^{N-1} c_n(t),x^n_{mathcal {J}}(t),quad c_n(t):= left{ begin{array}{ll} 0, &{} tnotin K_n,\ 1, &{} tin K_n.\ end{array}right. end{aligned}$$

In order to collocate a set of ((mathcal {J}+1)) linear equations to be obtained later at some suitable points, we consider the roots of (mathbb {U}_{mathcal {J}+1}(t)) on the subinterval (K_n). By modifying the points giveninEq. (17), we take the collocation nodes as

$$begin{aligned} t_{nu ,n}=frac{1}{2}left( t_n+t_{n+1}+h,cos left( frac{nu ,pi }{mathcal {J}+2}right) right) ,quad nu =1,2,ldots ,mathcal {J}+1. end{aligned}$$

(30)

At the end, we note that in the proposed splitting approach, the given initial conditions of the underlying model problem are prescribed on the first subinterval (K_0). Once the approximate solution on (K_0=[t_0,t_1]) is determined, we utilize it to assign the initial conditions on the next time interval (K_1). To do so, it is sufficient to evaluate the obtained approximation at (t_1). We repeat this idea on the next subintervals in order until we arrive at the last subinterval (K_{N-1}). Below, we illustrate the main steps of our matrix collocation algorithm on an arbitrary subinterval (K_n) for (n=0,1,ldots ,N-1).

Our chief aim is to solve the nonlinear COVID-19 systemEq. (1) efficiently by using the spectral method based on SCPSK basis. Towards this end, we first need to get rid of the nonlinearity of the model. This can be done by employing the Bellmans quasilinearization method (QLM)[39]. Thus we will get more advantages in terms of running time, especially for large values of J in comparison to the performance of directly applied collocation methods to nonlinear models, see cf.[40,41,42]. By combining the idea of QLM and the splitting of the domain we will obtain more gains in terms of accuracy for the approximate solutions of nonlinear modelEq. (1). Let us first describe the technique of QLM. For more information, we may refer the readers to the above-mentioned works.

By reformulating the original COVID-19 modelEq. (1) in a compact form we get

$$begin{aligned} frac{d}{dt} varvec{z}(t)=varvec{G}(t,varvec{z}(t)), end{aligned}$$

(31)

where

$$begin{aligned} varvec{z}(t)=left[ begin{array}{c} S(t)\ S_v(t)\ I(t)\ I_v(t)\ R(t)\ R_v(t)\ J(t)\ J_v(t) end{array}right] ,quad varvec{G}(t,varvec{z}(t))=left[ begin{array}{c} g_1(t)\ g_2(t)\ g_3(t)\ g_4(t)\ g_5(t)\ g_6(t)\ g_7(t)\ g_8(t) end{array}right] = left[ begin{array}{c} Lambda - beta S(I+I_v)- (lambda +mu ) S+ theta _1 R\ -beta ' S_v(I+ I_v)+ theta _2R_v+ lambda S- (delta +mu ) S_v\ beta S(I+ I_v)- (gamma _1+alpha _1+mu ) I \ beta ' S_v(I+ I_v)- (gamma _2+alpha _2+mu )I_v\ gamma _1 I-(theta _1+mu ) R+ eta _1 J \ gamma _2 I_v- (theta _2+mu ) R_v+ eta _2J_v+ delta S_v\ alpha _1 I- (eta _1+mu _1)J\ alpha _2I_v- (eta _2+mu _2) J_v end{array}right] . end{aligned}$$

To begin the QLM process, we assume (varvec{z}_0(t)) is available as an initial rough approximation for the solution (varvec{z}(t)) of the COVID-19 systemEq. (31). Through an iterative manner, the QLM procedure reads as follows

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t)approx varvec{G}(t,varvec{z}_{s-1}(t))+varvec{G}_{varvec{z}}(t,varvec{z}_{s-1}(t)),left( varvec{z}_{s}(t)-varvec{z}_{s-1}(t)right) ,quad s=1,2,ldots . end{aligned}$$

Here, the notation (varvec{G}_{varvec{z}}) stands for the Jacobian matrix of the COVID-19 systemEq. (31), which is of size 8 by 8. By performing some calculations we reach the linearized equivalent model form as

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t)+varvec{M}_{s-1}(t),varvec{z}_{s}(t)=varvec{r}_{s-1}(t),qquad s=1,2,ldots , end{aligned}$$

(32)

where (varvec{M}_{s-1}(t):=varvec{J}(S_{s-1}(t), (S_v)_{s-1}(t), I_{s-1}(t), (I_v)_{s-1}(t))) as the Jacobian matrix (varvec{J}) previously constructed inEq. (7). Also we have

$$begin{aligned} varvec{z}_{s}(t)= left[ begin{array}{c} S_{s-1}(t)\ (S_v)_{s-1}(t)\ I_{s-1}(t)\ (I_v)_{s-1}(t)\ R_{s-1}(t)\ (R_v)_{s-1}(t)\ J_{s-1}(t)\ (J_v)_{s-1}(t) end{array}right] ,quad varvec{r}_{s-1}(t)= left[ begin{array}{c} Lambda +beta ,S_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ beta ',(S_v)_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ -beta ,S_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ -beta ',(S_v)_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ 0\ 0\ 0\ 0 end{array}right] . end{aligned}$$

Along with the systemEq. (32) the initial conditions

$$begin{aligned} varvec{z}_{s}(0)=left[ begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} end{array}right] ^T, end{aligned}$$

(33)

are given due toEq. (2). We now are able to solve the family of linearized initial-value problemsEqs. (32)-(33) numerically by our proposed matrix collocation method on an arbitrary (long) domain ([t_a,t_b]). For this purpose and for clarity of exposition, we restrict our illustrations to a local subinterval (K_n) for (n=0,1,ldots ,N-1).

In view ofEq. (29) by utilizing only ((mathcal {J}+1)) SCPSK basis functions, we assume that the eight solutions of systemEq. (32) can be represented in terms ofEq. (29). Thus, we take these solutions at iteration (sge 1) as

$$begin{aligned} left{ begin{array}{l} S^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,1},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},1},quad (S_v)^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,2},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},2},\ I^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,3},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},3},quad (I_v)^{n}_{mathcal {J},s}(t)~=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,4},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},4},\ R^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,5},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},5},quad (R_v)^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,6},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},6},\ J^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,7},mathbb {U}_j(t)=varvec{U}_{J}(t),varvec{W}^{n,s}_{mathcal {J},7},quad (J_v)^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,8},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},8},\ end{array}right. end{aligned}$$

(34)

for (tin K_n). Moreover, by (varvec{W}^{n,s}_{mathcal {J},i}= left[ begin{array}{cccc} omega ^{n,s}_{0,i}&omega ^{n,s}_{1,i}&dots&omega ^{n,s}_{mathcal {J},i} end{array}right] ^T) we denote the vectors of unknowns for (1le ile 8) at the iteration (sge 1). Also, the vector of SCPSK basis, i.e., (varvec{U}_mathcal {J}(t)) is defined inEq. (29). We next provide a decomposition for (varvec{U}_mathcal {J}(t)) given by

$$begin{aligned} varvec{U}_mathcal {J}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J}. end{aligned}$$

(35)

Here, the vector (varvec{Q}_mathcal {J}(t)) including the powers of ((t-t_n)) introduced by

$$begin{aligned} varvec{Q}_mathcal {J}(t)=left[ 1quad t-t_nquad (t-t_n)^{2}quad ldots quad (t-t_n)^{mathcal {J}}right] . end{aligned}$$

The next object is the matrix (varvec{F}_mathcal {J}=(f_{i,j})_{i,j=0}^{mathcal {J}}) of size ((mathcal {J}+1)times (mathcal {J}+1)). The entries of the latter matrix are given inEq. (15). One can also show that (det (varvec{F}_mathcal {J})ne 0) and it is a triangular matrix. It follows that

$$begin{aligned} f_{i,j}:= left{ begin{array}{ll} o_{i,j}, &{} textrm{if}~ ile j,\ 0, &{} textrm{if}~ i> j. end{array}right. end{aligned}$$

We then insert the obtained term (varvec{U}_mathcal {J}(t)) inEq. (35) intoEq. (34). The resulting expansions are

$$begin{aligned} left{ begin{array}{l} S^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (S_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ I^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_J,varvec{W}^{n,s}_{mathcal {J},3},quad (I_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ R^{n}_{mathcal {J},s}(t) =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (R_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ J^{n}_{mathcal {J},s}(t), =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (J_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$

(36)

We then proceed by nothing that the derivative of the vector (varvec{Q}_mathcal {J}(t)) can be stated in terms of itself. A vivid calculation reveals that

$$begin{aligned} dot{varvec{Q}}_{mathcal {J}}(t)=varvec{Q}_{mathcal {J}}(t),varvec{D}_mathcal {J},quad varvec{D}_mathcal {J}=left[ begin{array}{lllll} 0 &{} 1 &{} 0 &{}ldots &{} 0\ 0 &{} 0 &{} 2 &{}ldots &{} 0\ vdots &{} vdots &{} ddots &{}vdots &{} vdots \ 0 &{} 0 &{} 0 &{}ddots &{} mathcal {J}\ 0 &{} 0 &{} 0 &{} ldots &{} 0 end{array}right] _{(mathcal {J}+1)times (mathcal {J}+1)}. end{aligned}$$

(37)

From this relation, we are able to derive a matrix forms of the derivatives of the unknown solutions inEq. (36).

$$begin{aligned} left{ begin{array}{l} dot{S}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (dot{S}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ dot{I}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},3},quad (dot{I}_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ dot{R}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (dot{R}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ dot{J}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (dot{J}_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$

(38)

The exact solutions of the linearized systemEq. (32) can be written in a vectorized form as

$$begin{aligned} varvec{z}_s(t)approx varvec{z}^n_{mathcal {J},s}(t):= left[ begin{array}{l} S^{n}_{mathcal {J},s}(t)\ (S_v)^{n}_{mathcal {J},s}(t)\ I^{n}_{mathcal {J},s}(t)\ (I_v)^{n}_{mathcal {J},s}(t)\ R^{n}_{mathcal {J},s}(t)\ (R_v)^{n}_{mathcal {J},s}(t)\ J^{n}_{mathcal {J},s}(t)\ (J_v)^{n}_{mathcal {J},s}(t) end{array}right] ,quad dot{varvec{z}}_s(t)approx frac{d}{dt}varvec{z}^n_{mathcal {J},s}(t):= left[ begin{array}{l} dot{S}^{n}_{mathcal {J},s}(t)\ (dot{S}_v)^{n}_{mathcal {J},s}(t)\ dot{I}^{n}_{mathcal {J},s}(t)\ (dot{I}_v)^{n}_{mathcal {J},s}(t)\ dot{R}^{n}_{mathcal {J},s}(t)\ (dot{R}_v)^{n}_{mathcal {J},s}(t)\ dot{J}^{n}_{mathcal {J},s}(t)\ (dot{J}_v)^{n}_{mathcal {J},s}(t) end{array}right] . end{aligned}$$

(39)

We next introduce the following block diagonal matrices of dimensions (8(mathcal {J}+1)times 8(mathcal {J}+1)) as

$$begin{aligned} widehat{varvec{Q}}(t){} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t) end{array}right) ,\ widehat{varvec{D}}{} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J} end{array}right) ,\ widehat{varvec{F}}{} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J} end{array}right) . end{aligned}$$

By the aid of the former definitions, the matrix formats of (varvec{z}^n_{mathcal {J},s}(t)) and (dot{varvec{z}}^n_{mathcal {J},s}(t)) will rewrite concisely as

$$begin{aligned} varvec{z}^n_{mathcal {J},s}(t)=widehat{varvec{Q}}(t),widehat{varvec{F}},varvec{W}^n,quad dot{varvec{z}}^n_{mathcal {J},s}(t)=widehat{varvec{Q}}(t),widehat{varvec{F}},widehat{varvec{D}},varvec{W}^n. end{aligned}$$

(40)

Here, (varvec{W}^n) is the successive vector of eight previously defined vector of unknowns

$$begin{aligned} varvec{W}^n=left[ begin{array}{cccc} varvec{W}^{n,s}_{mathcal {J},1}&varvec{W}^{n,s}_{mathcal {J},2}&ldots&varvec{W}^{n,s}_{mathcal {J},8} end{array}right] ^T. end{aligned}$$

We now can collocate the linearized Eq.(32) at the zeros of SCPSK given inEq. (17) on the subdomain (K_n). We get

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t_{nu ,n})+varvec{M}_{s-1}(t_{nu ,n}),varvec{z}_{s}(t_{nu ,n})=varvec{r}_{s-1}(t_{nu ,n}),qquad nu =1,2,ldots ,mathcal {J}, end{aligned}$$

(41)

for (s=1,2,ldots). Denote the coefficient matrix by (widehat{varvec{M}}^n_{s-1}) and the right-hand-side vector as (widehat{varvec{R}}^n_{s-1}). These are defined by

$$begin{aligned} widehat{varvec{M}}^n_{s-1}= left[ begin{array}{cccc} varvec{M}_{s-1}(t_{0,n})&{}textbf{0}&{}ldots &{}textbf{0}\ textbf{0}&{}varvec{M}_{s-1}(t_{1,n})&{}ldots &{}textbf{0}\ vdots &{}vdots &{}ddots &{}vdots \ textbf{0}&{}textbf{0}&{}ldots &{}varvec{M}_{s-1}(t_{mathcal {J},n}) end{array}right] ,quad widehat{varvec{R}}^n_{s-1}= left[ begin{array}{c} varvec{r}_{s-1}(t_{0,n})\ varvec{r}_{s-1}(t_{1,n})\ vdots \ varvec{r}_{s-1}(t_{mathcal {J},n}) end{array}right] . end{aligned}$$

Let us define further the vectors of unknowns as

$$begin{aligned} dot{varvec{Z}}^n_s= left[ begin{array}{c} dot{varvec{z}}_{s}(t_{0,n})\ dot{varvec{z}}_{s}(t_{1,n})\ vdots \ dot{varvec{z}}_{s}(t_{mathcal {J},n}) end{array}right] ,quad varvec{Z}^n_s= left[ begin{array}{c} dot{varvec{z}}_{s}(t_{0,n})\ dot{varvec{z}}_{s}(t_{1,n})\ vdots \ dot{varvec{z}}_{s}(t_{mathcal {J},n}) end{array}right] . end{aligned}$$

Consequently, the system of Eq.(41) can be stated briefly as

$$begin{aligned} dot{varvec{Z}}^{n}_{s}+widehat{varvec{M}}^n_{s-1},varvec{Z}^n_s=widehat{varvec{R}}^n_{s-1},quad n=0,1,ldots ,N-1, end{aligned}$$

(42)

and with (s=1,2,ldots). Before we talk about the fundamental matrix equation, we need to state two vectors (varvec{Z}^n_s) and (dot{varvec{Z}}^{n}_{s})inEq. (42) in the matrix representation forms. The proof is easy by just considering the definitions of the involved matrices and vectors inEq. (40).

If two vectors (varvec{z}^n_{mathcal {J},s}(t)) and (dot{varvec{z}}^n_{mathcal {J},s}(t)) inEq. (40) computed at the collocation pointsEq. (30), we arrive at the next matrix forms

$$begin{aligned} varvec{Z}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},varvec{W}^n,qquad dot{varvec{Z}}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},widehat{varvec{D}},varvec{W}^n, end{aligned}$$

(43)

where the matrix (bar{widehat{varvec{Q}}}) is given by

$$begin{aligned} bar{widehat{varvec{Q}}}=[widehat{varvec{Q}}(t_{0,n})quad widehat{varvec{Q}}(t_{1,n})quad ldots quad widehat{varvec{Q}}(t_{mathcal {J},n}) ]^T. end{aligned}$$

Moreover, two matrices (widehat{varvec{Q}}, widehat{varvec{F}}) are defined inEq. (40). Similarly, the vector (varvec{W}^n) is given inEq. (40).

By turning to relationEq. (40) we substitute the derived matrix formats into it. Precisely speaking, after replacing (varvec{Z}^n_s) and (dot{varvec{Z}}^n_s) we gain the so-called fundamental matrix equation (FME) of the form

$$begin{aligned} varvec{B}_n,varvec{W}^n=widehat{varvec{R}}^n_{s-1}, quad textrm{or}quad left[ varvec{B}_n;widehat{varvec{R}}^n_{s-1}right] ,quad sge 1,~0le nle N-1, end{aligned}$$

(44)

where

$$begin{aligned} varvec{B}_n:=bar{widehat{varvec{Q}}},widehat{varvec{F}}+widehat{varvec{M}}^n_{s-1},bar{widehat{varvec{Q}}},widehat{varvec{F}},widehat{varvec{D}}. end{aligned}$$

To complete the process of QLM-SCPSK approach, it is necessary to implement the initial conditionsinEq. (2) and add them intoEq. (44). So, the next task is to constitute the matrix representation ofEq. (2). Let us approach (trightarrow 0) in the first relation ofEq. (40). It gives us

$$begin{aligned} varvec{B}_{0,n},varvec{W}^n=widehat{varvec{R}}^n_{s-1,0},qquad varvec{B}_{0,n}:=widehat{varvec{Q}}(0),widehat{varvec{F}},quad widehat{varvec{R}}^n_{s-1,0}=left[ begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} end{array}right] ^T. end{aligned}$$

We then replace eight rows of the augmented matrix ([varvec{B}_n;widehat{varvec{R}}^n_{s-1}]) by the already obtained row matrix ([varvec{B}_{0,n};widehat{varvec{R}}^n_{s-1,0}]). Denote the modified FME by

$$begin{aligned} check{varvec{B}_{n}},varvec{W}^n=check{textbf{R}}^n_{s-1},quad textrm{or} quad left[ check{varvec{B}_{n}};check{textbf{R}}^n_{s-1}right] . end{aligned}$$

(45)

This implies that the solution of the modelEq. (1) is obtainable on each subdomain (K_n) by iterating (n=0,1,ldots ,N-1). On (K_0) as the first subdomain, the given initial conditionsinEq. (2) will be used to find the corresponding approximations for the systemEq. (1). Hence, this approximate solutions on (K_0) evaluated at the starting point of (K_1) will be utilized for the initial conditions on (K_1). By repeating this process we acquire all approximations on all (K_n) for (0le nle N-1).

Generally, finding the true solutions of the COVID-19 systemEq. (1) is not possible practically. In this case, the residual error functions (REFs) help us to measure the quality of approximations obtained by the QLM-SCPSK technique. Once we calculate the eight approximations by the illustrated method, we substitute them into the model systemEq. (1). In fact, the REFs are defined as the difference between the left-hand side and the right-hand side of the considered equation. On the subdomain (K_n) we set the REFs as

$$begin{aligned}{} & {} mathbb {R}_{1,mathcal {J}}^{n}(t):=left| dot{S}^{n}_{mathcal {J},s}(t)-Lambda +beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+(lambda +mu ) S^{n}_{mathcal {J},s}(t)- theta _1 R^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{2,mathcal {J}}^{n}(t):=left| (dot{S_v})^{n}_{mathcal {J},s}(t)+beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)- theta _2(R_v)^{n}_{mathcal {J},s}(t)- lambda S^{n}_{mathcal {J},s}(t)+ (delta +mu ) (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{3,mathcal {J}}^{n}(t):=left| dot{I}^{n}_{mathcal {J},s}(t)-beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _1+alpha _1+mu ) I^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{4,mathcal {J}}^{n}(t):=left| (dot{I_v})^{n}_{mathcal {J},s}(t)- beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _2+alpha _2+mu )(I_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{5,mathcal {J}}^{n}(t):=left| dot{R}^{n}_{mathcal {J},s}(t) - gamma _1 I^{n}_{mathcal {J},s}(t)+(theta _1+mu ) R^{n}_{mathcal {J},s}(t)- eta _1 J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{6,mathcal {J}}^{n}(t):=left| (dot{R_v})^{n}_{mathcal {J},s}(t)- gamma _2 (I_v)^{n}_{mathcal {J},s}(t)+ (theta _2+mu ) (R_v)^{n}_{mathcal {J},s}(t)- eta _2(J_v)^{n}_{mathcal {J},s}(t)- delta (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{7,mathcal {J}}^{n}(t):=left| dot{J}^{n}_{mathcal {J},s}(t) -alpha _1 I^{n}_{mathcal {J},s}(t)+ (eta _1+mu _1)J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{8,mathcal {J}}^{n}(t):=left| (dot{J_v})^{n}_{mathcal {J},s}(t)- alpha _2(I_v)^{n}_{mathcal {J},s}(t) +(eta _2+mu _2) (J_v)^{n}_{mathcal {J},s}(t)right| cong 0, end{aligned}$$

(46)

for a fixed iteration number s and we have defined (L^n_{mathcal {J},s}:=I^{n}_{mathcal {J},s}(t)+ (I_v)^{n}_{mathcal {J},s}(t)) for brevity.

Analogously, at the fixed iteration s, the numerical order of convergence associated with the obtained REFs can be defined in the infinity norm. These are given by

$$begin{aligned} L^{infty }_{ell }equiv L^{infty }_{ell }(mathcal {J}):=max _{0le nle N-1}left( max _{tin K_n},left| mathbb {R}_{ell ,mathcal {J}}^{n}(t)right| right) ,quad ell =1,2,ldots ,8. end{aligned}$$

Therefore, the convergence order (Co) for each solution is defined by

$$begin{aligned} textrm{Co}_{mathcal {J}}^{ell }:=log _2left( frac{L^{infty }_{ell }(mathcal {J})}{L^{infty }_{ell }(2mathcal {J})}right) ,quad ell =1,2,ldots ,8. end{aligned}$$

(47)

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Stability analysis and numerical evaluations of a COVID-19 model with vaccination - BMC Medical Research ... - BMC Medical Research Methodology

For years, researchers have been working on vaccines that aim to prevent viral infections by strengthening immune defenses at viruses doorway to the body: the nose.

A small study recently published in PNAS presents a similar, if lower-tech, idea. Coating the inside of the nose with the over-the-counter antibiotic ointment Neosporin seems to trigger an immune response that may help the body repel respiratory viruses like those that cause COVID-19 and the flu, the study suggests.

The research raises the idea that Neosporin could serve as an extra layer of protection against respiratory illnesses, on top of existing tools like vaccines and masks, says study co-author Akiko Iwasaki, an immunobiologist at the Yale School of Medicine and one of the U.S. leading nasal vaccine researchers.

The study builds upon some of Iwasakis prior researchwhich has shown that similar antibiotics can trigger potentially protective immune changes in the bodybut its still preliminary, she cautions. For the new study, her team had 12 people apply Neosporin inside their nostrils twice a day for a week, while another seven people used Vaseline for comparison. At several points during the study, the researchers swabbed the participants noses and ran PCR tests to see what was going on inside.

Read More: What to Do About Your Bunions

They found that Neosporinand specifically one of its active ingredients, the antibiotic neomycin sulfateseems to stimulate receptors in the nose that are fooled into thinking theres a viral infection and in turn create a barrier thats put up against any virus, Iwasaki explains. In theory, she says, that means it could protect against a range of different infections.

Right now, though, thats just a theory. For this study, Iwasakis team didnt take the next step of testing whether that immune response actually prevents people from getting infected when theyre exposed to virusesin part because its ethically questionable to intentionally expose people to pathogens for research. (They did, however, demonstrate that rodents whose noses were coated with neomycin were protected from the virus that causes COVID-19.)

On its website, the maker of Neosporin says that the product has not "been tested or formulated to prevent against COVID-19 or any other virus," and also note that they do not advise putting the product inside the eyes, nose, or mouth.

Dr. James Crowe, who directs the Vanderbilt Vaccine Center and was not involved in the research, says the study is intriguing, but hed need to see more human data before he gets excited. Im skeptical it would be strongly effective in people, Crowe says. If you have a modest effect on the virus, is that enough to really benefit you clinically?

It is somewhat counterintuitive to think that an antibiotic, which kills bacteria, could do anything to protect people from viruses. Its not that the antibiotic has a direct effect against viruses, Iwasaki explains. Instead, it seems that neomycin, when applied topically, provoke changes in the body that help it fight off virusesessentially, triggering a natural antiviral effect.

So should you smear Neosporin in your nose next time a COVID-19 wave hits? Not so fast, says Dr. Benjamin Bleier, who specializes in nasal disorders at Massachusetts Eye and Ear and has studied nasal immunity.

Read More: COVID-Cautious Americans Feel Abandoned

Bleier, who was not involved in the new study, calls the research very well done, but says there are questions that need to be answered before it hits clinical prime time. First, could the body develop tolerance or resistance to neomycin if the antibiotic were regularly used in this way? (Antibiotic resistance is a growing concern, and overusing or inappropriately prescribing antibiotics is a contributor to the problem.) Second, could the average person apply neomycin deeply and thoroughly enough for meaningful protection? And finally, could this approach damage the delicate inner nose or have other side effects over time? (Even in the small study, one of the people who used intranasal Neosporin dropped out due to minor side effects, apparently related to a drug allergy.)

Its great science, but theres still a long way to go before we should put it in our noses, agrees Dr. Sean Liu, an infectious disease physician at New Yorks Mount Sinai health system who was also not involved in the study.

Iwasaki agrees that more research is necessary. She says the next step is testing higher doses of neomycin, since Neosporin contains a fairly small amount that may not be enough to provide robust protection for humans. To gather more data, she says, researchers could track people going about their normal livesexcept that some apply neomycin to their noses and some apply Vaselineand see if one group gets sick less often than the other, though that would require a lot of time and people.

Despite the difficulties, Liu says theres good reason for further study. Finding new uses for affordable, widely accessible medications is good for public health, and any progress toward neutralizing viruses is welcome. If the approach is proven to work, it could also be useful to have a tool that's effective against a broad range of viruses and could potentially be paired with other drugs to strengthen its efficacy, Crowe adds.

Plus, Iwasaki says, additional disease-prevention tools could help people who are especially vulnerable to respiratory diseasessuch as those who are immunocompromisedand need additional protection to feel safe. If further research proves promising, Iwasaki says, she could imagine neomycin serving as an additional disease-fighting tool when people are in particularly germy places, like a crowded party or an airport.

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Can Neosporin in the Nose Help Prevent Viral Infections? - TIME

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Long COVID: plasma levels of neurofilament light chain in mild COVID-19 patients with neurocognitive symptoms ... - Nature.com

In a recent study published in the journal JAMA Network Open, researchers investigated whether the coronavirus disease 2019 (COVID-19) vaccine for adolescents between the ages of 12 and 15 years, which was approved in May 2021, was associated with changes in the incidence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and related hospitalizations among the pediatric population in California, United States (U.S.).

Study:COVID-19 Vaccination and Incidence of Pediatric SARS-CoV-2 Infection and Hospitalization. Image Credit:Prostock-studio/Shutterstock.com

The spread and severity of the COVID-19 pandemic have been successfully controlled due to the rapid development of vaccines against SARS-CoV-2 and concerted efforts worldwide to vaccinate adult and at-risk populations.

The messenger ribonucleic acid (mRNA) vaccines against SARS-CoV-2, developed largely by Moderna and Pfizer BioNTech, were widely used for the adult populations.

Since children and adolescents were not found to be at a high risk of severe COVID-19, developing vaccines for the younger populations was of secondary priority during the peak periods of the pandemic.

However, in May 2021, the first mRNA COVID-19 vaccine for adolescents between 12 and 15 years was approved. In the subsequent months, vaccines for children between the ages of five and 11 years and six months and five years were also approved.

Although these vaccines are safe, vaccine hesitancy because of parental concerns about safety and adverse effects, and perceptions of reduced severity of the infection among children have resulted in low vaccine uptake among the younger populations.

In the present study, the researchers examined whether the COVID-19 vaccine for adolescents impacted the incidence of SARS-CoV-2 infection and hospitalizations among the pediatric population in California.

A better understanding of the impact of the vaccine in lowering incidence rates, reducing the severity of the disease, and mitigating the need for hospitalization is essential in formulating future public health policies on booster doses and developing vaccines against emerging SARS-CoV-2 variants.

The researchers analyzed deidentified data for close to four million pediatric COVID-19 cases and over 12,000 hospitalizations from California.

The outcomes associated with COVID-19 vaccination, including the incidence of SARS-CoV-2 infections and hospitalizations, were analyzed for each county and state according to the vaccine introduction phases for the three age groups.

The deidentified data contained age, county of residence, and hospitalization status information. A polymerase chain reaction (PCR) test was required to confirm COVID-19.

For the statistical analyses, the researchers grouped the cases based on county of residence, as well as age groups according to vaccine eligibility.

Furthermore, the data for each age group was also divided into periods of vaccine ineligibility and eligibility, and the outcomes were evaluated from a month after the vaccination until the analysis of the data or the beginning of the vaccine eligibility period for the next age group.

The results suggested that the COVID-19 vaccine effectively limited the transmission of SARS-CoV-2 among the pediatric population in California.

The analysis found that close to 380,000 COVID-19 cases and 273 hospitalizations among children between the ages of six months and 15 years were averted in four to seven months after the availability of the vaccine. These numbers represent 26% of the cases in the pediatric population.

The researchers stated that their results among the pediatric population were similar to those from various U.S. and Israeli studies reporting the effectiveness of the COVID-19 vaccine in averting a substantial number of COVID-19 cases among the adult population.

The positive impact of the COVID-19 vaccine was found to be the highest among children between the ages of 12 and 15 years.

Among children ages six months to five years, the reduction in the number of COVID-19 cases was not found to be significant. However, the researchers believe this could be because of low transmission rates of the variant circulating during the evaluation period for that age group.

Notably, despite the vaccination coverage being just above half (53.5%) among adolescents between the ages of 12 and 15 years and even lower among children below 12, a total of 376,085 cases of COVID-19 were averted in California.

These findings highlight the effectiveness of the COVID-19 vaccine in lowering the incidence and severity of SARS-CoV-2 infections and limiting the transmission of the virus among children and adolescents.

To conclude, the study found that despite just over 50% vaccination coverage, the COVID-19 mRNA vaccine approved for use among adolescents and children in the U.S. averted close to 400,000 cases among the pediatric population.

These results highlighted the importance of the COVID-19 vaccine in protecting individuals of all age groups against SARS-CoV-2 infections. Furthermore, these findings also support future public health decisions to administer booster doses.

Journal reference:

Head, J. R., Collender, P. A., Len, Toms M, White, L. A., Sud, S. R., Camponuri, S. K., Lee, V., Lewnard, J. A., & Remais, J. V. (2024). COVID-19 Vaccination and Incidence of Pediatric SARS-CoV-2 Infection and Hospitalization. JAMA Network Open. doi:https://doi.org/10.1001/jamanetworkopen.2024.7822. https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2817868

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Did California's pediatric COVID-19 vaccination program reduce reported cases and hospitalizations? - News-Medical.Net

Following the availability of COVID-19 vaccines, there was a noticeable surge in negativity surrounding vaccines on Twitter. Researchers at the ESCMID Global Congress (formerly ECCMID) in Barcelona, Spain (held from 27th to 30th April) made this observation.

The analysis revealed that spikes in negative tweets coincided with announcements from governments and healthcare authorities regarding vaccination. To address this issue, the researchers propose adopting a fresh approach to discussing vaccinesone that avoids using the term anti-vaxxers.

Dr. Guillermo Rodriguez-Nava, lead researcher from Stanford University School of Medicine, Stanford, USA, emphasized the significance of vaccines as one of humanitys greatest achievements. Vaccines have the potential to eradicate dangerous diseases like smallpox, prevent deaths from illnesses with 100% mortality rates (such as rabies), and even protect against cancers caused by HPV.

Despite these benefits, opposition to vaccine use has grown in recent years. Negative voices have already had consequences, with measles re-emerging in countries where it was once considered eradicated. This situation not only affects children who cannot decide for themselves but also impacts immunocompromised patients who cannot receive vaccinations.

Dr Rodriguez-Nava and colleagues analyzed COVID-19 vaccine-related posts on Twitter. They used open-source software (the Snscrape library in Python) to download tweets with the hashtag vaccine from 1 January 2018 to 31 December 2022. Cutting-edge AI methods were then employed for sentiment analysis, classifying tweets as either positive or negative. Additionally, they created a counterfactual scenario to understand how tweet patterns would have looked if COVID-19 vaccines hadnt been introduced in December 2020.

The results showed that both before and after vaccine availability, negative sentiment tweets dominated. For instance, one negative tweet read: The EU Commission should immediately terminate contracts for new doses of fake #vaccines against #COVID19 and demand the return of the 2.5bn paid so far. Everyone who lied that #vaccines prevent the spread of the virus must be held accountable.

In contrast, positive tweets celebrated vaccination milestones. For example: Two-month shots! #vaccines are always a reason to celebrate in our house. #VaccinesWork.

Since the introduction of COVID-19 vaccines, the number of vaccine-related tweets has increased significantly10,201 more per month on average than expected if vaccination hadnt started. Negativity also rose, with approximately 12,420 negative sentiment tweets per month after 11 December 202027% more than expected without vaccination.

The proportion of positive tweets decreased slightly (from 20.3% to 18.8%), while negative tweets increased (from 79.6% to 81.1%) after COVID vaccine introduction.

Notably, negative activity spiked during vaccination announcements. For instance, April 2021 saw the highest number of negative tweetsthe same month the White House announced COVID-19 vaccine eligibility for all people aged 16 and older.

Interestingly, the lowest number of negative tweets occurred in April 2022, the month Elon Musk acquired Twitter. While the exact reason is unknown, it may be related to seasonal patterns (higher negativity during winter) or users focusing on platform changes under new ownership.

In summary, negative sentiments about vaccines were already prevalent on social media before COVID-19 vaccines arrived. Their introduction significantly amplified negative sentiments on X (formerly Twitter) regarding vaccines.

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Negativity about vaccination on Twitter increases after COVID-19 vaccines become available - Dailynewsegypt - Daily News Egypt

National Eye Institute / Wikimedia Commons

The incidence of uveitis in the year after COVID-19 was 17% among nearly 474,000 Korean adults with a history of the inflammatory eye condition, according to areport in JAMA Ophthalmology.

Researchers at the Hanyang University College of Medicine in Seoul mined theKorean National Health Insurance Service and Korea Disease Control and Prevention Agency databases for information on473,934 patients diagnosed as having uveitis from January 2015 to February 2021.

The patients had previously had uveitis and had received at least one dose of an mRNA (Pfizer/BioNTech or Moderna) or adenovirus vectorbased (AstraZeneca or Johnson & Johnson) COVID-19 vaccine. The average patient age was 58.9 years, 51.3% were women, and none tested positive for COVID-19 during the study period.

Uveitisis a potentially serious inflammation of the eye's middle layer of tissue that can cause symptoms such as pain, redness, and blurry vision.

The incidence of uveitis was 8.6% at 3 months, 12.5% at 6 months, and 16.8% at 1 year. The odds of uveitis were increased among recipients of all four vaccines, including Pfizer (hazard ratio [HR], 1.68), Moderna (HR, 1.51),AstraZeneca (HR, 1.60), andJohnson & Johnson(HR, 2.07). The risk was highest in the first 30 days after vaccination and peaked between the first and second doses (HR, 1.64).

These results emphasize the importance of vigilance and monitoring for uveitis in the context of vaccinations, including COVID-19 vaccinations, particularly in individuals with a history of uveitis.

"Although uveitis following vaccination is rare, our findings support an increased risk after COVID-19 vaccination, particularly in the early postvaccination period," the study authors wrote. "These results emphasize the importance of vigilance and monitoring for uveitis in the context of vaccinations, including COVID-19 vaccinations, particularly in individuals with a history of uveitis."

In a relatedcommentary, Anika Kumar and Nisha Acharya, MD, said it's important to weigh the risk of uveitis with that of remaining unvaccinated against COVID-19. "Indeed, other investigations of postvaccine NIU [noninfectious uveitis] that similarly identified increased risks of NIU after vaccination noted that effect sizes were small and attributable risks were low; thus, the findings should not preclude individuals from receiving a vaccination," they wrote.

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Risk of uveitis recurrence higher in year after COVID vaccination - University of Minnesota Twin Cities

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Multi-omics analysis reveals COVID-19 vaccine induced attenuation of inflammatory responses during breakthrough ... - Nature.com

Anthony S. Fauci has agreed to testify in front of the House panel investigating the nations coronavirus response, the first time the prominent infectious-disease expert will publicly face Congress since leaving government nearly 1 years ago.

Fauci, who helped steer the Trump and Biden administrations efforts to fight the virus, is scheduled to testify June 3 in front of the House Oversight select subcommittee on the coronavirus pandemic, with lawmakers expected to press him on the still-unknown origins of the pandemic, the governments vaccine mandates and other issues that remain politically divisive, more than four years after the outbreak began.

The GOP-led panel includes some of Faucis most persistent critics in Congress, such as Reps. Marjorie Taylor Greene (R-Ga.) and Ronny Jackson (R-Tex.), who have repeatedly alleged that the pandemic began with an accident at a lab in China funded by Faucis agency and covered up by U.S. officials.

Retirement from public service does not excuse Dr. Fauci from accountability to the American people, Rep. Brad Wenstrup (R-Ohio), who chairs the panel, said in a statement. On June 3, Americans will have an opportunity to hear directly from Dr. Fauci about his role in overseeing our nations pandemic response, shaping pandemic-era policies, and promoting singular questionable narratives about the origins of COVID-19.

Fauci has denied wrongdoing, and public health leaders have praised his work and said Republicans have unfairly targeted him.

Debate about the origins of the SARS-CoV-2 virus remains, with evolutionary biologists and virologists saying the outbreak probably began because of a spillover from infected animals, but some scientists suggesting a leak from a lab was the likely source. A declassified government intelligence report released last year said U.S. intelligence officials were unaware of a lab leak that could have caused the pandemic, adding that officials had no evidence that the Wuhan Institute of Virology, the Chinese laboratory where researchers were studying coronaviruses, had SARS-CoV-2 or a close progenitor in its possession before the outbreak.

The select subcommittee has not uncovered any evidence that directly implicates Dr. Fauci and [former National Institutes of Health director Francis] Collins in a coverup of the pandemics origin or collusion with scientific journals to suppress the lab-leak hypothesis, Rep. Raul Ruiz (Calif.), the panels top Democrat, said at a hearing last week.

The 83-year-old Fauci led the National Institute of Allergy and Infectious Diseases for nearly 40 years, where he forged relationships with prominent politicians, such as then-President George W. Bush, and helped shape the nations response to HIV/AIDS, Ebola and other infectious diseases.

The longtime government official quickly became a household name in the early days of the coronavirus pandemic, routinely appearing at White House briefings where he urged Americans to wear masks, get vaccinated and take other precautions. Many Americans said they were reassured by his briefings particularly in contrast to then-President Donald Trumps freewheeling medical advice, such as endorsing anti-malaria drugs to fight covid. President Biden named Fauci his chief medical adviser, and the Biden administration relied on him as a key spokesman during its vaccine rollout in 2021.

But public confidence in Fauci and other health officials deteriorated amid frustrations about pandemic-era policies such as remote schooling and attacks from GOP lawmakers. Fifty-three percent of Americans in April 2022 said they trusted Faucis recommendations on coronavirus vaccines, down from 68 percent in December 2020, according to polling by KFF, a nonpartisan health research organization. The dip was driven by growing Republican skepticism; just 25 percent of Republicans said they trusted Faucis coronavirus vaccine recommendations in April 2022, down from 47 percent in December 2020, while Democrats trust in Fauci remained largely unchanged.

After leaving government in December 2022, Fauci joined the Georgetown University faculty as a distinguished professor and wrote a memoir set to publish in June.

Fauci privately testified in front of House lawmakers in January, answering questions about his role in the nations coronavirus response, whether scientists should experiment with risky viruses in their labs and how his agency funded research abroad. Attendees offered starkly different representations of the closed-door hearings, with Republicans saying Fauci dodged direct questions and changed his answers on lab leak-related issues, while Democrats countered that Fauci was helpful and the GOP-led questions broke little ground.

The Republicans have totally distorted Dr. Faucis testimony, Rep. Kathy Castor (D-Fla.) said after the first day of closed-door hearings, adding that she hoped Faucis private comments would be quickly made publicly available. The transcript has not yet been published.

Although Fauci has been out of government since late 2022, and many Americans focus on the pandemic has dwindled as the virus has receded, he continues to be routinely invoked by Republicans as a political symbol. The coronavirus panels GOP lawmakers and their witnesses maintain that the longtime government official exerted too much control over the nations pandemic response and should have been regarded with more skepticism.

The media parroted whatever Fauci and the CDC fed them, Marty Makary, a Johns Hopkins transplant surgeon and Fox News analyst, said at a hearing last year.

Other notable figures are set to soon face the House panel, including Peter Daszak, president of the New York-based research organization EcoHealth Alliance, who is scheduled to testify next week. EcoHealth has defended its research with the Wuhan Institute of Virology and denied any connection between that work and the emergence of SARS-CoV-2.

correction

An earlier version of this article incorrectly said that Rep. Marjorie Taylor Greene represents Louisiana in the House. She represents Georgia. The article also misstated the year when Anthony S. Fauci privately testified before House lawmakers. He testified in January 2024. The article has been corrected.

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Fauci agrees to testify in Congress on covid origins, pandemic policies - The Washington Post

Shortly after winning a stay on a COVID-19 vaccine patent lawsuit brought by Moderna, Pfizer and its German partner BioNTech are being dragged into another round of mRNA litigation by GSK.

GSK on Thursday filed a lawsuit in Delaware federal court accusing Pfizer and BioNTech of infringing five patents related to the mRNA technology behind the partners COVID shot Comirnaty. The relevant mRNA research was conducted more than a decade before the COVID-19 pandemic and was picked up by GSK in 2015 when the company acquired a substantial portion of Novartis global vaccine business, according to the lawsuit.

GSK claims Pfizer and BioNTech subsequently reaped billions of dollars in revenue from infringing GSKs Patents-in-Suit and continue to benefit, without ever obtaining a license.

Now, the British drugmaker is seeking a reasonable royalty from Pfizer and BioNTech, as well as a compulsory ongoing licensing fee for the partners alleged use of GSKs patented mRNA technology.

A GSK spokesperson confirmed over email that the company filed suit and said its committed to taking appropriate action where necessary to protect the companys intellectual property. She described the GSK patents in question as foundational to Pfizer and BioNTechs mRNA vaccines.

GSK would be willing to license these patents on commercially reasonable grounds to make sure patients continue to enjoy access to Pfizer and BioNTechs shots, the spokesperson added.

Pfizer, for its part, remains confident in its intellectual property (IP) position and intends to vigorously defend against [GSKs] claims, a company spokesperson said in a statement.

The latest lawsuit in Delaware marks the opening of a new front in GSKs vaccine patent litigation against Pfizer.

Last August, GSK claimed in a separate court filing that Pfizers respiratory syncytial virus (RSV) shot Abrysvo violated four patents related to its own RSV immunization Arexvy. GSK argued that Pfizer began the project that led to Abrysvo no earlier than 2013at least seven years after GSK kickstarted its own RSV program.

Separately, Pfizer and BioNTech have been locking horns with Moderna over mRNA IP for nearly two years.

The kerfuffle started in late August of 2022 when Moderna filed lawsuits in the United States and Germany contending that Pfizer and BioNTech copied key features of its patented mRNA technology. Pfizer and BioNTech filed a countersuit and demanded a jury trial in December of that same year, with lawyers for the partners arguing that Moderna was attempting to put itself in the single, starring role of the response to the COVID-19 pandemic.

More recently, Pfizer and BioNTech last summer pressed the U.S. Patent and Trademark Offices (PTOs) Patent Trial and Appeals Board to invalidate a pair of Moderna patents around mRNA vaccine production.

Earlier this month, Massachusetts federal judge Richard G. Stearns granted a stay on the case to provide more time for the Patent Trial and Appeal Board to review Pfizer and BioNTechs patent challenges.

COVID-19 vaccine revenues have fallen precipitously for both Pfizer-BioNTech and Moderna over the last two years.

In 2023, Comirnaty broughtin $11.2 billion in sales, a steep 70% decline from the $37.8 billion generated by the shot in 2022.

Link:

GSK pulls Pfizer and BioNTech into yet another COVID-19 vaccine patent lawsuit - FiercePharma

By VCU News staff

Jeanine Guidry, Ph.D., an affiliate faculty member at the Department of Social and Behavioral Sciences in theSchool of Population Healthand at theRichard T. Robertson School of Media and Culturein theCollege of Humanities and Sciencesat Virginia Commonwealth University, is the recipient of the APIC/AJIC Award for Publication Excellence for her study Willingness to Get the COVID Vaccine With and Without Emergency Use Approval.

The award recognizes an author who has published an article in the American Journal of Infection Control which was widely read and cited during the previous year.

Guidry's study, conducted at VCU with colleaguesLinnea Laestadius, Ph.D.;Emily Vraga, Ph.D.;Carrie Miller, Ph.D.;Paul Perrin, Ph.D.;Candace Burton, Ph.D.;Mark Ryan, M.D.;Bernard Fuemmeler, Ph.D.; andKellie Carlyle, Ph.D., involved a July 2020 survey of 788 adults that found that 59.9% of respondents were definitely or probably planning to receive a future coronavirus vaccine, but fewer than half of respondents planned to get it under emergency use authorization. It also found that white people were more likely than Black people to be vaccinated.

With the ongoing presence of COVID-19 in our society, the risk of long COVID and the long-term threat of vaccine hesitancy, continuing to improve our understanding of why people may or may not vaccinate is of great importance, said Guidry, who is now a faculty member atTilburg Universityin the Netherlands.My co-authors and I are grateful for APIC/AJICs support and commitment to deeper understanding in this field.

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Professor honored for paper on willingness to get the COVID vaccine - VCU News